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

doi:10.1007/s42967-025-00508-z

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (4): 1450-1477.doi: 10.1007/s42967-025-00508-z

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

Zhengyang Xue, Yinhua Xia

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

收稿日期:2024-07-06 修回日期:2025-05-27 接受日期:2025-05-28 出版日期:2026-08-20 发布日期:2026-09-17
通讯作者: Yinhua Xia, Email: yhxia@ustc.edu.cn E-mail:yhxia@ustc.edu.cn
基金资助:
The research of Yinhua Xia was partially supported by Shanghai Institute for Mathematics and Interdisciplinary Sciences (Grant No. SIMIS-ID-2024-(XS)) and the NSFC (Grant No. 12271498).

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

Zhengyang Xue, Yinhua Xia

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

Received:2024-07-06 Revised:2025-05-27 Accepted:2025-05-28 Online:2026-08-20 Published:2026-09-17
Supported by:
The research of Yinhua Xia was partially supported by Shanghai Institute for Mathematics and Interdisciplinary Sciences (Grant No. SIMIS-ID-2024-(XS)) and the NSFC (Grant No. 12271498).

摘要/Abstract

摘要： 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.

关键词: Incompressible Navier-Stokes equations, Ultra-weak discontinuous Galerkin (UWDG) method·H(div)-conforming, Exactly divergence-free, Spectral deferred correction scheme

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.

Key words: Incompressible Navier-Stokes equations, Ultra-weak discontinuous Galerkin (UWDG) method·H(div)-conforming, Exactly divergence-free, Spectral deferred correction scheme

Zhengyang Xue, Yinhua Xia. Exactly Divergence-Free Ultra-Weak Discontinuous Galerkin Method for Incompressible Flow[J]. Communications on Applied Mathematics and Computation, 2026, 8(4): 1450-1477.

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

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

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