the first spacecraft to leave the inner solar system , pioneers 10 and 11 were designed to conduct an exploration of the interplanetary medium beyond the orbit of mars and perform close - up observations of jupiter during the 1972 - 73 jovian opportunities .  the spacecraft were launched in march 1972 ( pioneer 10 ) and april 1973 ( pioneer 11 ) on top of identical three - stage atlas - centaur launch vehicles . after passing through the asteroid belt , pioneer 10 reached jupiter in december 1973 .the trajectory of its sister craft , pioneer 11 , in addition to visiting jupiter in 1974 , also included an encounter with saturn in 1979 ( see for more details ) .after the planetary encounters and successful completion of their primary missions , both pioneers continued to explore the outer solar system . due to their excellent health and navigational capabilities , the pioneers were used to search for trans - neptunian objects and to establish limits on the presence of low - frequency gravitational radiation .eventually , pioneer 10 became the first man - made object to leave the solar system , with its official mission ending in march 1997 .since then , nasa s deep space network ( dsn ) made occasional contact with the spacecraft .the last successful communication from pioneer 10 was received by the dsn on 27 april 2002 .pioneer 11 sent its last coherent doppler data in october 1990 ; the last scientific observations were returned by pioneer 11 in september 1995 .the orbits of pioneers 10 and 11 were reconstructed based primarily on radio - metric ( doppler ) tracking data .the reconstruction between heliocentric distances of 2070 au yielded a persistent small discrepancy between observed and computed values .after accounting for known systematic effects , the unmodeled change in the doppler residual for pioneer 10 and 11 is equivalent to an approximately sunward constant acceleration of the magnitude of this effect remains approximately constant within the 3 db gain bandwidth of the high - gain antenna ( hga ) .the nature of this anomalous acceleration remains unexplained ; this signal has become known as the pioneer anomaly .there were numerous attempts in recent years to provide an explanation for the anomalous acceleration of pioneers 10 and 11 .these can be broadly categorized as either invoking conventional mechanisms or utilizing principles of `` new physics '' .initial efforts to explain the pioneer anomaly focused on the possibility of on - board systematic forces . while these can not be conclusively excluded , the evidence to date does not support these mechanisms : the magnitude of the anomaly exceeds the acceleration that these mechanisms would likely produce , and the temporal evolution of the anomaly differs from that which one would expect , for instance , if the anomaly were due to thermal radiation of a decaying nuclear power source .conventional mechanisms external to the spacecraft were also considered .first among these was the possibility that the anomaly may be due to perturbations of the spacecrafts orbits by as yet unknown objects in the kuiper belt .another possibility is that dust in the solar system may exert a drag force , or it may cause a frequency shift , proportional to distance , in the radio signal .these proposals could not produce a model that is consistent with the known properties of the pioneer anomaly , and may also be in contradiction with the known properties of planetary orbits .the value of the pioneer anomaly happens to be approximately , where is the speed of light and is the hubble constant at the present epoch .attempts were made to exploit this numerical coincidence to provide a cosmological explanation for the anomaly , but it has been demonstrated that not only this approach would produce an effect with the opposite sign , it will also will have a much smaller magnitude .as the search for a conventional explanation for the anomaly appeared unsuccessful , this provided a motivation to seek an explanation in `` new physics '' .no such attempt to date produced a clearly viable mechanism for the anomaly .the inability to explain the anomalous behavior of the pioneers with conventional physics has resulted in a growing discussion about the origin of the detected signal .the limited size of the previously analyzed data set , also limits our current knowledge of the anomaly .we emphasize that in order to determine the origin of and especially before any serious discussion of new physics can take place , one must analyze the entire set of radio - metric doppler data received from the pioneers .in this paper we report on the progress of the recovery of the pioneers radio - metric doppler data and flight telemetry and the status of the analysis of the pioneer anomaly .the paper is organized as follows . in section [ sec : recovery ]we discuss the recovery of the extended doppler data set and its current status . in section [ sec : tel - recovery ] we present the flight telemetry and discuss its value for the upcoming investigation . in section [ sec : rec - force ] we discuss modeling the thermal recoil force and present preliminary results of this effort . in section [ sec : summ ] we conclude with a summary and outline the next sets in the study of the pioneer anomaly .spacecraft&time span & distances ( au ) + pioneer 10 & 03.01.87 22.07.98 & 40.0 70.5 + pioneer 11 & 05.01.87 01.10.90 & 22.4 31.7 +  + spacecraft&time span & distances ( au ) + pioneer 10 & 08.09.73 27.04.02 & 4.56 80.2 + pioneer 11 & 10.04.73 11.10.94 & 1.01 41.7 +  [ tb : new - doppler - data ]  as of october 2007 , an effort to recover all archived pioneer 10 and 11 doppler data , initiated at jpl in june 2005 , has been completed ; there is now almost 30 years of pioneer 10 and 20 years of pioneer 11 doppler data , most of which was never used in the investigation of the anomaly ( table [ tb : new - doppler - data ] ) .the recovery process presented many unanticipated challenges .uncertainty in spin calibration and missing ramp information are two issues that created a major setback in the data recovery project , as most of the newly recovered data files were not properly conditioned . in the following sections , we address these issues and briefly describe the techniques used for orbital analysis .radio communication with the pioneer spacecraft was maintained using s - band microwave transmissions .the transmission frequency was .1 ghz .the frequency of the return transmission from the spacecraft was .3 ghz .when the spacecraft s radio communication subsystem was in coherent mode , its receiver and transmitter were coupled , and the ratio of the frequency of the signal transmitted vs. the signal received by the spacecraft was exactly .it is this phase - coherent mode of operation that allowed precision doppler measurements by ground stations , often with mhz accuracy .  at this accuracy, one must take into account the combined effect of the spacecraft s spin and the polarization of the radio signal .the radio signal transmitted to , and returned by , the pioneer spacecraft was circularly polarized .the spacecraft was physically oriented such that its high - gain antenna always pointed towards the earth ; the spacecraft was spinning around its spin axis , which approximately coincided with the antenna axis .  as a result ,each revolution of the spacecraft added a full cycle to the signal in each direction .  at a nominal spin rate of 4.5 revolutions per minute ( rpm ) , this resulted a shift of 0.15 cycles per second in the return frequency . knowing the actual spin rate of the spacecraft from telemetry , it is possible to calculate , and account for , the spacecrafts spin more accurately .when a data file containing radio - metric information has been modified to account for this effect due to spin , it is said to be spin calibrated .when we assembled historical pioneer doppler data files for the first time , alarmingly we found that some of the files were spin - calibrated , whereas other files were not .it soon became evident that , in order to render these files suitable for precision orbit calculations , their spin calibration had to be reconstructed from historical records ( see discussion in sec . [sec : spin ] ) .several years after the launch of the twin pioneer spacecraft , the deep space network ( dsn ) began experimenting with `` ramping '' its transmission frequencies , in order to follow the variable frequency shift that is a result of the changing relative velocities of a distant spacecraft and a station on the earth s surface , during the course of a data transmission session .ramping is characterized by a starting frequency and a ramp rate , in hz / s .unfortunately , older data file formats used by the dsn had no provisions for storing this ramp data , therefore this information was stored separately . further compoundingthe problem was the fact that ramp data records were often used to store the transmission frequency even when no actual ramping took place ( i.e. , the ramp rate was 0 ) ; the transmission frequency recorded in the doppler record may or may not have coincided with the actual transmission frequency .when the recently recovered doppler records were first assembled and analyzed , a strange `` banding '' appeared in the data , with data points offset by several hundred to several thousands hz from each other .missing ramp records were identified as a likely cause for this effect in many cases . in order to reconstruct a usable data set , it was necessary to search historical records that contained the necessary ramping information , or discard records for which no reliable ramp records could be found .we realized that unusual behavior of the recovered raw doppler data discussed in the previous section was due to the fact that these files were not submitted to a proper radio - metric data conditioning ( rmdc ) procedure .this procedure is a standard part of preparing orbit tracking data for a trajectory analysis . as part of the standard processrmdc it includes both spin calibration and application of the correct ramp information to raw doppler data received from dsn stations .it turned out that rmdc work was necessary for all pioneer archival tracking data files ( atdfs ; see discussion in ) that we received from national space science data center ( nssdc ) , as all of these files were raw unconditioned atdfs .below is a description of the rmdc process required for making pioneer atdfs usable for our investigation .the process is very detail - oriented , tedious , and labor - intensive .it consists of two phases analysis and implementation that are discussed below .analysis of tracking data files comprises the following critical steps :  * generate a summary of the contents of each atdf ; * separate and print the tracking data , ramp data , and transmitter parameters of each file summary ; * compare ramp data to transmitter summaries to identify erroneous / extraneous transmitter on periods , and uplink periods which are not covered by ramp records , then transfer this information to the data summary ; * check data summaries for erroneous reference frequencies ( receiver and exciter ) ; times when the reference frequencies are being ramped , but there are no corresponding ramp records ; erroneous status ( validity , transmitter on / off , etc . ) flags ; erroneous identifier ( exciter band , receiver band , doppler mode , synthesizer , exciter type , receiver type , etc . ) flags ; erroneous doppler bias values ;  sometimes , the validation process can be an almost point - by - point exercise .after the analysis of tracking data files has been completed , the necessary processing steps are implemented as follows :  * translate the analysis information into change commands ( some of the necessary fixes could require as many as 4 commands to implement ) to be used with the appropriate program ; * execute the program to apply the correction commands ( at least 250 + commands for each atdf ) ; * generate a summary of the updated atdf ; * check the new summaries to confirm that the corrections have been properly applied , and no new problems have been introduced ( or `` come to the surface '' ) by making the changes.  