Central random choice methods for hyperbolic conservation laws

Abstract In this article, we briefly review the random choice method (RCM) and ADER methods for solving one two-dimensional hyperbolic conservation laws. The main advantage of RCM is that it computes discontinuities with infinite resolution. method, original problem reduced to a set local Riemann problems (RPs). exact solutions these RPs are used form solution problem. However, has following disadvantages: (1) should solve RP exactly, however, usually complex unavailable many problems. (2) accuracy smooth region flow poor. explicit, one-step schemes very high order in time space. They depend on generalized (GRP) exactly. Zahran (J Math Anal Appl 346:120–140, 2008), an improved version (central ADER) was introduced where were solved numerically central fluxes, instead upwind fluxes. more accurate, faster, simple implement, solver free, need less computer memory. To fade drawbacks above keep their advantages, propose, paper, RCM. We merge technique resulting scheme called Central (CRCM). improvements listed as follows: use WENO reconstruction initial data constant RCM, by using finite difference sampling update solution, Here staggered non-staggered enhance new methods, third-order TVD flux (Zahran Bull Belg Soc Simon Stevin 14:259–275, 2007), first-order flux. Compared ADER, combine advantages memory, free. Moreover, capture resolution improve parts. have CPU than due evaluation CRCM. An extension general systems nonlinear laws two dimensions presented. present several numerical examples results confirm presented superior schemes.