Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations

Abstract

This paper develops and analyzes a new family of dual-wind discontinuous Galerkin (DG) methods for stationary Hamilton-Jacobi equations and their vanishing viscosity regularizations. The new DG methods are designed using the DG finite element discrete calculus framework of [17] that defines discrete differential operators to replace continuous differential operators when discretizing a partial differential equation (PDE). The proposed methods, which are non-monotone, utilize a dual-winding methodology and a new skew-symmetric DG derivative operator that, when combined, eliminate the need for choosing indeterminable penalty constants. The relationship between these new methods and the local DG methods proposed in [38] for Hamilton-Jacobi equations as well as the generalized-monotone finite difference methods proposed in [13] and corresponding DG methods proposed in [12] for fully nonlinear second order PDEs is also examined. Admissibility and stability are established for the proposed dual-wind DG methods. The stability results are shown to hold independent of the scaling of the stabilizer allowing for choices that go beyond the Godunov barrier for monotone schemes. Numerical experiments are provided to gauge the performance of the new methods.

Funding

Directorate for Mathematical and Physical Sciences(1620168)