once the steps above are implemented , the tracking data files are ready to be used for generating standard format orbit data files ( odfs ) that can be submitted for further analysis with the orbit determination program ( odp ) software .after the two - stage process above is complete , another person takes a `` conditioned '' odf and runs it through odp to verify the quality of the file .experience has shown that this step sometimes identifies further problems , resulting in some additional time and effort being needed before a fully - usable tracking data file is available . if the quality is satisfactory then the file is added to the collection of good files , otherwise the file is returned for further rmdc .the process repeats until all files are found acceptable .the work on data conditioning of nearly 600 + pioneer atdfs is progressing well and will be completed by the end of 2007 .the result of this analysis is that at long last , a usable data set is now available for analysis , covering in particular the early parts of the pioneer 10 and 11 missions , for which no doppler data records were available in the past .this is especially important because there has been speculation that there may have been an `` onset '' of the anomalous acceleration of pioneer 10 and 11 when they reached approximately the orbit of saturn ( see discussion in ) .is this onset a real effect , or just an artifact of incomplete past analyses ?if the effect is real , was it caused by , for instance , incorrect solar pressure calibration , or does it represent a hint about the real cause of the pioneer anomaly ?our upcoming analysis is aimed to answer these and other questions .doppler data are the recorded measurements of the shift in frequency of the signal received from the spacecraft , as a result of the combined relative motion of the spacecraft and the earth station(s ) .the position of the transmitting station changes as a function of time as a result of the combined effects of the earth s rotation , precession , and orbital motion .the position of the spacecraft can be calculated if its orbital parameters are known .after these calculations , the doppler shift in frequency can be estimated and compared with the observed frequency shift ( for more details on the analysis of radio - metric doppler data and formats used for these purposes consult and references therein ) .  in principle ,an orbit determination algorithm employs a generalized equation of motion ( for extensive details , see ) , incorporating not just the newtonian gravitational effects of massive solar system bodies , but also general relativistic corrections , non - gravitational forces , and equations describing the propagation of the radio signals through the interplanetary medium , the gravitational field of the sun and planets , and the earth s atmosphere and ionosphere .if needed , other forces may be incorporated into the model and used for hypothesis testing ( see details in ) .the aim of the orbital analysis is to determine the initial position and velocity ( state vector ) of the spacecraft , as well as the values of additional , model - dependent parameters that yield a best match with observational data . in our case ,the goal is to reduce the difference between calculated and measured doppler values ( i.e. , the doppler residual ) and then adjust the initial state vector and other parameters until the residual is minimized ( for details on models , methods , and formats consult ) .the anomalous acceleration of the pioneer spacecraft is very small .only  w of anisotropically emitted thermal power is sufficient to produce an acceleration of comparable magnitude .as the pioneer spacecraft utilized radioisotope thermoelectric generators ( rtgs ) that produced several kw of waste heat , an anisotropy as small as % in the pattern of thermal radiation can produce acceleration of the needed magnitude .for this reason , it is very important to characterize the thermal emissions of the pioneer spacecraft as accurately as possible .a unique feature of the current effort is the use of telemetry files documenting the thermal and electrical state of the spacecraft .this information was not available previously ; however , by may 2006 , the telemetry files for the entire durations of both missions were recovered , pre - processed and are ready for the upcoming study . both of the newly assembled data sets are pivotal to establishing the origin of the detected signal .all transmissions of both pioneer spacecraft , including all engineering telemetry , were archived in the form of files containing master data records ( mdrs ) .originally , mdrs were scheduled for limited retention .fortunately , the pioneers mission records avoided this fate : with the exception of a few gaps in the data the entire mission record has been saved , comprising ,000 data points for pioneer 10 , and ,000 data points for pioneer 11 , a total of  gb .these recently recovered telemetry readings are important in reconstructing a complete history of the thermal , electrical , and propulsion systems for both spacecraft .this , it is hoped , may in turn lead to a better determination of the spacecrafts acceleration due to on - board systematic effects .telemetry formats can be broadly categorized as science formats versus engineering formats .telemetry words included both analog and digital values .digital values were used to represent sensor states , switch states , counters , timers , and logic states .analog readings , from sensors measuring temperatures , voltages , currents and more , were encoded using 6-bit words .this necessarily limited the sensor resolution and introduced a significant amount of quantization noise .furthermore , the analog - to - digital conversion was not necessarily linear ; prior to launch , analog sensors were calibrated using a fifth - order polynomial .calibration ranges were also established ; outside these ranges , the calibration polynomials are known to yield nonsensical results .  with the help of the information contained in these words , it is possible to reconstruct the history of rtg temperatures and power , radio beam power , electrically generated heat inside the spacecraft , spacecraft temperatures , and propulsion system .the exterior temperatures of the rtgs were measured by one sensor on each of the four rtgs : the so - called `` fin root temperature '' sensor .figure [ fig : c201 ] depicts the evolution of the rtg 1 fin root temperature for pioneer 10 .a best fit analysis confirms that the rtg temperature indeed evolves in a manner consistent with the radioactive decay of the nuclear fuel on board .the results for all the other rtgs on both spacecraft are similar , confirming that the rtgs were performing thermally in accordance with design expectations .rtg electrical power can be estimated using two sensor readings per rtg , measuring rtg current and voltage , from which power can be computed by direct calculation .all this electrical power is eventually converted to waste heat by the spacecrafts instruments , with the exception of power radiated away by transmitters .whatever remains of electrical energy ( fig .[ fig : elec ] ) after accounting for the power of the transmitted radio beam is converted to heat on - board .some of it is converted to heat outside the spacecraft body .the pioneer electrical system is designed to maximize the lifetime of the rtg thermocouples by ensuring that the current draw from the rtgs is always optimal .this means that power supplied by the rtgs may be more than that required for spacecraft operations .excess electrical energy is absorbed by a shunt circuit that includes an externally mounted radiator plate . especially early in the mission , when plenty of rtg power was still available , this radiator plate was the most significant component external to the spacecraft body that radiated heat .a specific telemetry word tells us the shunt circuit current , from which the amount of power dissipated by the external radiator can be computed using the known ohmic resistance of the radiator plate .other externally mounted components that consume electrical power are the plasma analyzer , the cosmic ray telescope , and the asteroid / meteoroid detector .though these instruments exact power consumption is not telemetered , we know their average power consumption from design documentation , and the telemetry bits tell us when these instruments were powered .two additional external loads are the battery heater and the propellant line heaters .the power state of these loads is not telemetered . according to mission logs ,the battery heater was commanded off on both spacecraft on 12 may 1993 .  yeta further external load is the set of cables connecting the rtgs to the inverters .the resistance of these cables is known . using the rtg current readings it is possible to accurately determine the amount of power dissipated by these cables in the form of heat .after accounting for all these external loads , whatever remains of the available electrical power on board is converted to heat inside the spacecraft .so long as the body of the spacecraft is in equilibrium with its surroundings , heat dissipated through its walls has to be equal to the heat generated inside .       as evident from fig .[ fig : pioneer ] , the appearance of the pioneer spacecraft is dominated by the 2.74 m diameter high gain antenna ( hga ) .the spacecraft body , located behind the hga , consists of a larger , regular hexagonal compartment housing the propellant tank and spacecraft electronics ; an adjacent , smaller compartment housed science instruments .the spacecraft body is covered by multilayer thermal insulating blankets , except for a louver system located on the side opposite the hga , which was activated by bimetallic springs to expel excess heat from the spacecraft .each spacecraft was powered by four radioisotope thermoelectric generators ( rtgs ) mounted in pairs at the end of two booms , approximately three meters in length , extended from two sides of the spacecraft body at an angle of 120 . a third boom , approximately 6 m long , held a magnetometer .the total ( design ) mass of the spacecraft was  kg at launch , of which 27 kg was propellant .  