Exactly Divergence-Free Ultra-Weak Discontinuous Galerkin Method for Incompressible Flow

Abstract

In this paper, we present an ultra-weak discontinuous Galerkin (UWDG) method that employs H(div)-conforming spaces for solving incompressible flows, ensuring exact divergence-free solutions. Leveraging the exact divergence-free property and the continuity of the normal velocity at cell interfaces, the convection term is formulated in conservative form and discretized as the DG method for conservation laws. To achieve a high-order time discretization, we first implement a predictor-corrector procedure utilizing either one-step or multi-step methods. Then, we integrate the spectral deferred correction scheme to develop a linear semi-implicit high-order fully discrete scheme. Numerical experiments demonstrate the efficiency and accuracy of this high-order method.

Keywords

Incompressible Navier-Stokes equations / Ultra-weak discontinuous Galerkin (UWDG) method / $H(\text {div})$-conforming')">$H(\text {div})$-conforming / Exactly divergence-free / Spectral deferred correction scheme / 65M60 / 65M12 / 65M20 / 76D05

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Funding

National Natural Science Foundation of China(12271498)

Shanghai Institute for Mathematics and Interdisciplinary Sciences(SIMIS-ID-2024-(XS))

RIGHTS & PERMISSIONS

Shanghai University

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