An Alternative Formulation of Discontinous Galerkin Schemes for Solving Hamilton–Jacobi Equations

Abstract

The aim of this paper is to develop an alternative formulation of discontinuous Galerkin (DG) schemes for approximating the viscosity solutions to nonlinear Hamilton–Jacobi (HJ) equations. The main difficulty in designing DG schemes lies in the inherent non-divergence form of HJ equations. One effective approach is to explore the elegant relationship between HJ equations and hyperbolic conservation laws: the standard DG scheme is applied to solve a conservation law system satisfied by the derivatives of the solution of the HJ equation. In this paper, we consider an alternative approach to directly solving the HJ equations, motivated by a class of successful direct DG schemes by Cheng and Shu (J Comput Phys 223(1):398–415, 2007), Cheng and Wang (J Comput Phys 268:134–153, 2014). The proposed scheme is derived based on the idea from the central-upwind scheme by Kurganov et al. (SIAM J Sci Comput 23(3):707–740, 2001). In particular, we make use of precise information of the local speeds of propagation at discontinuous element interface with the goal of adding adequate numerical viscosity and hence naturally capturing the viscosity solutions. A collection of numerical experiments is presented to demonstrate the performance of the method for solving general HJ equations with linear, nonlinear, smooth, non-smooth, convex, or non-convex Hamiltonians.