for the purposes of attitude control , the spacecraft were designed to spin at the nominal rate of 4.8 rpm .six small monopropellant ( hydrazine ) thrusters , mounted in three thruster cluster assemblies , were used for spin correction , attitude control , and trajectory correction maneuvers ( see fig . [fig : pioneer ] ) .the passive thermal control system consisted of a series of spring - activated louvers ( see fig .[ fig : louvers ] ) .the springs were bimetallic , and thermally ( radiatively ) coupled to the electronics platform beneath the louvers .the louver blades were highly reflective in the infrared .the assembly was designed so that the louvers fully open when temperatures reach 30 , and fully close when temperatures drop below 5 .  as the total exterior area of the spacecraft body and the effective emissivity of the thermal blankets covering the spacecraftare known , a crude initial calculation can be made determining the amount of heat radiated by the spacecraft walls and through the louver system .these estimates turn out to be consistent with pre - launch thermal vacuum chamber tests of the louver system , and with temperature readings obtained from on - board temperature sensors .there are 6 platform temperature sensors ( fig .[ fig : tempsens ] ) inside the spacecraft body : 4 are located inside the main compartment , 2 sensors are in the science instrument compartment .the main compartment has a total of 12 2-blade louver blade assemblies ; the science compartment has 2 3-blade assemblies .the thermal vacuum chamber tests provide values for emitted thermal power per louver assembly as a function of the temperature of the electronics platform behind the louver .this allows us to estimate the amount of thermal power leaving the spacecraft body through the louvers , as a function of platform temperatures , providing means to estimate the amount of heat radiated by the louver system .-6pt  0.001  -6pt  0.001  the spin rate of the pioneer spacecraft was measured independently by three instruments : a star sensor and two sun sensors .each of these instruments was capable of providing a roll reference pulse , which in turn was used to synchronize other spacecraft activities .the spin rate was telemetered to the ground using 3 6-bit words .the resulting 18-bit value measured the time of one full revolution of the spacecraft in units of 1/8192 seconds using an on - board crystal oscillator as a reference clock .unfortunately , the star sensor on board pioneer 10 stopped functioning shortly after jupiter encounter .the sun sensors , in turn , could only operate if the sun was bright enough and appeared sufficiently far from the spacecraft spin axis . consequently , the sun sensors stopped providing a useful roll reference pulse after the spacecraft reached a heliocentric distance of astronomical units .the pioneer support team devised an alternate procedure , utilizing the imaging photo - polarimeter ( ipp ) instrument and its ability to image faint objects in a precisely timed manner .later , when there was no longer sufficient power available on board to operate this instrument , the spin rate was estimated using other means , such as relying on the previously established rate of spin change , and on the spin introducing a slight modulation of the spacecraft s radio signal when the spin axis did not exactly coincide with the spacecraft - earth line .the star sensor and sun sensors remained fully operational on pioneer 11 throughout its mission .as a result , we have high accuracy spin information for both spacecraft for much of their missions .the spin behavior of the two spacecraft was not identical ( fig .[ fig : spin ] ) .the spin rate of pioneer 10 was decreasing steadily , and does not appear to be affected by maneuvers .the spin rate of pioneer 11 increased rapidly early in the mission as a result of a spin thruster malfunction , but it continued to increase afterwards . a close examination of the spin behavior reveals , however , that the increases in spin rate are discontinuous and coincide with maneuvers ; between maneuvers , the spin rate was decreasing ( fig .[ fig : spinzoom ] ) .each of the pioneer spacecraft was equipped with six monopropellant hydrazine thrusters , designed to perform three types of maneuvers : spin - up / spin - down maneuvers , delta- maneuvers , and precession maneuvers .the first of these was used early in their missions to adjust the spin rate after the deployment of the rtg and magnetometer booms .the second type of maneuver was used to correct the spacecrafts trajectories , to achieve the desired planetary encounters with jupiter and ( for pioneer 11 ) saturn .precession maneuvers , in turn , were used regularly throughout the mission ; their goal was to adjust the spacecrafts spin axis , to ensure that the hga continuously points in the direction of the earth .  in theory ,precession maneuvers do not alter the velocity of the spacecraft , only change their orientation . in practice , due to small differences between thrusters , uncertainties in the duration of firing pulses , thruster misalignment , and outgassing after maneuvers , a small , essentially random change in the spacecrafts motionis expected .these changes can be modeled , for instance , as an instantaneous change in the line - of - sight velocity of the spacecraft . however , before one can do such modeling , the time when the maneuvers occurred must be known .flight telemetry contains information about the number of pulses fired by each of the six thrusters on board . the number of firing pulses recorded , and the thrusters participating in the maneuver unambiguously identify the type of maneuver that was performed .although thruster pulse counts are monitored continuously , due to low data rates , the corresponding data words are telemetered as infrequently as once every 51.2 minutes at the lowest telemetry data rate . due to gaps in ground station coverage or missing data records , there are instances when an increase in the thruster pulse count is bracketed by two telemetry readings that are many hours , perhaps even more than a day apart .however , due to the smallness of the velocity change as a result of a precession maneuver , even such an uncertainty in the timing of the maneuver does not adversely affect the accuracy of orbital modeling .even with all the recently recovered telemetry information , modeling the thermal recoil force on the pioneer spacecraft remains a difficult task .the main reason for this can be summed up as follows : what we seek is not just the amount of heat emitted by the spacecraft ( which we know with good precision ) , but the small anisotropy in the pattern of its thermal emissions , an inherently second - order effect .the traditional way of estimating thermal recoil forces is to build a detailed thermal model using , for instance , finite element analysis tools .we are also exploring an alternative approach that combines the use of flight telemetry with precision orbit calculations , and provides a more direct and , it is hoped , more accurate estimate of the anisotropy of thermal radiation .the radiative exchange of thermal energy between two surfaces is described by the equation where is the radiative power emitted by the surface that is absorbed by the surface , is the intensity of radiation at surface element , is the distance between surface elements and , and and are the angles between the line connecting surface elements and and their respective normals .  the intensity of radiation is related to the radiant emittance by the formula .the radiant emittance , in turn , can be calculated from the surface temperature and emissivity coefficient using the stefan - boltzmann law : where  js is the stefan - boltzmann constant .  in principle, these equations completely determine the thermal emissions of an object .the anisotropy of thermal emissions can be determined , for instance , by surrounding the object with an infinite ( or , at least , sufficiently large ) black sphere and calculating the amount of radiation absorbed by this sphere using eq .( [ eq : rad ] ) .  in order to carry out the calculations ,however , one must have accurate knowledge of  * the emitting object s detailed geometry ; * the object s temperature at all points across its surface ; * the infrared emissivity coefficient at all points across the surface , and its dependance on the temperature .this information is not always readily available . in the case of pioneer 10 and 11 ,only a few detailed drawings have been found from which the spacecraft s geometry could be estimated .surface properties are not known very accurately , and very little is known about the effects of aging in the deep space environment .although the spacecraft had several temperature sensors on board , these were designed to measure interior temperatures ; the external temperature distribution can only be calculated by constructing a complex model that takes into account the internal structure and heat conductive properties of the spacecraft .  despite these difficulties , a highly detailed mathematical model of the spacecraft has been constructed successfully .this model is used to calculate radiative heat exchange between spacecraft surfaces , absorbed solar loads , heat flows , and predicted temperatures .the finite element model incorporates nodes and 2600 plate elements , using 3.4 million radiation conductors and linear conductors .results from this model can be validated against the readings from the spacecraft s temperature sensors , as seen in figs .[ fig : tempmap ] and [ fig : rtgtemp ] .flight telemetry can provide another important means of model verification .in addition to temperature sensors , flight telemetry also offers readings from which the thermal power generated on board can be calculated .the principle of energy conservation dictates that in a spacecraft that s in steady state , the amount of thermal power generated must equal the amount of heat emitted by the spacecraft . because of this, temperature and power readings can be viewed as redundant parameters characterizing the same physical processes .our analysis of the geometry and thermal properties of the pioneer spacecraft yielded the following findings :  * the thermal output of the pioneer spacecraft can be accurately modeled using as few as two heat sources , electrical and rtg heat ; * as the spacecraft is spinning , lateral forces result in no significant long - term acceleration ; the spin axis , in turn , points approximately in the direction of the earth at all times ; * the acceleration of the spacecraft due to thermal radiation is a linear function of the power of the internal power sources .in other words , the thermal acceleration of the pioneer 10 and 11 spacecraft due to heat anisotropically rejected off the vehicles can be modeled by a simple equation ( see details in ) : where is the speed of light , is the spacecraft s mass , is the -th heat source , and is the associated dimensionless efficiency factor .( the additional factor of arises as a result of modeling the spacecraft s exterior surfaces as lambertian emitters . )the values of are known from flight telemetry or design documentation ( fig .[ fig : power ] ) .the values of can be determined approximately by analyzing the spacecraft s geometry and surface properties , as outlined in the previous section .the simplicity of the relationship between heat and acceleration suggests that another approach may be possible .