Numerical Simulation of Hyperbolic Conservation Laws Using High Resolution Schemes with the Indulgence of Fuzzy Logic
The aim of this paper is to solve numerically a class of problems on conservation laws, modelled by hyperbolic partial differential equations. In this paper, primary focus is over the idea of fuzzy logic-based operators for the simulation of problems related to hyperbolic conservation laws. Present approach considers a novel computational procedure which relies on using some operators from fuzzy logic to reconstruct several higher-order numerical methods known as the flux-limited methods. Further optimization of the flux limiters is discussed. The approach ensures better convergence of the approximation and preserves the basic properties of the solution of the problem under consideration. The new limiters are further applied to several real-life problems like the advection problem to demonstrate that the optimized schemes ensure better results. Simulation results are included wherever required.
We describe preliminary results from an effort to quantify the uncertainties in parton distribution functions and the resulting uncertainties in predicted physical quantities. The production cross section of the $W$ boson is given as a first example. Constraints due to the full data sets of the CTEQ global analysis are used in this study. Two complementary approaches, based on the Hessian and the Lagrange multiplier method respectively, are outlined. We discuss issues on obtaining meaningful uncertainty estimates that include the effect of correlated experimental systematic uncertainties and illustrate them with detailed calculations using one set of precision DIS data.