A Second-Order Maximum-Principle-Preserving Crouzeix-Raviart Finite Element Method for Time-dependent Transport Equation
This work provides a computationally efficient and robust numerical method for solving transport equations, which is important for applications in fluid dynamics and other fields requiring maximum principle preservation.
The authors developed an explicit, second-order, maximum-principle-preserving Crouzeix-Raviart finite element method for the 2D time-dependent transport equation, achieving second-order accuracy while preserving the maximum principle, as confirmed by numerical experiments on smooth and discontinuous test cases.
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.