( [ eq : thermal ] ) may be incorporated directly into the equations of motion , allowing one to designate the efficiency factors as parameters to be fitted .this , radically different approach requires no knowledge of the geometry or thermal properties of the spacecraft .question is , does it yield believable results ?recently , one of us ( vtt ) constructed an orbit determination program specifically to incorporate into the equations of motion eq .( [ eq : thermal ] ) , with the values of supplied directly from flight telemetry .the software allows us to test a variety of hypotheses , attributing some , or all , of the anomalous acceleration to thermal radiation , but also incorporating other fictitious forces .this program could also be used to guide our investigation with jpl s orbit determination program , similar to the earlier efforts in the study of the effect ( see discussion in ) .this work is aimed to answer three questions :  * does such an approach yield values for the efficiency factors that are consistent with the values calculated using conventional methods ? * is it possible to obtain a good orbital solution incorporating the thermal recoil force , without resorting to the use of other forces ? * is it possible to distinguish between solutions with a thermal recoil force vs. solutions that incorporate other forces , such as a constant acceleration pointing towards the sun ?  the results of this work to date , while not conclusive , are encouraging .preliminary estimates of are consistent with the results obtained from finite - element thermal analyses , including the sophisticated on - going efforts at jpl and our earlier , simpler numerical integrations .it is our hope that the two approaches will complement each other , and that the accuracy necessary to determine the extent to which anisotropic thermal radiation from the spacecraft can be responsible for the anomalous acceleration ( i.e. , an acceleration accuracy of  m / s will be achievable .by 2007 , the existence of the pioneer anomaly is no longer in doubt .a steadily growing part of the community has concluded that the anomaly should be subject to further investigation and interpretation .our continuing effort to process and analyze pioneer radio - metric and telemetry data is part of a broader strategy ( see discussion in ) .based on the information obtained from flight telemetry , we were able to develop a high accuracy thermal , electrical , and dynamical model of the pioneer spacecraft .this model will be used to further improve our understanding of the anomalous acceleration and especially to study the contribution from the on - board thermal environment to the anomaly .it is clear that a thermal model for the pioneer spacecraft would have to account for all heat radiation produced by the spacecraft .one can use telemetry information to accurately estimate the amount of heat produced by the spacecrafts major components .the next step is to utilize this result along with information on the spacecrafts design to estimate the amount of heat radiated in various directions .this entails , on the one hand , an analysis of all available radio - metric data , to characterize the anomalous acceleration beyond the periods that were examined in previous studies .telemetry , on the other hand , enables us to reconstruct a thermal , electrical , and propulsion system profile of the spacecraft .soon , we should be able to estimate effects due to on - board systematic acceleration sources , expressed as a function of telemetry readings .this provides a new and unique way to refine orbital predictions and may also lead to an unambiguous determination of the origin of the pioneer anomaly .we would like to express our gratitude to our many colleagues who have either collaborated with us on this manuscript or given us their wisdom .we specifically thank sami asmar , gene l. goltz , kyong j. lee , timothy p. mcelrath , and neil mottinger of jpl for their help in understanding and conditiononing of the pioneer doppler data .our gratitude also goes to gary kinsella of jpl and louis k. scheffer of cadence design systems who benefited us with their insightful comments and suggestions regarding thermal modeling of the pioneers .part of this work was initiated during our visit to the perimeter institute for theoretical physics , waterloo , canada . in this respectwe would like to thank john moffat for his hospitality and support .we also thank the planetary society for support and , in particular , louis d. freidman , charlene m. anderson , and bruce betts for their interest , stimulating conversations and encouragement .the work described here , in part , was carried out at the jet propulsion laboratory , california institute of technology , under a contract with the national aeronautics and space administration .j. d. anderson , p. a. laing , e. l. lau , a. s. liu , m. m. nieto , and s. g. turyshev , _ indication , from pioneer 10/11 , galileo , and ulysses data , of an apparent anomalous , weak , long - range acceleration _ ,lett . * 81 * 2858 - 2861 ( 1998 ) [ arxiv : gr - qc/9808081 ] .s. g. turyshev , j. d. anderson , p. a. laing , e. l. lau , a. s. liu , m. m. nieto , _ the apparent anomalous , weak , long - range acceleration of pioneer 10 and 11_. in : _ gravitational waves and experimental gravity _ , proc . of the xviiith workshop of the rencontres de moriond ,les arcs , savoi , france ( jan . 23 - 30 , 1999 ) , j. dumarchez , j. tran thanh van , eds .( world publishers , hanoi - vietnam , 2000 ) , 481 - 486 ( 1999 ) [ arxiv : gr - qc/9903024 ] .j. d. anderson , p. a. laing , e. l. lau , a. s. liu , m. m. nieto , and s. g. turyshev , _ study of the anomalous acceleration of pioneer 10 and 11 _ , phys .d * 65 * 082004/1 - 50 ( 2002 ) [ arxiv : gr - qc/0104064 ] .s. g. turyshev , v. t. toth , l. r. kellogg , e. l. lau , and k. j. lee , _ the study of the pioneer anomaly : new data and objectives for new investigation _ , international journ .d * 15*(1 ) 1 - 55 ( 2006 ) [ arxiv : gr - qc/0512121 ] .

Abstract: the pioneer 10/11 spacecraft yielded the most precise navigation in deep space to date . however , their radio - metric tracking data received from the distances between 2070 astronomical units from the sun consistently indicated the presence of a small , anomalous , doppler frequency drift . the drift is a blue frequency shift that can be interpreted as a sunward acceleration of  m / s for each particular spacecraft . this signal has become known as the pioneer anomaly ; the nature of this anomaly remains unexplained .   new pioneer 10 and 11 radio - metric doppler data recently became available . the much extended set of pioneer doppler data is the primary source for new upcoming investigation of the anomaly . we also have almost entire records of flight telemetry files received from the the pioneers . together with original project documentation and newly developed software tools , this additional information is now used to reconstruct the engineering history of both spacecraft . to that extent , a thermal model of the pioneer vehicles is being developed to study possible contribution of thermal recoil force acting on the two spacecraft . in addition , to improve the accuracy of orbital reconstruction , we developed a new approach that uses actual flight telemetry data during trajectory analysis of radio - metric doppler files . the ultimate goal of these efforts is to investigate possible contributions of the thermal recoil force to the detected anomalous acceleration .


nonlinear hyperbolic systems of partial differential equations , arising in continuum physics and especially in compressible fluid dynamics , admit discontinuous solutions containing shock waves that may _ depend on underlying small scale mechanisms _ such as the coefficients of viscosity , capillarity , hall effect , relaxation , heat conduction , etc .such small - scale dependent shocks exist for a variety of problems of physical interest , for instance , those modeled by conservative hyperbolic systems with dispersive phenomena ( e.g. with capillary effects ) and nonconservative hyperbolic systems ( e.g. for two - phase fluid flows ) , as well as boundary layer problems ( e.g. with viscosity terms ) . in the past twenty years , it is has been successively recognized that a standard entropy inequality ( after lax , oleinik , kruzkov , and others ) does not suffice for the unique characterization of physically meaningful solutions to such problems , so that _ additional criteria _ are required in order to characterize these small - scale dependent shock waves uniquely .these criteria are based on the prescription of a _kinetic relation _ for conservative hyperbolic systems , a _ family of paths _ for nonconservative hyperbolic systems , and an _ admissible boundary set _ for boundary value problems .although standard finite difference , finite volume and finite element methods have been very successful in computing solutions to hyperbolic conservation laws , including those containing shock waves , these well - established methods are found to be inadequate for the approximation of small - scale dependent shocks . starting with hou and lefloch ( 1994 ) and hayes and lefloch ( 1996 ) ,this lack of convergence was explained in terms of the equivalent equation associated with discrete schemes through a formal taylor expansion .the leading terms in the equivalent equation represent the numerical viscosity of the scheme and need not match the physically relevant small scale mechanisms that have been neglected in the hyperbolic modeling . consequently , standard shock - capturing schemes , in general , fail to converge to physically meaningful solutions .our purpose here is to review the methods developed in the past twenty years which accurately describe and compute small scale dependent shocks .the challenge is , both , to develop the proper theoretical tools for the description of such solutions and to ensure the convergence of numerical methods toward physically meaningful solutions .we built here on several earlier reviews by lefloch ( 1989 , 1999 , 2002 , 2010 ) and include the most recent developments and material on numerical methods that were not presented until now especially the theory of schemes with well - controlled dissipation ( wcd , in short ) recently developed by the authors .   