A new discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:268

Article

Cheng, Yingda¹  Wang, Zixuan¹ 

[1] Michigan State Univ, Dept Math, E Lansing, MI 48824 USA

关键词: Hamilton-Jacobi equation;    Discontinuous Galerkin methods;    Entropy fix;    Unstructured mesh;   

DOI  :  10.1016/j.jcp.2014.02.041

来源: Elsevier

【 摘 要 】

In this paper, we improve upon the discontinuous Galerkin (DG) method for Hamilton-Jacobi (HJ) equation with convex Hamiltonians in [5] and develop a new DG method for directly solving the general HJ equations. The new method avoids the reconstruction of the solution across elements by utilizing the Roe speed at the cell interface. Besides, we propose an entropy fix by adding penalty terms proportional to the jump of the normal derivative of the numerical solution. The particular form of the entropy fix was inspired by Harten and Hyman's entropy fix [12] for Roe scheme for the conservation laws. The resulting scheme is compact, simple to implement even on unstructured meshes, and is demonstrated to work for nonconvex Hamiltonians. Benchmark numerical experiments in one dimension and two dimensions are provided to validate the performance of the method. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free