Runge-Kutta Central Discontinuous Galerkin Methods for the Special Relativistic Hydrodynamics (Communications in Computational Physics, 2017)
This paper develops Runge-Kutta PK-based central discontinuous Galerkin(CDG) methods with WENO limiter for the one- and two-dimensional special relativistichydrodynamical (RHD) equations, K = 1,2,3. Different from the non-centralDG methods, the Runge-Kutta CDG methods have to find two approximate solutionsdefined on mutually dual meshes. For each mesh, the CDG approximate solutions onits dual mesh are used to calculate the flux values in the cell and on the cell boundary sothat the approximate solutions on mutually dual meshes are coupled with each other,and the use of numerical flux will be avoided. The WENO limiter is adaptively implementedvia two steps: the "troubled" cells are first identified by using a modifiedTVB minmod function, and then the WENO technique is used to locally reconstructnew polynomials of degree (2K+1) replacing the CDG solutions inside the "troubled" cells by the cell average values of the CDG solutions in the neighboring cells as wellas the original cell averages of the "troubled" cells. Because the WENO limiter is onlyemployed for finite "troubled" cells, the computational cost can be as little as possible.The accuracy of the CDG without the numerical dissipation is analyzed and calculationof the flux integrals over the cells is also addressed. Several test problems in oneand two dimensions are solved by using our Runge-Kutta CDG methods with WENOlimiter. The computations demonstrate that our methods are stable, accurate, and robustin solving complex RHD problems.