in one space dimension ,the classes of hyperbolic systems under review either admit the conservative form eq:109 u_t + f(u)_x = 0 or the nonconservative form eq:111 u_t + a(u ) u_x = 0 , in which the map is the unknown .in , the given flux is defined on a ( possibly non - connected ) open set and satisfies the following strict hyperbolicity condition : for every , the matrix admits real and distinct eigenvalues and a basis of right - eigenvectors . in , the given matrix - valued field need not be a jacobian matrix and , also , is required to possess real and distinct eigenvalues and a complete set of eigenvectors .small - scale dependent solutions arise with systems of the form or and it is our objective to present and investigate specific models of interest in physical applications , especially :  * _ strictly hyperbolic systems . _the simplest example of interest is provided by the scalar conservation law with cubic flux and added second- and third - order terms and is presented in section [ sec:21 ] below .more challenging models arise in the dynamics of fluids and nonlinear elastic material with viscosity and capillarity effects ( cf . section [ sec:23 ] ) . * _ non - strictly hyperbolic systems . _ in presence of certain phase transition phenomena , the models of fluids and elastic materials fail to be globaly strictly hyperbolic .furthermore , the system of magnetohydrodynamics with viscosity and hall effects is probably the most challenging model and plays also an essential role in applications , for instance , in the modeling of the solar wind . *_ nonconservative hyperbolic systems ._ as will be discussed in section [ sec:5 ] below , one of the simplest ( yet challenging ) example in the class of nonconservative hyperbolic systems is obtained by coupling two burgers equations , while the most challenging model of interest in the application contains five equations ( or seven equations , if two thermodynamical variables are introduced ) for the evolution of two ( fluid and vapor ) phases of a fluid mixture .other important models are the multi - layer shallow water system and the lagrangian gas dynamics with internal energy taken as an independent variable .* _ initial and boundary value problems . _models of interest include , both , the linearized and nonlinear euler equations with artificial viscosity or physical viscosity .  a ( mathematical ) entropy inequality can be naturally associated with all ( conservative or nonconservative ) systems under consideration , that is , eq:109-ineq u(u)_t + f(u)_x 0 , where and are refered to as the entropy and entropy - flux , respectively .however , in constrast with more classical problems arising in fluid dynamics , is often insufficiently discriminating in order to characterize physically meaningful solutions to the initial value problem associated with or .therefore , additional admissibility criteria are required , as follows :  * _ kinetic relations _ provide a general tool to define nonclassical entropy solutions to strictly hyperbolic systems of conservation laws ( lefloch 1993 , hayes and lefloch 1996 ) and are relevant when the characteristic fields of the systems do not satisfy lax s genuine nonlinearity condition ( lax 1957 , 1973 ) .they were introduced first for an hyperbolic - elliptic model of phase transitions in solids ( abeyaratne and knowles 1991a , 1991b , truskinovsky 1987 , 1993 , 1994 ) .roughly speaking , a kinetic relation for a nonclassical ( undercompressive ) shock in the -characteristic family prescribes the right - hand state as a function of the left - hand state . * _ families of paths _\mapsto \varphi(s ; u_- , u_+) ] connect left - hand states to right - hand states and are derived by analyzing the trajectories of traveling solutions , once an augmented model taking higher - order effects into account is selected . * _ admissible boundary sets _ ( dubois and lefloch 1988 ) are required in order to formulate well - posed initial boundary value problems associated with nonlinear hyperbolic systems .waves propagating in weak solutions may collapse near a boundary and generate a boundary layer connecting a given boundary state with an actual state constrained to lie in a prescribed boundary set .it is well now recognized ( hou and lefloch 1994 , hayes and lefloch 1996 , lefloch and rohde 2000 , lefloch and mohamadian 2008 , fjordholm and mishra 2012 ) that a finite difference or volume scheme ( leveque 2003 ) may not converge to physically - relevant weak solutions , unless one can ensure a certain _ consistency property with small - scale effects _ , that is , a consistency with the prescribed kinetic relation , family of paths , or admissible boundary set associated with any specific problem under consideration .it was suggested in lefloch ( 2010 ) to call the numerical methods satisfying this requirement as the schemes with `` controlled dissipation '' and , recent work by the authors ( covering the treatment of shocks with arbitrary strength ) , it was proposed to refer to them as _ schemes with well - controlled dissipation_.  in particular , the role of the _ equivalent equation _ associated with a given finite difference scheme has been emphasized and analyzed .the leading terms of the equivalent equations for standard finite difference schemes like the lax - friedrichs scheme , for instance , significantly differ from the physically - relevant small - scale mechanisms .for small - scale dependent shocks , the approximate solutions , say converge ( when the discretization parameter approaches zero ) toward a limit , say , which is _ distinct from the physical solution , _ say .in other words , standard finite difference or finite volume techniques , in general , lead to the non - convergence property v : = _ h 0 u^h u this holds for a variety of strictly hyperbolic systems , nonconservative hyperbolic systems , and boundary layer problems .  on the other hand , ` schemes with well - controlled dissipation 'are precisely designed to overcome this challenge and are built by analyzing discrete dissipation operators arising in equivalent equations , imposing that the latter should match the small - scale mechanisms in the underlying augmented system , at least to leading order .it was emphasized by hou and lefloch ( 1994 ) and hayes and lefloch ( 1996 ) that , in fact , the objective need not be to ensure the convergence of the schemes , but rather to control the error term ( in a suitable norm ) in terms of the physical parameters arising in the problem : shock strength , order of accuracy of the scheme , ratio of capillarity over viscosity , etc .  in recent years , extensive studies have demonstrated the relevance of the equivalent equation , as a tool for designing numerical methods for computing small - scale dependent shocks , and have included numerical experiments in physically realistic set - ups .this paper is organized as follows .section 2 is concerned with strictly hyperbolic systems and the discussion of nonclassical undercompressive shocks to such systems , when the associated augmented models contain diffusive and dispersive terms .then , section 3 discusses the numerical methods adapted to these problems .next , in section 4 we turn our attention to nonconservative hyperbolic systems .the boundary value problems are discussed in section 5 and some concluding remarks are made in section 6 .we consider first the class of _ hyperbolic conservation laws with vanishing diffusion , _ i.e. [ eq:356 ] u^_t + f(u^)_x = ( b(u^ ) u^_x)_x , where is the unknown , the flux is a given smooth function , and the diffusion coefficient is bounded above and below . for any given initial data and , solutions to the initial value problem associated with converge strongly ( when ) toward a limit satisfying the hyperbolic conservation law [ eq:356 - 0 ] u_t + f(u)_x = 0 in the weak sense of distributions .weak solutions to are not uniquely characterized by their initial data , but must also be constrained to satisfy a certain entropy condition , ensuring that they be achieved as limits of ( oleinik 1963 , kruzkov 1970 , volpert 1967 ) .more precisely , solutions to satisfy , for every convex function , in which and is refered to as an entropy - entropy flux pair .hence , satisfies the so - called entropy inequalities [ eq:357 ] u(u)_t + f(u)_x 0 , u 0 .weak solutions to that satisfy _ all _ inequalities ( i.e for every convex ) are refered to as _ classical entropy solutions ._ it is also customary to reformulate the entropy condition in the kruzkov form : for all .nonclassical shock waves arise in weak solutions when , both , diffusion and dispersion are included . the simplest model of interest is provided by the _linear diffusion - dispersion model _ [ eq : cons2 ] u^_t + f(u^)_x = u^_xx + ( ) u^_xxx , which depends upon two parameters and refered to as the diffusion and the dispersion coefficients .this equation was studied first by jacobs , mckinney , and shearer ( 1993 ) , hayes and lefloch ( 1996 , 1997 ) , and bedjaoui and lefloch ( 2002abc , 2004 ) .importantly , the relative scaling between and determines the limiting behavior of solutions , and we can distinguish between three cases :  * in the _ diffusion - dominant regime _ , the qualitative behavior of solutions to is similar to the behavior of solutions to and , in fact , the limit is then _ independent _ upon , and is a _ classical _ entropy solution characterized by the infinite family of entropy inequalities . * in the _ dispersion - dominant regime _ , high oscillations develop ( as approaches zero ) especially in regions of steep gradients of the solutions and only weak convergence of is observed . the vanishing dispersion method developed by lax and levermore ( 1983 ) for the korteweg - de vries equationis the relevant theory in this regime , which can not be covered by the techniques under consideration in the present review .* in the regime of _ balanced diffusion - dispersion , _ corresponding typically to for a fixed , the limit does exist ( in a strong topology ) and only _ mild oscillations _ are observed near shocks , so that the limit is a weak solution to the hyperbolic conservation law .most importantly , when , the solutions exhibit nonclassical behavior , as they contain undercompressive shocks ( as defined below ) and _ strongly depend _ upon the coefficient .  from nowon , we focus our attention on the `` critical regime '' where the small - scale terms are kept in balance and , for instance , we write as eq:240 u^_t + f(u^)_x = u^_xx + ^2 u^_xxx , in which is a fixed parameter and . for this augmented model, we easily derive the identity the diffusive contribution decomposes into a non - positive term and a conservative one , while the dispersive contribution is entirely conservative ; hence , formally at least , as we recover the entropy inequality , _ but only _ for the specific entropy function . in other words, we obtain the _ ( single ) quadratic entropy inequality _eq:250 ( u^2/2 ) _ t + f(u)_x 0 , f : = u f. in general , no specific sign is available for arbitrary convex entropies ( unlike what we observed in the diffusion - only regime ) .   more generally , we can consider the _ nonlinear diffusion - dispersion model _eq:129 u^_t + f(u^)_x = ( b(u^ ) u^_x ) _ x + ^2 ( c_1(u^ ) ( c_2(u^ ) u^_x ) _ x ) _ x , where are given smooth and positive functions . for this model ,the formal limit satisfies the following _ entropy inequality _ determined by the function [ eq : ucc ] & u(u)_t + f(u)_x 0 , + & u = c_2 c_1 > 0 , f : = f u. namely , in the entropy variable , the dispersive term takes the form so that any solution to satisfies & u(u^)_t + f(u^)_x = - d^+ c^_x , d^:= b(u ) u(u ) |u_x|^2 , + & c^:= b(u ) u(u ) u_x + ( c_1(u ) u ( c_1(u ) u_x ) _ x - | c_2(u ) u_x|^2 /2 ) .nonlinear augmented terms also arise in the _ model of thin liquid films _ [ thin ] u^_t + ( u_^2 - u_^3)_x = ( u_^3 u_x^)_x - ^3 ( u_^3 u^_xxx ) _ x with fixed and .the scaling is natural since the augmented term now involves four derivatives .the right - hand side describes the effects of surface tension in a thin liquid film moving on a surface and  ] satisfies if we also include the effect of a ( nonlinear ) viscosity , we arrive at a fully nonlinear phase transition model with viscosity and capillarity : again , the total energy plays the role of a mathematical entropy , and we find and once more , a single entropy inequality is obtained .  