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.
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.
- Content Type Journal Article
- DOI 10.1007/s10915-009-9340-y
- Authors
- Fengyan Li, Rensselaer Polytechnic Institute Department of Mathematical Sciences Troy NY 12180 USA
- Sergey Yakovlev, Rensselaer Polytechnic Institute Department of Mathematical Sciences Troy NY 12180 USA
- Journal Journal of Scientific Computing
- Online ISSN 1573-7691
- Print ISSN 0885-7474
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