Adaptive mesh refinement techniques for high‐order shock capturing schemes for multi‐dimensional hydrodynamic simulations

Abstract

AbstractThe numerical simulation of physical phenomena represented by non‐linear hyperbolic systems of conservation laws presents specific difficulties 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 adaption, 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 programming 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 (J. Comput. Phys. 1988; 77:439–471; 1989; 83:32–78), a fifth‐order weighted essentially non‐oscillatory (WENO) interpolatory technique (J. Comput. Phys. 1996; 126:202–228) and Donat–Marquina's flux‐splitting method (J. Comput. Phys. 1996; 125:42–58), with the adaptive mesh refinement (AMR) technique of Berger and collaborators (Adaptive mesh refinement for hyperbolic partial differential equations. Ph.D. Thesis, Computer Science Department, Stanford University, 1982; J. Comput. Phys. 1989; 82:64–84; 1984; 53:484–512). Copyright © 2006 John Wiley & Sons, Ltd.