in the case that is quadratic in , for some positive capillarity coefficient , we set , and the total stress decomposes as the evolution equations then take the form [ 666 ] & w_t - v_x = 0 , + & v_t - ( w)_x = ( (w ) w_x^2 2 - ( ( w ) w_x ) _x ) _ x + ( ( w ) v_x ) _ x .we then obtain which , again , leads to the entropy inequality .when the viscosity and capillarity are taken to be constants , we recover the previous model and , again , the entropy inequality is identical for both regularizations .all the models in this section admit shock wave solutions that do satisfy a single entropy inequality in the sense of distributions ( that is , , , , or ) , but fail to satisfy standard entropy conditions ( oleinik , kruzkov , lax , wendroff , etc . ) .however , these entropy conditions played an essential role in the design of efficient shock - capturing schemes for standard fluid dynamics problems ( leveque 2003 ) .the new shocks are referred to as as nonclassical shocks and can be checked to be undercompressive , in the sense that they are linearly unstable , since a perturbation passes through them rather than impinging on them ( as is the case of compressive shocks ) .for this reason , an additional admissibility condition is necessary , which is called a kinetic relation .it takes the form of an additional jump relation at shock discontinuities .kinetic relations can be obtained analytically only for the simpler models ( scalar conservation laws with linear diffusion and dispersion , nonlinear elasticity model with linear viscosity and capillarity ) , so numerical approaches are necessary to tackle these problems . in the referencesalready cited , numerical investigations have established that kinetic functions exist ( and often satisfy certain monotonicity properties ) for a large class of physically relevant models including thin liquid films , generalized camassa - holm , nonlinear phase transitions , van der waals fluids ( for small shocks ) , and magnetohydrodynamics .the methods and techniques to be presented in this paper are also relevant for other classes of problems .for instance , the buckley - leverett equation for two - phase flows in porous media provides another model of interest for the applications , and was studied hayes and shearer ( 1999 ) and van duijn , peletier , and pop ( 2007 ) .there are also other models of great physical interest which , however , have not yet received as much attention . since the hyperbolic flux of these models does admit an inflection point and that dispersive - type effects are important in the modeling of such problems , it is expected that undercompressive shocks shoud occur , at least in certain regimes of applications .this is the case of the quantum hydrodynamics models ( marcati , jerome ) , phase field models ( caginalp , ratz ) , suliciu - type models ( carboux , frid , suliciu ) , non - local models involving fractional integrals ( kissiling et al . ,rohde ) discrete molecular models based on potentials ( for instance of lennard - jones type ) ( bohm , dreyer , nt , tv , weinan ) .the references at the end of this paper .another rich direction of research where analogue challenges are met is provided by the coupling techniques involving two hyperbolic systems with distinct flux - functions , for which we refer to lefloch ( 1993 ) , seguin and vovelle ( 2003 ) , godlevski and raviart ( 2004 ) , adimurthi , mishra , and gowda ( 2005 ) , godlevski , le thanh , and raviart ( 2005 ) , bachman and vovelle ( 2006 ) , brger and karlsen ( 2008 ) , chalons , raviart , and seguin ( 2008 ) , holden , karlsen , mitrovic , and panov ( 2009 ) , boutin , coquel , and lefloch ( 2011 , 2012 , 2013 ) , and the references cited therein .we consider a nonlinear hyperbolic system in the conservative form and we now discretize it ( in space ) on a grid consisting of points , with being a uniform mesh width .( the grid is assumed to be uniform for the sake of simplicity in the exposition . ) on this grid , a standard ( semi - discrete ) finite difference scheme provides an approximation of the point values of solutions to , defined by [ eq : fdm1 ] u_i + ( g_i+1/2(t ) - g_i-1/2(t ) ) = 0 . here, is a consistent _ numerical flux _ associated with of the flux , that is , for all relevant .alternatively , by considering the cell averages one formulates a finite volume scheme which has the same form .a well - known theorem due to lax and wendroff ( 1960 ) establishes that if the numerical approximation converges ( in a suitable sense ) , it can converge only towards a weak solution of the underlying system .furthermore , if some structural conditions on the numerical flux are assumed as in ( tadmor 1987 , 2003 ) , one can show that the scheme in the limit also satisfies a discrete version of the entropy inequality , that is , [ eq : efdm1 ] u(u_i ) + ( g_i+1/2(t ) - g_i-1/2(t ) ) 0 . here ,the _ numerical entropy flux _ is consistent with the entropy flux in , in the sense that for all relevant .when such a discrete entropy inequality is available , one can readily modify lax and wendroff s argument and show that if the approximations generated by the finite difference ( or finite volume ) scheme converge , then they can only converge toward an entropy solution of .a variety of numerical fluxes that satisfy the entropy stability criteria ( as stated in tadmor 1987 , 2003 ) have been designed in the last three decades .theses classes of schemes include exact riemann solvers of godunov - type , approximate riemann solvers such as roe and harten - lax - van leer ( hll ) solvers , as well a central difference schemes such as lax - friedrichs and rusanov schemes .a comprehensive description of these schemes and their properties are available in the literature , for instance in the textbook by leveque ( 2003 ) .   as mentioned in the introduction , standard conservative and entropy stable schemes fail to approximate nonclassical shocks ( and other small - scale dependent solutions ) . as an illustrative example, we consider here the cubic scalar conservation law with linear diffusion and dispersion , that is , with and a fixed .the underlying conservation law is approximated with the standard lax - friedrichs and rusanov schemes and the resulting solutions are plotted in figure [ fig:1 ] .the figure clearly demonstrates that godunov and lax - friedrichs schemes , both , converge to the _ classical _ entropy solution to the scalar conservation law and , therefore , do not approximate the nonclassical entropy solution , realized as the vanishing diffusion - dispersion limit of and also plotted in the same figure .the latter consists of three distinct constant states separated by two shocks , while the classical solution contains a single shock .as pointed out in the introduction , this failure of standard schemes in approximating small - scale dependent shocks ( in various contexts ) can be explained in terms of the equivalent equation of the scheme ( as was first observed by hou and lefloch ( 1994 ) and hayes and lefloch ( 1996 , 1998 ) .the equivalent equation is derived via a ( formal ) taylor expansion of the discrete scheme and contains mesh - dependent terms and high - order derivatives of the solution .  for a first - order scheme like, the equivalent equation has the typical form [ eq : eqen ] u_t + f(u)_x = ( ( u)u_x)_x + ^2 ( _ 1(u)(_2(u)u_x)_x)_x + ( ^3 ) .interestingly enough , this equation is of the augmented form with being now the small - scale parameter .standard schemes often have , , and , with being small - scale terms prescribed in the nonlinear physical model like .as the shocks realized as the limit of ( [ eq:129 ] ) depend explicitly on the expressions of the diffusion and dispersion terms , this difference in the diffusion and dispersion terms between ( [ eq : eqen ] ) and ( [ eq:129 ] ) is the _ crucial _ reason as to why standard schemes fail to correctly approximate small - scale dependent shocks.   the equivalent equation of the finite difference scheme also suggests a way for modifying the scheme such that the correct small - scale dependent shock waves can be approximated .following hayes and lefloch ( 1997 , 1998 ) , lefloch and rohde ( 2000 ) , and lefloch and mohamadian ( 2008 ) , the key idea is to design finite difference schemes whose equivalent equation matches , both , the diffusive and the dispersive terms in the augmented model ( for instance ) .namely , the schemes are designed so that their equivalent equation ( [ eq : eqen ] ) has , , .such schemes are refered to as _ schemes with controlled dissipation_.  in order to now proceed with the derivation of a class of schemes with controlled dissipation , we focus attention on the prototypical example of the scalar conservation law , and we consider nonclassical shocks generated in the limit of balanced vanishing diffusion and dispersion ; cf . . on the uniform grid presented in the previous subsection and for any integer , we approximate the conservation law ( [ eq:109 ] ) with the following -th order consistent finite difference scheme : [ eq : fds2p ] + _ j =- p^j = p _ jf_i+j = _ j =- p^j = p _ j u_i+j + _ j =- p^j = p _ j u_i+j . here , is the cell nodal value , is the flux , the constant is the coefficient of capillarity ( given by the physics of the problem ) and being a positive constant .the coefficients and need to satisfy the following _ -order conditions _ : these conditions define a set of linear equations for unknowns and can be solved explicitly .similarly , the coefficients must satisfy while , for the coefficients ,   the proposed finite difference scheme ( [ eq : fds2p ] ) is a conservative and consistent discretization of the conservation law ( [ eq:109 ] ) .it is formally only first - order accurate since the diffusive terms are proportional to .this scheme need not preserve the monotonicity of the solutions .the equivalent equation associated with the scheme ( [ eq : fds2p ] ) reads [ eq : ee ] + f(u)_x = & c u_xx + c^2 ( ) ^2 u_xxx - _k=2p+1^ a^p_k ( f(u))^[k ] + & + c _k=2p+1^ b^p_k u^[k ] + c^2 _k=2p+1^ c^p_k u^[k ] . here , } = \frac{d^k g}{dx^k} ] is prescribed , which must satisfy certain regularity and compatibility conditions , in particular [ cond1 ] ( 0;u_l , u_r ) = u_l , ( 1;u_l , u_r ) = u_r , ( s ; u ,u ) = u. we refer to dal maso , lefloch , and murat ( 1990 , 1995 ) for a full presentation of the theory .once the nonconservative product has been defined , one may define the weak solutions of . according to this theory , across a discontinuitya weak solution has to satisfy the generalized rankine - hugoniot condition [ gen_r - h ] = _ 0 ^ 1 a((s;u_-,u_+ ) ) _s(s;u_-,u_+ ) ds , where is the speed of propagation of the discontinuity , and are the left- and right - hand limits of the solution at the discontinuity , and = u_+ - u_- ] , ^\top ] with 1500 points and cfl = 0.4 . as shown in figure [ fig : godex ] ,the location of the discontinuities is correctly approximated , whereas the intermediate states approximated by the godunov scheme are _ incorrect ._ the error in these intermediate states does vanish as tends to 0 . even though the