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A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations

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Abstract

In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of Hamilton-Jacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since Hamilton-Jacobi equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to directly solve such equations. By recognizing and following a “weighted-residual” or “stabilization-based” formulation of central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method for Hamilton-Jacobi equations. The L 2 stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance of the method in approximating the viscosity solutions of general Hamilton-Jacobi equations are demonstrated through extensive numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Fengyan Li & Sergey Yakovlev

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  1. Fengyan Li
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  2. Sergey Yakovlev
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Correspondence to Fengyan Li.

Additional information

The research of F. Li was supported in part by the NSF under the grant DMS-0652481, NSF CAREER award DMS-0847241 and by an Alfred P. Sloan Research Fellowship. Additional support was provided by NSFC grant 10671091 while Li was visiting Department of Mathematics at Nanjing University, China.

The research of S. Yakovlev was supported in part by the NSF CAREER award DMS-0847241.

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Li, F., Yakovlev, S. A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations. J Sci Comput 45, 404–428 (2010). https://doi.org/10.1007/s10915-009-9340-y

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