A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows.

*(English)*Zbl 1349.76553Summary: In [Z. Xu, Math. Comput. 83, No. 289, 2213–2238 (2014; Zbl 1300.65063)], a class of parametrized flux limiters is developed for high order finite difference/volume essentially non-oscillatory (ENO) and Weighted ENO (WENO) schemes coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration for solving scalar hyperbolic conservation laws to achieve strict maximum principle preserving (MPP). In this paper, we continue along this line of research, but propose to apply the parametrized MPP flux limiter only to the final stage of any explicit RK method. Compared with the original work [loc. cit.], the proposed new approach has several advantages: First, the MPP property is preserved with high order accuracy without as much time step restriction; Second, the implementation of the parametrized flux limiters is significantly simplified. Analysis is performed to justify the maintenance of third order spatial/temporal accuracy when the MPP flux limiters are applied to third order finite difference schemes solving general nonlinear problems. We further apply the limiting procedure to the simulation of the incompressible flow: the numerical fluxes of a high order scheme are limited toward that of a first order MPP scheme which was discussed in [D. Levy, IMA J. Numer. Anal. 25, No. 3, 507–522 (2005; Zbl 1074.76035)]. The MPP property is guaranteed, while designed high order of spatial and temporal accuracy for the incompressible flow computation is not affected via extensive numerical experiments. The efficiency and effectiveness of the proposed scheme are demonstrated via several test examples.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

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

35L65 | Hyperbolic conservation laws |

##### Keywords:

hyperbolic conservation laws; high order scheme; WENO; parametrized flux limiter; maximum principle preserving; Runge-Kutta method; incompressible flow
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