Anti-diffusive high order WENO schemes for Hamilton-Jacobi equations. (English) Zbl 1119.65378

Xu, Zhengfu; Shu, Chiwang

Methods Appl. Anal. 12, No. 2, 169-190 (2005).

Summary: We generalize the technique of anti-diffusive flux corrections for high order finite difference weighted essentially non-oscillatory (WENO) schemes solving conservation laws, to solve Hamilton-Jacobi equations. The objective is to obtain sharp resolution for kinks, which are derivative discontinuities in the viscosity solutions of Hamilton-Jacobi equations. We would like to resolve kinks better while maintaining high order accuracy in smooth regions. Numerical examples for one and two space dimensional problems demonstrate the good quality of these Hamiltonian corrected WENO schemes.

MSC:

65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L60	First-order nonlinear hyperbolic equations
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games

Keywords:

anti-diffusive flux correction; Hamiltonian correction; high order accuracy; finite difference; weighted essentially non-oscillatory (WENO) schemes; Hamilton-Jacobi equations; viscosity solutions; numerical examples