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A third-order accurate direct Eulerian GRP scheme for the Euler equations in gas dynamics. (English) Zbl 1349.76551
Summary: 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 [M. Ben-Artzi et al., ibid. 218, No. 1, 19–43 (2006; Zbl 1158.76375)]. 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.

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
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