Abstract
In this paper, we propose a new positivity-preserving (new PP) technique for fifth-order finite volume unequal-sized WENO schemes when solving some extreme problems of compressible Euler equations on structured meshes. After spatial reconstruction in each time marching, a detective process is used to examine the positivity of density and pressure at some checking points in each target cell. If the negativity happens at one checking point, a new compression limiter is carried out to ensure that the modified polynomials can achieve the positivity of density and pressure in the whole target cell instead of only at some discrete points. This treatment is easily implemented, owing to a quick and simple way to overestimate the minimum and maximum values of the polynomials of density and pressure in the target cell. Some numerical experiments for classical extreme problems show that the proposed method has an advantage in computational efficiency and robustness in comparison to the classical positivity-preserving techniques (Zhang and Shu in J Comput Phys 229:8918–8934, 2010), since the CFL number of 0.6 is allowed for all tests in this paper.
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References
Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191–3211 (2008)
Cai, X., Zhang, X., Qiu, J.: Positivity-preserving high order finite volume HWENO schemes for compressible Euler equations. J. Sci. Comput. 68, 464–483 (2016)
Capdeville, G.: A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes. J. Comput. Phys. 227, 2977–3014 (2008)
Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011)
Christlieb, A.J., Liu, Y., Tang, Q., Xu, Z.: High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes. J. Comput. Phys. 281, 334–351 (2015)
Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)
Fan, C., Zhang, X., Qiu, J.: Positivity-preserving high order finite volume hybrid Hermite WENO schemes for compressible Navier–Stokes equations. J. Comput. Phys. 445, 110596 (2021)
Fan, C., Zhang, X., Qiu, J.: Positivity-preserving high order finite difference WENO schemes for compressible Navier–Stokes equations. J. Comput. Phys. 467, 111446 (2022)
Gardner, C., Dwyer, S.: Numerical simulation of the XZ Tauri supersonic astrophysical jet. Acta Math. Sci. 29, 1677–1683 (2009)
Guo, Y., Xiong, T., Shi, Y.: A positivity-preserving high order finite volume compact-WENO scheme for compressible Euler equations. J. Comput. Phys. 274, 505–523 (2014)
Ha, Y., Gardner, C.: Positive scheme numerical simulation of high Mach number astrophysical jets. J. Sci. Comput. 34, 247–259 (2008)
Ha, Y., Gardner, C., Gelb, A., Shu, C.-W.: Numerical simulation of high Mach number astrophysical jets with radiative cooling. J. Sci. Comput. 24, 597–612 (2005)
Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005)
Hu, X.Y., Adams, N., Shu, C.-W.: Positivity-preserving method for high-order conservative schemes solving compressible Euler equations. J. Comput. Phys. 242, 169–180 (2013)
Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Korobeinikov, V.P.: Problems of Point-Blast Theory. American Institute of Physics (1991)
Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws, M2AN. Math. Model. Numer. Anal. 33, 547–571 (1999)
Levy, D., Puppo, G., Russo, G.: Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22(2), 656–672 (2000)
Liang, C., Xu, Z.: Parametrized maximum principle preserving flux limiters for high order schemes solving multi-dimensional scalar hyperbolic conservation laws. J. Sci. Comput. 83, 2213–2238 (2013)
Linde, T., Roe, P.L.: Robust Euler codes. In: 13th Computational Fluid Dynamics Conference, AIAA Paper-97-2098
Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)
Perthame, B., Shu, C.-W.: On positivity preserving finite volume schemes for Euler equations. Numer. Math. 73, 119–130 (1996)
Sedov, L.I.: Similarity and Dimensional Methods in Mechanics. Academic Press, New York (1959)
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
Wang, C., Zhang, X., Shu, C.-W., Ning, J.: Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations. J. Comput. Phys. 231, 653–665 (2012)
Xiong, T., Qiu, J.-M., Xu, Z.: Parametrized positivity preserving flux limiters for the high order finite difference WENO scheme solving compressible Euler equations. J. Sci. Comput. 67, 1066–1088 (2015)
Xu, Z.: Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem. Math. Comput. 83, 2213–2238 (2014)
Zhang, X.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations. J. Comput. Phys. 328, 301–343 (2017)
Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091–3120 (2010)
Zhang, X., Shu, C.-W.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)
Zhang, X., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments. Proc. R. Soc. A Math. Phys. Eng. Sci. 467, 2752–2776 (2011)
Zhang, X., Shu, C.-W.: Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231, 2245–2258 (2012)
Zhu, J., Qiu, J.: A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016)
Zhu, J., Qiu, J.: A new type of finite volume WENO schemes for hyperbolic conservation laws. J. Sci. Comput. 73, 1338–1359 (2017)
Zhu, J., Qiu, J.: A new third order finite volume weighted essentially non-oscillatory scheme on tetrahedral meshes. J. Comput. Phys. 349, 220–232 (2017)
Zhu, J., Qiu, J.: New finite volume weighted essentially nonoscillatory schemes on triangular meshes. SIAM J. Sci. Comput. 40, A903–A928 (2018)
Funding
The work of Y. Tan and J. Zhu is partially supported by NSFC grant 11872210. The work of Q. Zhang is partially supported by NSFC grant 12071214.
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Tan, Y., Zhang, Q. & Zhu, J. A New Positivity-Preserving Technique for High-Order Schemes to Solve Extreme Problems of Euler Equations on Structured Meshes. J Sci Comput 99, 27 (2024). https://doi.org/10.1007/s10915-024-02493-6
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DOI: https://doi.org/10.1007/s10915-024-02493-6
Keywords
- Positivity-preserving technique
- Finite volume
- Unequal-sized WENO scheme
- Extreme problem of Euler equations