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

In Xu [11], 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 RK method. Compared with the original work [11], 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. Local truncation error analysis is performed to justify the maintenance of high order spatial/temporal accuracy when the MPP flux limiters are applied to high order finite difference schemes solving linear problems. We further apply the limiting procedure to the simulation of the incompressible flow: we first design a first order MPP scheme for the incompressible flow, then apply an MPP flux limiter to the flux of high order scheme toward that of a first order MPP scheme. The MPP property is guaranteed without affecting the designed order of spatial and temporal accuracy for the incompressible flow computation. The efficiency and effectiveness of the proposed scheme is demonstrated via several test examples.