Simplified second-order Godunov-type methods.

*(English)*Zbl 0645.65050A second order extension of Godunov’s classical numerical method for the solution of systems of conservation laws, namely the Riemann problem, is investigated. The exact Riemann solution is replaced by the simple Riemann solution of A. Harten, P. D. Lax and B. van Leer [SIAM Rev. 25, 35-61 (1983; Zbl 0565.65051)] which contains only one intermediate state. This construction assumes that a priori bounds on the smallest and largest signal velocities in the exact Riemann solution are given.

A number of algorithms for obtaining these bounds are suggested. Heuristic arguments are presented to support the given choice of bounds. A first order scheme is derived which uses these approximate Riemann solutions, and their relationship to known finite difference schemes is shown.

The approach of B. van Leer, G. D. van Albada and W. W. Roberto jun. [Astron. Astrophys. 108, 76-84 (1982; Zbl 0492.76117)] to construct second order schemes based on these approximate Riemann solutions is used. On particular interest is a central difference scheme requiring no upwind switches. This scheme is only slightly more complex than standard predictor-corrector finite difference schemes. Preliminary numerical results are presented which show that these schemes are nonoscillatory, have good shock resolution and produce results which are competitive with those produced by more complex second order Godunov-type schemes.

A number of algorithms for obtaining these bounds are suggested. Heuristic arguments are presented to support the given choice of bounds. A first order scheme is derived which uses these approximate Riemann solutions, and their relationship to known finite difference schemes is shown.

The approach of B. van Leer, G. D. van Albada and W. W. Roberto jun. [Astron. Astrophys. 108, 76-84 (1982; Zbl 0492.76117)] to construct second order schemes based on these approximate Riemann solutions is used. On particular interest is a central difference scheme requiring no upwind switches. This scheme is only slightly more complex than standard predictor-corrector finite difference schemes. Preliminary numerical results are presented which show that these schemes are nonoscillatory, have good shock resolution and produce results which are competitive with those produced by more complex second order Godunov-type schemes.

Reviewer: J.Vaníček

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |