Runge-Kutta Central Discontinuous Galerkin Methods for the Special Relativistic Hydrodynamics. (6th July 2017)

• Record Type: Journal Article
• Authors: Zhao, Jian; Tang, Huazhong
• Is Part Of: Communications in computational physics. Volume 22:Number 3(2017:Sep.)
• Page Start: 643
• Page End: 682
• Publication Date: 2017-07-06
• Subjects: 76M10, -- 76M25, -- 76Y05, -- 76N15; Central discontinuous Galerkin method, -- WENO limiter, -- Runge-Kutta time discretization, -- relativistic hydrodynamics; Mathematical physics -- Data processing -- Periodicals; Physics -- Data processing -- Periodicals; 530.150285
• DOI: 10.4208/cicp.OA-2016-0192
• Languages: English
• ISSNs: 1815-2406

Abstract

Abstract: This paper develops Runge-Kutta P K -based central discontinuous Galerkin (CDG) methods with WENO limiter for the one- and two-dimensional special relativistic hydrodynamical (RHD) equations, K = 1, 2, 3. Different from the non-central DG methods, the Runge-Kutta CDG methods have to find two approximate solutions defined on mutually dual meshes. For each mesh, the CDG approximate solutions on its dual mesh are used to calculate the flux values in the cell and on the cell boundary so that 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 implemented via two steps: the "troubled" cells are first identified by using a modified TVB minmod function, and then the WENO technique is used to locally reconstruct new polynomials of degree (2 K +1) replacing the CDG solutions inside the "troubled" cells by the cell average values of the CDG solutions in the neighboring cells as well as the original cell averages of the "troubled" cells. Because the WENO limiter is only employed 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 calculation of the flux integrals over the cells is also addressed. Several test problems in one and two dimensions are solved by using our Runge-Kutta CDG methods with WENO limiter. The computations demonstrate that our methods are stable, accurate, and robust in solving complex RHD problems.