Runge–Kutta discontinuous Galerkin methods with WENO limiter for the special relativistic hydrodynamics

Abstract

This paper develops the PK-based Runge–Kutta discontinuous Galerkin (RKDG) methods with WENO limiter for the one- and two-dimensional special relativistic hydrodynamics, K=1,2,3, which is an extension of the work [J.X. Qiu, C.-W. Shu, Runge–Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci. Comput. 26 (2005) 907–929]. The WENO limiter for the RKDG method is adaptively implemented via two following steps: the “troubled” cells are first identified by using a TVB modified minmod function, and then a new polynomial solution inside the “troubled” cells is locally reconstructed to replace the RKDG solution by using the WENO technique based on the cell average values of the RKDG solution in the neighboring cells as well as the original cell averages of the “troubled” cells. Several test problems in one and two dimensions are computed using the developed RKDG methods with WENO limiter. The computations demonstrate that our methods are stable, accurate, and robust in maintaining accuracy for simulating flows in the special relativistic hydrodynamics.