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Runge-Kutta Central Discontinuous Galerkin Methods for the Special Relativistic Hydrodynamics

Part of:Basic methods in fluid mechanicsCompressible fluids and gas dynamics, generalQuantum hydrodynamics and relativistic hydrodynamics

Published online by Cambridge University Press:  06 July 2017

Jian Zhao and
Huazhong Tang
Jian Zhao*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
Huazhong Tang*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China; School of Mathematics and Computational Science, Xiangtan University, Hunan Province, Xiangtan 411105, P.R. China
*
*Corresponding author. Email addresses:everease@163.com (J. Zhao); hztang@math.pku.edu.cn (H. Z. Tang)
*Corresponding author. Email addresses:everease@163.com (J. Zhao); hztang@math.pku.edu.cn (H. Z. Tang)

Abstract

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This paper develops Runge-Kutta PK -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 (2K+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.

Keywords

Central discontinuous Galerkin methodWENO limiterRunge-Kutta time discretizationrelativistic hydrodynamics

MSC classification

Secondary:76M10: Finite element methods76M25: Other numerical methods76Y05: Quantum hydrodynamics and relativistic hydrodynamics76N15: Gas dynamics, general

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017, pp. 643 - 682
Copyright
Copyright © Global-Science Press 2017 

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