godunov method takes into account the exact expression of the viscous profiles ( paths ) and the exact solutions of the riemann problems , the numerical solutions provided by the method _ do not converge _ to the expected weak solutions , due to the numerical viscosity added in the projection step .following the procedure outlined in the construction of the entropy stable scheme above , the following entropy stable fluctuations were derived in castro et al ( 2013 ) for the coupled burgers equation : [ eq : cpd ]  _i+1/2 ^ - & = [ [ ] ] _i+1/2  2w_i + w_i+1 + 2v_i + v_i+1  -  _ i+1/2 + [ [ w]]_i+1/2  , + _ i+1/2^+ & = [ [ ] ] _ i+1/2  2w_i + w_i+1 + 2v_i + v_i+1  +_ i+1/2 + [ [ w]]_i+1/2  .in order to validate these numerical schemes , we consider again the riemann problem with initial data ^\top ] and compare the exact solution with the numerical one provided by the espc scheme ( the relevant scheme with controlled dissipation ) in the interval  ] and a series of left - hand states belonging to the exact shock curve .the riemann problem is solved in the interval  ] .all the simulations are performed with mesh points . for the sake of comparison , a reference solution computed with the eddy viscosity system and a fixed on a very fine mesh of mesh pointsis also shown . as seen in this figure , the solutions computed with all the schemes are quite close to the reference solution . as seen in the closeup, there is a minor difference in the intermediate state computed by the espc - nv and roe schemes .the espc scheme contains oscillations .this is to be expected as the mass conservation equations contains no numerical viscosity .however , the approximate solution computed by this scheme is still quite close to the reference solution .  in order to compare the performance of the schemes for a large set of initial data , we compute a _ numerical _ hugoniot locus by fixing the same right - hand state as in , and then varying the left - hand state .a reference hugoniot locus , calculated from a numerical approximation of the mixed hyperbolic - parabolic system , is shown .we also display the hugoniot locus corresponding to the family of straight line segments .all the hugoniot loci in the - plane and the - plane are shown in figure [ fig : tl2 ] .  from figure[ fig : tl2 ] , we observe that the hugoniot locus calculated using straight line segments is clearly different from the one calculated from the underlying viscous two - layer shallow water equations . on the other hand ,all the three numerical schemes lead to hugoniot loci that are very close to each other and to the reference hugoniot locus .minor differences are visible when we zoom in ; see the bottom row of figure [ fig : tl2 ] .we see that , among the three schemes , the espc scheme provides the best overall approximation , to the reference hugoniot locus .however , both the espc - nv and the roe schemes also provide a good approximation to the reference hugoniot locus .the results show that ( rather surprisingly ) the numerical approximation of two - layer shallow water equations is not as sensitive to the viscous terms as the coupled burgers system .the path - consistent roe scheme performs adequately in approximating the correct solution . at the same time , the espc schemes provide a slightly more accurate approximation .[ fig : tl1 ]   and planes for the two - layer shallow water equations , computed with the espc , espc - nv and roe schemes .a reference hugoniot locus computed from the viscous shallow water system and a hugoniot locus computed using straight line paths are also shown.,title="fig : " ]  and planes for the two - layer shallow water equations , computed with the espc , espc - nv and roe schemes .a reference hugoniot locus computed from the viscous shallow water system and a hugoniot locus computed using straight line paths are also shown.,title="fig : " ]  and planes for the two - layer shallow water equations , computed with the espc , espc - nv and roe schemes .a reference hugoniot locus computed from the viscous shallow water system and a hugoniot locus computed using straight line paths are also shown.,title="fig : " ]  and planes for the two - layer shallow water equations , computed with the espc , espc - nv and roe schemes .a reference hugoniot locus computed from the viscous shallow water system and a hugoniot locus computed using straight line paths are also shown.,title="fig : " ]  [ fig : tl2 ]   when approximating shocks to nonconservative hyperbolic systems , the above numerical experiments clearly indicate the superior performance of the _ schemes with controlled dissipation , _ in contrast with solely imposing a formal path - consistent condition ( directy imposed on , for instance , godunov or roe flux ) .however , as in the case of nonclassical shocks to conservation laws , schemes with controlled dissipation can fail to approximate strong ( large amplitude ) shocks .evidence for this fact has been presented in the numerical experiments for the coupled burgers system ( see figure [ fig : cp2 ] ) , as well as in fjordholm and mishra ( 2012 ) who considered the system of gas dynamics in lagrangian coordinates .  therefore , again following the extensive discussion in section 3 , it is now clear that we need _ schemes with well - controlled dissipation _ ( wcd ) in order to approximate shocks with arbitrary large amplitude to nonconservative systems .schemes with controlled dissipation provide a starting point but need to be further improved .the `` full '' equivalent equation , that is , a specification of the high - order terms , analogous to the equivalent equation for conservative systems , needs to be taken into account so that the asymptotics for large shocks are taken into account in the design of the schemes .furthermore , higher - order finite difference discretizations of the nonconservative term are required .only then can we proceed analogously to the conservative case , and balance the leading - order terms of the equivalent equation to the high - order terms in order to design a suitable _ wcd condition . _the specification of this wcd condition and extensive experiments with the resulting schemes are the subject of the forthcoming paper ( beljadid , lefloch , mishra , and pares , 2014 ) .we now turn our attention to the _ initial and boundary value problem _ for nonlinear hyperbolic systems of conservation laws with prescribed data : [ eq : cl : bc ] u(0 , x ) & = u_0(x ) , x = ( x_l , ) , + u(t , x_l)&= u(t ) , t 0 , for some given boundary point .this one - half boundary value problem can be readily generalized to include two boundaries ( that is , for some ) .the study of the initial and boundary value problem poses additional difficulties as compared to the study of the cauchy problem : most importantly , the problem  and is ill - posed in the sense that , in general , it admits no solution unless the boundary condition is understood in the weaker sense eq : df u(t , x_l ) ( u(t ) ) , t 0 , as proposed by dubois and lefloch ( 1988 ) . here , is a set containing , which _ depends _ upon the way that the boundary value problem is handled , for instance via the riemann problem ( sharp boundary layers ) or by the vanishing viscosity method ( viscous boundary layers .dubois and lefloch ( 1988 ) studied sharp boundary layers for the euler equations : by solving the so - called boundary riemann problem and defining as the set of all boundary values of riemann problems with fixed left - hand state at , these authors show that can be decomposed into several strata ( or submanifolds ) in the state space whose local dimension increases with the strength of the shock layer .  alternatively , following ( benabdallah 1986 , dubois and lefloch 1988 , gisclon 1996 , gisclon and serre 1994 , joseph and lefloch 1996 , joseph and lefloch 1999 ), one may consider the viscous approximations ( with ) [ eq:3560 ] u^_t + f(u^)_x = ( b(u^ ) u^_x)_x , x = ( x_l , ) , with given viscosity matrix , and supplement these equations with initial and boundary conditions : [ e : vcl : bd ] u^(0 , x ) & = u_0(x ) , x = ( x_l , ) , + u^(t , x_l ) & = u_l ( t ) , t 0 .we assume that this initial and boundary value problem - is locally well - posed ( which is the case under mild conditions on the viscosity matrix and with sufficiently regularity on the data ) and that for the solutions converge ( in a suitable topology ) to a limit . due to a boundary layer phenomena, the limit , in general , _ does not _ satisfy the prescribed boundary condition pointwisely .instead , dubois and lefloch ( 1988 ) showed that , if the system of conservation laws is endowed with an entropy - entropy flux pair , then the solutions of the viscous approximation  converge as ) to a solution of the initial and boundary problem  and in a sufficiently strong topology , then the following _ entropy boundary inequality _ [ eq : dlf ] f((t ) ) - f(u_l(t ) ) - d_u u(u_l(t ) ) ( f((t ) ) -f(u_l(t ) ) ) 0 .for the case of scalar conservatio laws , a version of this inequality was also derived earlier by le roux ( 1977 ) and bardos , le roux , and ndelec ( 1979 ) .  as for other small - scale dependent models before , a major difficulty in the study of initial and boundary value problems was pointed out by gisclon and serre ( 1994 ) and joseph and lefloch ( 1996 ) : the limit of the viscous approximation  _ depends _ on the underlying viscosity mechanism . in other words , the limit of  in general changes if one changes the viscosity matrix .  as an example , we consider the linearized shallow water equations with initial data and boundary data .the system is a _ linear _ , strictly hyperbolic , system and is the simplest possible problem that can be considered in this context .we consider two different viscosity operators : an _ artificial _ uniform ( laplacian ) viscosity and the _ physical _ eddy viscosity .the resulting limit solutions are shown in the left of figure [ fig:71 ] .as shown in the figure , there is a significant difference in solutions ( near the boundary ) corresponding to different viscosity operators .  for an extended discussion of the initial boundary value problem for systems of conservation laws and its viscous approximation, we refer to serre ( 2000 , 2007 ) and the bibliography therein , including for the theory of discrete shock profiles .we stress that analytically establishing the convergence in the possibly characteristic regime and general diffusion matrices was addressed only rather recently in the successive works : joseph and lefloch ( 1999 , 2002 , 2006 ) , ancona and bianchini ( 2006 ) , bianchini and spinolo ( 2009 ) , and christoforou and spinolo ( 2011 ) .many other results are also available in more specific cases , which we do not attempt to review here .as in the previous sections , it is common to use a finite difference or a finite volume scheme of the form , with a suitable numerical flux . following goodman ( 1982 ) and dubois and lefloch ( 1988 ) ( see also the textbook by leveque 2003 ) , the dirichlet boundary conditions at are imposed by setting in the _ ghost _ cell  ] with initial data [ eq : eulinit ] ( _ 0,u_0,p_0 ) =  ( 3.0,1.0,3.0 ) , & x < 0 , + ( 1.0,1.0,1.0 ) , & x > 0 .we impose absorbing boundary conditions at the right - hand boundary and dirichlet boundary conditions at the left - hand boundary with boundary data [ eq : eulbd ] ( ( -1,t),u(-1,t),p(-1,t ) ) = ( 2.0,1.0,2.0 ) , and , for simplicity , we set . the results for the finite difference scheme approximating the uniform viscosity and the physical viscosity at time are presented in figure [ fig:73 ] .the figure shows that the there is a clear difference in the limit solutions of this problem , obtained from the compressible navier - stokes equations and the euler equations with artificial viscosity .the difference is more pronounced in the density variable near the left - hand boundary .both the limit solutions were computed by setting and on a very fine mesh of points .