A Maximum-Principle-Satisfying High-order Finite Volume Compact WENO Scheme for Scalar Conservation Laws

TL;DR

This paper introduces a high-order finite volume compact WENO scheme that preserves the maximum principle for scalar conservation laws, combining nonlinear WENO weights with compact stencils to achieve high accuracy and stability.

Contribution

It develops a novel maximum-principle-satisfying finite volume compact WENO scheme that maintains high order accuracy without oscillations for scalar hyperbolic conservation laws.

Findings

The scheme effectively preserves the maximum principle in numerical solutions.

It achieves high-order accuracy and non-oscillatory behavior.

Numerical tests demonstrate excellent performance in 1D and 2D problems.

Abstract

In this paper, a maximum-principle-satisfying finite volume compact scheme is proposed for solving scalar hyperbolic conservation laws. The scheme combines WENO schemes (Weighted Essentially Non-Oscillatory) with a class of compact schemes under a finite volume framework, in which the nonlinear WENO weights are coupled with lower order compact stencils. The maximum-principle-satisfying polynomial rescaling limiter in [Zhang and Shu, JCP, 2010] is adopted to construct the present schemes at each stage of an explicit Runge-Kutta method, without destroying high order accuracy and conservativity. Numerical examples for one and two dimensional problems including incompressible flows are presented to assess the good performance, maximum principle preserving, essentially non-oscillatory and highly accurate resolution of the proposed method.