Maximum principle and positivity-preserving high order spectral volume schemes with parametrized flux limiters for solving hyperbolic conservation laws

Abstract

In this paper, we use maximum principle preserving (MPP) and positivity-preserving (PP) parametrized flux limiters to achieve strict maximum principle and positivity-preserving property for the high order spectral volume (SV) scheme for solving hyperbolic conservation laws. This research is based on a generalization of the MPP and PP parametrized flux limiters in Xu (2014) and Christlieb et al. (2015) with Runge–Kutta (RK) time discretizations. For constructing MPP (PP) RK-SV schemes for hyperbolic conservation laws, we first focus on the RK-SV schemes to discuss how to apply MPP or PP parametrized flux limiters. Then we design and analyze high order MPP RK-SV schemes for scalar conservation laws, and high order PP RK-SV schemes for compressible Euler systems. The efficiency and effectiveness of the proposed schemes are demonstrated via a set of numerical experiments. Both the analysis and numerical experiments indicate that the proposed scheme without any additional time step restriction, not only preserves the maximum principle of the numerical approximation, but also maintains the designed high-order accuracy of the SV scheme for linear advection problems.