+  +  the above example also illustrates the limitations of using a mixed hyperbolic - parabolic system like the compressible navier - stokes equations . in order to resolve the viscous scales , we need to choose , with being the viscosity parameter . as is very small in practice ,the computational effort involved is prohibitively expensive . in the above example, we needed points to handle .such ultra fine grids are not feasible , particularly in several space dimensions .hence , we will use a scheme with controlled dissipation such as the cnd scheme .this scheme requires both entropy conservative fluxes as well as numerical diffusion operators .we use the entropy conservative fluxes for the euler equations that were recently developed by ismail and roe ( 2007 ) and a discrete version of the navier - stokes viscosity as in mishra and spinolo ( 2011 ) .we discretize the initial and boundary value problem for the compressible euler equations on the computational domain $ ] with initial data and dirichlet data .the results with the cnd scheme and a standard roe scheme at time are shown in figure [ fig:74 ] .we present approximate solutions , computed on a mesh of points , for both schemes . both the roe and the cnd schemes have converged at this resolution .as we are interested in approximating the physical - viscosity solutions of the euler equations , realized as a limit of the navier - stokes equations , we plot a reference solution computed on a mesh of points of the compressible navier - stokes equations with .the figure shows that the roe scheme clearly converges to an incorrect solution near the left - hand boundary .this lack of convergence is most pronounced in the density variable .similar results were also obtained with the standard rusanov , hll and hllc solvers ( see the book by leveque ( 2003 ) for a detailed description of these solvers ) .on the other hand , the cnd scheme converges to the physical - viscosity solution .there are slight oscillations with the cnd scheme as the numerical diffusion operator is singular . however , these oscillations do not impact on the convergence properties of this scheme .although , the roe scheme does not generate any spurious oscillations , yet it converges to an incorrect solution of the euler equations . on the other hand ,the cnd scheme does converge to the physical - viscosity solution ( the navier - stokes limit ) in spite of some spurious oscillations .it appears that the oscillations are the price that one has to pay in order to resolve the physical - viscosity solution correctly .moreover , the cnd scheme is slightly more accurate than the roe scheme when both of them converge to the same solution ( see near the interior contact ) .+  +   the cnd scheme of mishra and spinolo ( 2011 ) is a scheme with controlled dissipation in this case .it approximates shocks of moderate amplitude quite robustly as shown in the previous numerical experiments .however , there might be a deterioration of performance once the shock amplitude is very large . in this case, a scheme with well - controlled dissipation ( wcd ) can be readily designed using the construction involving the equivalent equation , as outlined in the previous section .another class of numerical methods for approximating small - scale dependent solutions to hyperbolic problems is based on front tracking schemes and random choice schemes . although they may not be always suited for physical applications where no information is a priori known on the solutions and high - order accuracyis sought , these methods have some definite advantages .most importantly , numerical shocks are not smeared and are represented as discontinuities and it is only the location of the shock and the left- and right - hand values that are approximated .front - traching and random choice methods have been first used to establish the existence of small - scale dependent solutions and allows to achieve the following results :  existence theory for the initial value problem for solutions with nonclassical shocks by amadori , baiti , piccoli , and lefloch ( 1999 ) , baiti , lefloch , and piccoli ( 1999 , 2000 , 2001 , 2004 ) , lefloch ( 2002 ) , laforest and lefloch ( 2010 , 2014 ) .existence theory for the initial value problem for nonconservative systems by lefloch and liu ( 1993 ) , which was based on the nonconservative riemann solver and the theory of nonconservative products develope by dal maso , lefloch , and murat ( 1990 , 1995 ) .existence theory for the initial value problem for hyperbolic systems of conservation laws by amadori ( 1997 ) , amadori and colombo ( 1997 ) , ancona and marson ( 1999 ) , karlsen , lie , and risebro ( 1999 ) , donadello and marson ( 2007 ) .in particular , this strategy was numerically implemented by chalons and lefloch ( 2003 ) ( nonclassical solutions via the random choice scheme ) and , in combination with a level set technique , by zhong , hou , and lefloch ( 1996 ) , hou , rosakis , and lefloch ( 1999 ) , and merkle and rohde ( 2006 , 2007 ) , who treated a nonlinear elasticity model with trilinear law in two spatial dimensions . as observed by zhong , hou and lefloch , this model exhibits complex interface needles attached to the boundary , whose computation is numerically very challenging since they are strongly small - scale dependent .in addition , methods combining differences and interface tracking were also developed which ensure that the interface is sharp and ( almost ) exactly propagated ; see boutin , chalons , lagoutire , and lefloch ( 2008 ) .   as described in the current review ,the design and numerical implementation of robust and efficient methods for small - scale dependent schemes are well established by now . however ,a complete theory , with convergence results , is available only for random choice and front tracking schemes ; this theory encompasses nonclassical undercompressive shocks , nonconservative hyperbolic systems , and the boundary value problem .however , only partial results are available concerning the rigorous convergence analysis of the proposed schemes , and the interested reader may consult and built upon the references cited in the bibliography .   in summary ,our main guidelines in order to design efficient and robust numerical methods for small - scale dependent shock waves can be summarized as follows :  * standard finite difference or finite volume schemes fail to properly approximate weak solutions containing small - scale dependent shocks , and a _ well - controlled dissipation _ requirement is necessary . *this lack of convergence to physically - relevant solutions arises with most nonlinear hyperbolic problems , including : * non - genuinely nonlinear systems ( when dispersive effects are present and may contribute to the dynamics of shocks ) , * nonconservative hyperbolic systems ( since products of discontinuous functions by measures are regularization - dependent ) , and * boundary value problems ( due to the formation of possibly characteristic boundary layers ) . * numerous applications lead to such problems , especially : * the dynamics of viscous - capillary fluids , * the dynamics of two - phase fluids such as liquid - vapor or liquid - solid or solid - solid mixtures .* the model of magnetohydrodynamics with hall effect incuded , which leads to the most challenging system , on account of lack of genuine nonlinearity , lack of strict hyperbolicity , and presence of dispersive terms .* although the classes of problems under consideration are quite distinct physical origin , a single source of difficulty was identified here , that is , the importance of properly computing the global effect of small - scale terms .standard schemes for all these problems fail because the leading terms in the equivalent equation are different from the small - scale mechanisms of the underlying pde .* schemes with well - controlled dissipation have been developed in the past fifteen years , in order to accurately compute small - scale dependent shock waves .the key ingredient was to systematically design numerical diffusion operators that lead to the equivalent equation of the scheme matching the small - scale mechanisms of the underlying pde at the leading order .* as the residual terms of the equivalent equation can become large with increasing shock strength , any scheme of a fixed order of accuracy will fail to converge to the correct solution for very large shocks , and this issue was addressed .the proper notion of convergence for small - scale dependent shock waves is that the approximate solutions converge in as well as in terms of kinetic relations , as the order of the scheme is increased . *computing kinetic functions , familes of paths , and admissible boundary sets is very useful in order to investigate the effects of the diffusion / dispersion ratio , regularization , order of accuracy of the schemes , the efficiency of the schemes , as well as to make comparisons between several physical models .the first author ( plf ) was supported by a dfg - cnrs collaborative grant between france and germany entitled `` micro - macro modeling and simulation of liquid - vapor flows '' , by the centre national de la recherche scientifique , and by the agence nationale de la recherche . the second author ( sm )was partially supported by erc starting grant 306279 ( sparccle ) from the european research council .r. abgrall and karni s. 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Abstract: we provide a ` user guide ' to the literature of the past twenty years concerning the modeling and approximation of discontinuous solutions to nonlinear hyperbolic systems that admit _ small - scale dependent _ shock waves . we cover several classes of problems and solutions : nonclassical undercompressive shocks , hyperbolic systems in nonconservative form , boundary layer problems . we review the relevant models arising in continuum physics and describe the numerical methods that have been proposed to capture small - scale dependent solutions . in agreement with the general well - posedness theory , small - scale dependent solutions are characterized by a _ kinetic relation , _ a _ family of paths , _ or an _ admissible boundary set . _ we provide a review of numerical methods ( front tracking schemes , finite difference schemes , finite volume schemes ) , which , at the discrete level , reproduce the effect of the physically - meaningful dissipation mechanisms of interest in the applications . an essential role is played by the _ equivalent equation _ associated with discrete schemes , which is found to be relevant even for solutions containing shock waves .  [ firstpage ]  1.cm