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An accurate and shock-stable genuinely multidimensional scheme for solving the Euler equations. (English) Zbl 1459.76087
Summary: In numerical simulations of multidimensional high Mach flows, the conventional low-diffusion upwind schemes built on the dimensional splitting method will encounter the shock instability which is mainly manifested as the infamous carbuncle phenomena. A linear stability analysis reveals that the shock instability is triggered by the unattenuated perturbations in the transverse direction of the flow field. In the present work, an accurate and shock-stable genuinely multidimensional numerical scheme based on the Toro-Vázquez splitting method is presented. Different from the conventional one-dimensional upwind schemes that just consider the waves propagating along the direction normal to an interface, the proposed multidimensional scheme whose multidimensional properties are achieved by calculating multidimensional numerical fluxes at each corner of an interface also takes into account the waves traveling along the transverse direction. For obtaining these multidimensional numerical fluxes at the corner, a simple multidimensional upwind method is used to solve the weakly hyperbolic convection subsystem and the pressure subsystem whose flux Jacobian has a complete set of linearly independent eigenvectors is calculated by a multidimensional HLLEM scheme. Based on Balsara’s framework for constructing multidimensional schemes, the total numerical flux across an interface is obtained by using the Simpson’s rule to assemble the conventional one-dimensional numerical flux with the multidimensional numerical fluxes at the corner. A series of benchmark test problems validate the accuracy, robustness and efficiency of the proposed multidimensional scheme.
MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76N15 Gas dynamics, general
76L05 Shock waves and blast waves in fluid mechanics
Software:
HLLE; RIEMANN
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