This work deals with the numerical solution of the monodomain and bidomain models of electrical activity of myocardial tissue. The bidomain model is a system consisting of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the so-called gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple models for the membrane and ionic currents are considered, the Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates thatthese methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, an optimalthreshold for discarding non-significant information in the multiresolution representation of the solution is derived, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.
 In recent years a variety of high–order schemes for the numerical solution of conservation laws has been developed. In general, these numerical methods involve expensive flux evaluations in order to resolve discontinuities accurately. But in large parts of the flow domain the solution is smooth. Hence in these regions an unexpensive finite difference scheme suffices. In order to reduce the number of expensive flux evaluations we employ a multiresolution strategy which is similar in spirit to an approach that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial dimensions and boundary fitted meshes essential deviations from previous investigations appear to be necessary though. This concerns handling the more complex interrelations of fluxes across cell interfaces, the derivation of appropriate evolution equations for multiscale representations of cell averages, stability and convergence, quantifying the compression effects by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude with some numerical results for the two–dimensional Euler equations modeling hypersonic flow around a blunt body. 1 Model Problem and Its Discretization.- 1.1 Conservation Laws.- 1.2 Finite Volume Methods.- 2 Multiscale Setting.- 2.1 Hierarchy of Meshes.- 2.2 Motivation.- 2.3 Box Wavelet.- 2.3.1 Box Wavelet on a Cartesian Grid Hierarchy.- 2.3.2 Box Wavelet on an Arbitrary Nested Grid Hierarchy.- 2.4 Change of Stable Completion.- 2.5 Box Wavelet with Higher Vanishing Moments.- 2.5.1 Definition and Construction.- 2.5.2 A Univariate Example.- 2.5.3 A Remark on Compression Rates.- 2.6 Multiscale Transformation.- 3 Locally Refined Spaces.- 3.1 Adaptive Grid and Significant Details.- 3.2 Grading.- 3.3 Local Multiscale Transformation.- 3.4 Grading Parameter.- 3.5 Locally Uniform Grids.- 3.6 Algorithms: Encoding, Thresholding, Grading, Decoding.- 3.7 Conservation Property.- 3.8 Application to Curvilinear Grids.- 4 Adaptive Finite Volume Scheme.- 4.1 Construction.- 4.1.1 Strategies for Local Flux Evaluation.- 4.1.2 Strategies for Prediction of Details.- 4.2 A gorithms: Initial data, Prediction, Fluxes and Evolution.- 5 Error Analysis.- 5.1 Perturbation Error.- 5.2 Stability of Approximation.- 5.3 Reliability of Prediction.- 6 Data Structures and Memory Management.- 6.1 Algorithmic Requirements and Design Criteria.- 6.2 Hashing.- 6.3 Data Structures.- 7 Numerical Experiments.- 7.1 Parameter Studies.- 7.1.1 Test Configurations.- 7.1.2 Discretization.- 7.1.3 Computational Complexity and Stability.- 7.1.4 Hash Parameters.- 7.2 Real World Application.- 7.2.1 Configurations.- 7.2.2 Discretization.- 7.2.3 Discussion of Results.- A Plots of Numerical Experiments.- B The Context of Biorthogonal Wavelets.- B.1 General Setting.- B.1.1 Multiscale Basis.- B.1.2 Stable Completion.- B.1.3 Multiscale Transformation.- B.2 Biorthogonal Wavelets of the Box Function.- B.2.1 Haar Wavelets.- B.2.2 Biorthogonal Wavelets on the Real Line.- References.- List of Figures.- List of Tables.- Notation.  In recent years the concept of fully adaptive multiscale finite volume schemes for conservation laws has been developed and analytically investigated. Here the grid adaptation is performed by means of a multiscale analysis. So far, all cells are evolved in time using the same time step size. In the present work this concept is extended incorporating locally varying time stepping. A general strategy is presented for explicit as well as implicit time discretization. The efficiency and the accuracy of the proposed concept is verified numerically. We present a new adaptive numerical scheme for solving parabolic PDEs in Cartesian geometry. Applying a finite volume discretization with explicit time integration, both of second order, we employ a fully adaptive multiresolution scheme to represent the solution on locally refined nested grids. The fluxes are evaluated on the adaptive grid. A dynamical adaption strategy to advance the grid in time and to follow the time evolution of the solution directly exploits the multiresolution representation. Applying this new method to several test problems in one, two and three space dimensions, like convection-diffusion, viscous Burgers and reaction-diffusion equations, we show its second-order accuracy and demonstrate its computational efficiency. Given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one-dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner. To do so we consider the grid-averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest, and introduce an equivalent multiresolution representation. The multiresolution representation of the numerical solution consists of its grid-averages for the coarsest grid and the set of errors in predicting the grid-averages of each level of resolution in this hierarchy from those of the next coarser one. Once the numerical solution is resolved to our satisfaction in a certain locality of some grid, then the prediction errors there are small for this particular grid and all finer ones; this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance. Therefore instead of computing the time-evolution of the numerical solution on the given grid we compute the time-evolution of its compressed multiresolution representation. Algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid. At each step of refinement we add the values of the numerical flux at the center of the coarser cells. The information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well-resolved. When this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid. The computational efficiency of this multiresolution algorithm is proportional to the rate of data compression (for a prescribed level of tolerance) that can be achieved for the numerical solution of the given scheme.  We present a fully adaptive numerical scheme for evolutionary PDEs in Cartesian geometry based on a second-order finite volume discretization. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. For time discretization we use an explicit Runge-Kutta scheme of second-order with a scale-dependent time step. On the finest scale the size of the time step is imposed by the stability condition of the explicit scheme. On larger scales, the time step can be increased without violating the stability requirement of the explicit scheme. The implementation uses a dynamic tree data structure. Numerical validations for test problems in one space dimension demonstrate the efficiency and accuracy of the local time-stepping scheme with respect to both multiresolution scheme with global time stepping and finite volume scheme on a regular grid. Fully adaptive three-dimensional computations for reaction-diffusion equations illustrate the additional speed-up of the local time stepping for a thermo-diffusive flame instability.