A kernel based high order explicit unconditionally stable scheme for time dependent Hamilton-Jacobi equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:379

Article

Christlieb, Andrew^1,2  Guo, Wei³  Jiang, Yan⁴ 

[1] Michigan State Univ, Dept Computat Math Sci & Engn, Dept Math, E Lansing, MI 48824 USA

[2] Michigan State Univ, Dept Elect Engn, E Lansing, MI 48824 USA

[3] Texas Tech Univ, Dept Math & Stat, Lubbock, TX 70409 USA

[4] Univ Sci & Technol China, Sch Math Sci, Hefei 230026, Anhui, Peoples R China

关键词: Hamilton-Jacobi equation;    Kernel based scheme;    Unconditionally stable;    High order accuracy;    Weighted essentially non-oscillatory methodology;    Viscosity solution;   

DOI  :  10.1016/j.jcp.2018.11.037

来源: Elsevier

【 摘 要 】

In this paper, a class of high order numerical schemes is proposed for solving Hamilton-Jacobi (H-J) equations. This work is regarded as an extension of our previous work for nonlinear degenerate parabolic equations, see Christlieb et al. [14], which relies on a special kernel-based formulation of the solutions and successive convolution. When applied to the H-J equations, the newly proposed scheme attains genuinely high order accuracy in both space and time, and more importantly, it is unconditionally stable, hence allowing for much larger time step evolution compared with other explicit schemes and saving computational cost. A high order weighted essentially non-oscillatory methodology and a novel nonlinear filter are further incorporated to capture the correct viscosity solution. Furthermore, by coupling the recently proposed inverse Lax-Wendroff boundary treatment technique, this method is very flexible in handing complex geometry as well as general boundary conditions. We perform numerical experiments on a collection of numerical examples, including H-J equations with linear, nonlinear, convex or non-convex Hamiltonians. The efficacy and efficiency of the proposed scheme in approximating the viscosity solution of general H-J equations is verified. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free