Exactly divergence-free ultra-weak discontinuous Galerkin method for Brinkman–Forchheimer equations

Abstract

This paper presents an exactly divergence-free ultra-weak discontinuous Galerkin (UWDG) method, utilizing $H\left(\text{div}\right)$-conforming spaces for the Brinkman–Forchheimer equations. We also develop an unconditionally stable first-order time discretization scheme based on the linearized convex-splitting approach by interpreting the Brinkman–Forchheimer equations as an $L^2$ gradient flow in the exactly divergence-free space. It is proven that the fully discrete scheme remains unconditionally stable in both the $L^2$ norm and the energy functional, with corresponding error estimates provided. Furthermore, we employ the IMEX-BDF method with the deferred correction (DC) approach to develop high-order linear semi-implicit schemes. Numerical experiments with various parameter configurations validate the proposed schemes, demonstrating their accuracy, stability, and computational efficiency. These results underscore the robustness and effectiveness of the method in solving the Brinkman–Forchheimer equation.