Volume 4, Issue 2
A Third-Order Accurate Direct Eulerian GRP Scheme for One-Dimensional Relativistic Hydrodynamics

Kailiang Wu, Zhicheng Yang & Huazhong Tang

East Asian J. Appl. Math., 4 (2014), pp. 95-131.

Published online: 2018-02

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  • Abstract

A third-order accurate direct Eulerian generalised Riemann problem (GRP) scheme is derived for the one-dimensional special relativistic hydrodynamical equations. In our GRP scheme, the higher-order WENO initial reconstruction is employed, and the local GRPs in the Eulerian formulation are directly and analytically resolved to thirdorder accuracy via the Riemann invariants and Rankine-Hugoniot jump conditions, to get the approximate states in numerical fluxes. Unlike a previous second-order accurate GRP scheme, for the non-sonic case the limiting values of the second-order time derivatives of the fluid variables at the singular point are also needed for the calculation of the approximate states; while for the sonic case, special attention is paid because the calculation of the second-order time derivatives at the sonic point is difficult. Several numerical examples are given to demonstrate the accuracy and effectiveness of our GRP scheme.

  • Keywords

Godunov-type scheme WENO generalised Riemann problem Riemann invariant Rankine-Hugoniot jump condition relativistic hydrodynamics

  • AMS Subject Headings

65M06 76M12 76Y05

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-4-95, author = {Kailiang Wu, Zhicheng Yang and Huazhong Tang}, title = {A Third-Order Accurate Direct Eulerian GRP Scheme for One-Dimensional Relativistic Hydrodynamics}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {4}, number = {2}, pages = {95--131}, abstract = {

A third-order accurate direct Eulerian generalised Riemann problem (GRP) scheme is derived for the one-dimensional special relativistic hydrodynamical equations. In our GRP scheme, the higher-order WENO initial reconstruction is employed, and the local GRPs in the Eulerian formulation are directly and analytically resolved to thirdorder accuracy via the Riemann invariants and Rankine-Hugoniot jump conditions, to get the approximate states in numerical fluxes. Unlike a previous second-order accurate GRP scheme, for the non-sonic case the limiting values of the second-order time derivatives of the fluid variables at the singular point are also needed for the calculation of the approximate states; while for the sonic case, special attention is paid because the calculation of the second-order time derivatives at the sonic point is difficult. Several numerical examples are given to demonstrate the accuracy and effectiveness of our GRP scheme.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.101013.100314a}, url = {http://global-sci.org/intro/article_detail/eajam/10825.html} }
TY - JOUR T1 - A Third-Order Accurate Direct Eulerian GRP Scheme for One-Dimensional Relativistic Hydrodynamics AU - Kailiang Wu, Zhicheng Yang & Huazhong Tang JO - East Asian Journal on Applied Mathematics VL - 2 SP - 95 EP - 131 PY - 2018 DA - 2018/02 SN - 4 DO - http://doi.org/10.4208/eajam.101013.100314a UR - https://global-sci.org/intro/article_detail/eajam/10825.html KW - Godunov-type scheme KW - WENO KW - generalised Riemann problem KW - Riemann invariant KW - Rankine-Hugoniot jump condition KW - relativistic hydrodynamics AB -

A third-order accurate direct Eulerian generalised Riemann problem (GRP) scheme is derived for the one-dimensional special relativistic hydrodynamical equations. In our GRP scheme, the higher-order WENO initial reconstruction is employed, and the local GRPs in the Eulerian formulation are directly and analytically resolved to thirdorder accuracy via the Riemann invariants and Rankine-Hugoniot jump conditions, to get the approximate states in numerical fluxes. Unlike a previous second-order accurate GRP scheme, for the non-sonic case the limiting values of the second-order time derivatives of the fluid variables at the singular point are also needed for the calculation of the approximate states; while for the sonic case, special attention is paid because the calculation of the second-order time derivatives at the sonic point is difficult. Several numerical examples are given to demonstrate the accuracy and effectiveness of our GRP scheme.

Kailiang Wu, Zhicheng Yang & Huazhong Tang. (1970). A Third-Order Accurate Direct Eulerian GRP Scheme for One-Dimensional Relativistic Hydrodynamics. East Asian Journal on Applied Mathematics. 4 (2). 95-131. doi:10.4208/eajam.101013.100314a
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