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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 third-order 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.

  • AMS Subject Headings

65M06, 76M12, 76Y05

  • Copyright

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 third-order 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, WENO, generalised Riemann problem, Riemann invariant, Rankine-Hugoniot jump condition, 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 third-order 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 and Huazhong Tang. (2018). 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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