##
**Parametrized maximum principle preserving flux limiters for high order schemes solving multi-dimensional scalar hyperbolic conservation laws.**
*(English)*
Zbl 1286.65102

Summary: We extend the strict maximum principle preserving flux limiting technique developed for one-dimensional scalar hyperbolic conservation laws to the two-dimensional scalar problems. The parametrized flux limiters and their determination from decoupling maximum principle preserving constraint are presented in a compact way for two-dimensional problems. With the compact fashion that the decoupling is carried out, the technique can be easily applied to high-order finite difference and finite volume schemes for multi-dimensional scalar hyperbolic problems. For the two-dimensional problem, the successively defined flux limiters are developed for the multi-stage total-variation-diminishing Runge-Kutta time-discretization to improve the efficiency of computation. The high-order schemes with successive flux limiters provide high-order approximation and maintain the strict maximum principle with the mild Courant-Friedrichs-Lewy constraint. Two-dimensional numerical evidence is given to demonstrate the capability of the proposed approach.

### MSC:

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

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

35B50 | Maximum principles in context of PDEs |

### Keywords:

hyperbolic conservation laws; maximum principle preserving; parametrized flux limiters; high-order scheme; finite difference method; finite volume method; numerical examples; Runge-Kutta time-discretization; Courant-Friedrichs-Lewy constraint
PDF
BibTeX
XML
Cite

\textit{C. Liang} and \textit{Z. Xu}, J. Sci. Comput. 58, No. 1, 41--60 (2014; Zbl 1286.65102)

Full Text:
DOI

### References:

[1] | Harten, A., Engquist, B., Osher, S., Chakravarty, S. R.: Uniformly high order accurate essentially non-oscillatory schemes. J. Comput. Phys. 231-303 (1987) · Zbl 0652.65067 |

[2] | Hu, X.Y., Adams, N.A., Shu, C.-W.: Positivity-preserving flux limiters for high-order conservative schemes. J. Comput. Phys., to appear · Zbl 0859.65091 |

[3] | Liu, X-D; Osher, S; Chan, T, Weighted essentially nonoscillatory schemes, J. Comput. Phys., 115, 200-212, (1994) · Zbl 0811.65076 |

[4] | Liu, X-D; Osher, S, Non-oscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes, SIAM J. Numer. Anal., 33, 760-779, (1996) · Zbl 0859.65091 |

[5] | Liu, Y; Shu, C-W; Zhang, M, On the positivity of linear weights in WENO approximations, Acta Math. Appl. Sinica Engl. Ser., 25, 503-538, (2009) · Zbl 1177.65126 |

[6] | Osher, S, Riemann solvers, the entropy condition and difference approximations, SIAM J. Numer. Anal., 21, 217-235, (1984) · Zbl 0592.65069 |

[7] | Osher, S; Chakravarthy, S, High resolution schemes and the entropy condition, SIAM J. Numer. Anal., 21, 955-984, (1984) · Zbl 0556.65074 |

[8] | Qiu, J; Shu, C-W, Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci. Comput., 26, 907-929, (2005) · Zbl 1077.65109 |

[9] | Sanders, R, A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation law, Math. Comput., 51, 535-558, (1988) · Zbl 0699.65069 |

[10] | Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, ICASE, Report No. 97-65 · Zbl 1177.65126 |

[11] | Shu, C-W; Osher, S, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471, (1988) · Zbl 0653.65072 |

[12] | Shu, C-W, High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Rev., 51, 82-126, (2009) · Zbl 1160.65330 |

[13] | Xu, Z.: Parametrized maximum principle preserving flux limiters for high order scheme solving hyperbolic conservation laws: one-dimensional scalar problem. Accepted by mathematics of computation after revision · Zbl 1226.65083 |

[14] | Xu, Z., Shu, C.-W.: Anti-diffusive flux corrections for high order finite difference WENO schemes. J. Comput. Phys. 205, 458-485 (2005) · Zbl 1087.76080 |

[15] | Zhang, X; Shu, C-W, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229, 3091-3120, (2010) · Zbl 1187.65096 |

[16] | 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) · Zbl 1282.76128 |

[17] | Zhang, X; Shu, C-W, A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws, SIAM J. Numer. Anal., 48, 772-795, (2010) · Zbl 1226.65083 |

[18] | Zhang, X; Shu, C-W, Positivity-preserving high order finite difference WENO schemes for compressible Euler equations, J. Comput. Phys., 231, 2245-2258, (2012) · Zbl 1426.76493 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.