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A positivity-preserving high order finite volume compact-WENO scheme for compressible Euler equations. (English) Zbl 1351.76108

Summary: In this paper, a positivity-preserving fifth-order finite volume compact-WENO scheme is proposed for solving compressible Euler equations. As it is known, conservative compact finite volume schemes have high resolution properties while WENO (Weighted Essentially Non-Oscillatory) schemes are essentially non-oscillatory near flow discontinuities. We extend the idea of WENO schemes to some classical finite volume compact schemes [S. Pirozzoli, ibid. 178, No. 1, 81–117 (2002; Zbl 1045.76029)], where lower order compact stencils are combined with WENO nonlinear weights to get a higher order finite volume compact-WENO scheme. The newly developed positivity-preserving limiter [X. Zhang and C.-W. Shu, ibid. 229, No. 23, 8918–8934 (2010; Zbl 1282.76128); Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 467, No. 2134, 2752–2776 (2011; Zbl 1222.65107)] is used to preserve positive density and internal energy for compressible Euler equations of fluid dynamics. The HLLC (Harten, Lax, and van Leer with Contact) approximate Riemann solver [E. F. Toro, Riemann solvers and numerical methods for fluid dynamics. A practical introduction. 3rd ed. Berlin: Springer (2009; Zbl 1227.76006); P. Batten et al., SIAM J. Sci. Comput. 18, No. 6, 1553–1570 (1997; Zbl 0992.65088)] is used to get the numerical flux at the cell interfaces. Numerical tests are presented to demonstrate the high-order accuracy, positivity-preserving, high-resolution and robustness of the proposed scheme.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76Nxx Compressible fluids and gas dynamics

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References:

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