Central random choice methods for hyperbolic conservation laws

Abstract

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