Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem

In this paper, we present a class of parametrized limiters used to achieve strict maximum principle for high order numerical schemes applied to hyperbolic conservation laws computation. By decoupling a sequence of parameters embedded in a group of explicit inequalities, the numerical fluxes are locally redefined in consistent and conservative formulation. We will show that the global maximum principle can be preserved while the high order accuracy of the underlying scheme is maintained. The parametrized limiters are less restrictive on the CFL number when applied to high order finite volume scheme. The less restrictive limiters allow for the development of the high order finite difference scheme which preserves the maximum principle. Within the proposed parametrized limiters framework, a successive sequence of limiters are designed to allow for significantly large CFL number by relaxing the limits on the intermediate values of the multistage Runge-Kutta method. Numerical results and preliminary analysis for linear and nonlinear scalar problems are presented to support the claim. The parametrized limiters are applied to the numerical fluxes directly. There is no increased complexity to apply the parametrized limiters to different kinds of monotone numerical fluxes.