A Second-Order Maximum-Principle-Preserving Crouzeix-Raviart Finite Element Method for Time-dependent Transport Equation

TL;DR

This paper introduces a second-order, maximum-principle-preserving Crouzeix-Raviart finite element method for 2D time-dependent transport equations, emphasizing efficiency and robustness.

Contribution

It develops a low-order scheme with diagonal mass matrix structure and enhances it to second-order accuracy using flux-corrected transport viscosities.

Findings

The scheme preserves the maximum principle on complex domains.

Numerical tests confirm high accuracy for smooth and discontinuous solutions.

The method is conservative under divergence-free velocity fields.

Abstract

In this paper, we construct an explicit, second-order, and maximum-principle-preserving Crouzeix-Raviart (CR) finite element method for two-dimensional time-dependent transport equation. The key observation is that the mass matrix of the CR element is with diagonal structure, which allows us to avoid the need to solve a large linear system for each time step and help to construct a low-order scheme that preserves the maximum principle in a simple way. We first introduce low-order schemes based on minimum and bilinear viscosities, and then recover second-order accuracy by means of greedy and flux-corrected transport viscosities. For inflow boundary conditions, we further design a modified FCT limiter. In addition, we propose a simple reconstruction based on Wachspress coordinates to obtain a continuous piecewise linear approximation on the $\frac{h}2$-mesh that satisfies the maximum principle on the whole domain. Under divergence-free velocity fields, the proposed schemes are conservative. Numerical experiments on both smooth and discontinuous test cases, with both solenoidal and non-solenoidal velocity fields, confirm the accuracy and robustness of the proposed schemes.