The generalized Riemann problem for reactive flows
Journal of Computational Physics
SIAM Journal on Mathematical Analysis
Numerical solution of the Riemann problem for two-dimensional gas dynamics
SIAM Journal on Scientific Computing
A two-dimensional conservation laws scheme for compressible flows with moving boundaries
Journal of Computational Physics
Solution of Two-Dimensional Riemann Problems of Gas Dynamics by Positive Schemes
SIAM Journal on Scientific Computing
Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws
SIAM Journal on Numerical Analysis
Resolution of high order WENO schemes for complicated flow structures
Journal of Computational Physics
Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations
SIAM Journal on Scientific Computing
A direct Eulerian GRP scheme for compressible fluid flows
Journal of Computational Physics
Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem
Numerische Mathematik
An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws
Journal of Computational Physics
High-order time-splitting Hermite and Fourier spectral methods
Journal of Computational Physics
An adaptive GRP scheme for compressible fluid flows
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes, II
Journal of Computational Physics
A direct Eulerian GRP scheme for relativistic hydrodynamics: One-dimensional case
Journal of Computational Physics
A direct Eulerian GRP scheme for relativistic hydrodynamics: Two-dimensional case
Journal of Computational Physics
Journal of Computational Physics
The generalized Riemann problems for compressible fluid flows: Towards high order
Journal of Computational Physics
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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.