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

Abstract

In this paper, we propose a new discontinuous Galerkin finite element method to solve the Hamilton–Jacobi equations. Unlike the discontinuous Galerkin method of [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690.] which applies the discontinuous Galerkin framework on the conservation law system satisfied by the derivatives of the solution, the method in this paper applies directly to the solution of the Hamilton–Jacobi equations. For the linear case, this method is equivalent to the traditional discontinuous Galerkin method for conservation laws with source terms. Thus, stability and error estimates are straightforward. For the nonlinear convex Hamiltonians, numerical experiments demonstrate that the method is stable and provides the optimal ( k + 1)th order of accuracy for smooth solutions when using piecewise kth degree polynomials. Singularities in derivatives can also be resolved sharply if the entropy condition is not violated. Special treatment is needed for the entropy violating cases. Both one and two-dimensional numerical results are provided to demonstrate the good qualities of the scheme.