Adaptive mesh refinement techniques for high order shock capturing schemes for hyperbolic systems of conservation laws

Resumen

The numerical simulation of physical phenomena represented by nonlinear hyperbolic systems of conservation laws presents specific difficulties, that are not present in other kind of systems of partial differential equations. These are mainly due to the presence of discontinuities in the solution. State of the art methods for the solution of such equations involve high resolution shock capturing schemes, which are able to produce sharp profiles at the discontinuities and high accuracy in smooth regions, together with some kind of grid adaptation, which reduces the computational cost by using finer grids near the discontinuities and coarser grids in smooth regions. The combination of both techniques presents intrinsic numerical and computational difficulties. In this work we present a method obtained by the combination of a high order shock capturing scheme, built from Shu-Osher's conservative formulation, a fifth order weighted essentially non-oscillatory (WENO) interpolatory technique, Donat-Marquina's flux-splitting method and a third order Runge-Kutta method, with the adaptive mesh refinement (AMR) technique of Berger and collaborators. We show how all these techniques can be merged together to build up a highly efficient numerical method, and we show how to parallelize such an algorithm. We also present a description of the AMR algorithm that is much more general that the actual descriptions found in the scientific literature and tries to approach to the fundations of the running algorithms that are described and implemented in practice. We make extensive testing of our implementation to determite its extent of applicability and relative benefits with respect to the non-adaptive algorithm.