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

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  • 1 Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA;
    2 Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC 27402, USA;
    3 Department of Mathematical Sciences, University of the Virgin Islands, Kingshill, USVI 00850-9781, US Virgin Islands

Received date: 2020-09-03

  Revised date: 2021-01-30

  Online published: 2022-04-29

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 fnite element discrete calculus framework of[17] that defnes discrete diferential operators to replace continuous differential operators when discretizing a partial diferential equation (PDE). The proposed methods, which are non-monotone, utilize a dual-winding methodology and a new skewsymmetric 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 fnite diference 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.

Cite this article

Xiaobing Feng, Thomas Lewis, Aaron Rapp . Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations[J]. Communications on Applied Mathematics and Computation, 2022 , 4(2) : 563 -596 . DOI: 10.1007/s42967-021-00130-9

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