A third-order accurate direct Eulerian GRP scheme for the Euler equations in gas dynamics

作者：Wu Kailiang; Yang Zhicheng; Tang Huazhong^*

来源：Journal of Computational Physics, 2014, 264: 177-208.

DOI：10.1016/j.jcp.2014.01.041

• Godunov-type scheme
• Generalized Riemann problem
• Riemann problem
• Riemann invariant
• Rankine-Hugoniot jump condition

The paper proposes and implements a third-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for one- and two-dimensional (1D & 2D) Euler equations in gas dynamics. It is an extension of the second-order accurate GRP scheme proposed in Ben-Artzi et al. (2006) [5]. The approximate states in numerical fluxes of the third-order accurate GRP scheme are derived by using the higher-order WENO reconstruction of the initial data, the limiting values of the time derivatives of the solutions at the singularity point, and the Jacobian matrix. Besides the limiting values of the first-order time derivatives of fluid variables, the second-order time derivatives are also needed in developing the present GRP scheme and obtained by directly and analytically resolving the local GRP in the Eulerian formulation via two main ingredients, i.e. the Riemann invariants and Rankine-Hugoniot jump conditions. Unfortunately, for the sonic case that the transonic rarefaction wave appears in the GRP, the Jacobian matrix is singular on the sonic line. To this end, those approximate states are given in a different way that is based on the analytical resolution of the transonic rarefaction wave and the local quadratic polynomial interpolation. The 2D GRP scheme is implemented by using the third-order accurate time-splitting method. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed GRP scheme, in comparison to the second-order accurate GRP scheme.

• 出版日期2014-5-1
• 单位北京大学