A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations

Yan, Jue; Osher, Stanley

doi:10.1016/j.jcp.2010.09.022

Cited by 74 publications

(54 citation statements)

References 27 publications

(40 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[1,3,7,8,10,[14][15][16]20,[23][24][25]). On the other hand, most of convergent schemes of them [7,[14][15][16]20,23] have proposed strong requirements or extra conditions on triangulations to ensure that the schemes are monotone or to construct the scheme itself.…”

Section: (13)mentioning

confidence: 99%

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

Li¹

2015

JCM

View full text Add to dashboard Cite

We study numerical methods for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. We first propose a new class of abstract monotone approximation schemes and get a convergence rate of 1/2 . Then, according to the abstract convergence results, by newly constructing monotone finite volume approximations on interior and boundary points, we obtain convergent finite volume schemes for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. Finally give some numerical results.Mathematics subject classification: 65N30.

show abstract

Section: (13)mentioning

confidence: 99%

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

Li¹

2015

JCM

View full text Add to dashboard Cite

show abstract

“…Besides, a key advantage of this scheme is the local solvability, that is, the auxiliary variables approximating the gradient of the solution can be locally eliminated. Over the past twenty years, there has been extensive study of the LDG methods for steady problems or in the semi-discrete framework, such as for elliptic problems [4], convection-diffusion problems [5], the Stokes system [7], the KdV type equations [18], Hamilton-Jacobi equations [17], time-dependent fourth order problems [10], and so on.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems

Wang

Shu

Zhang

2016

Applied Mathematics and Computation

View full text Add to dashboard Cite

a b s t r a c tThe main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving one-dimensional convection-diffusion equations with a nonlinear convection. Both Runge-Kutta and multi-step IMEX methods are considered. By the aid of the energy method, we show that the IMEX LDG schemes are unconditionally stable for the nonlinear problems, in the sense that the time-step s is only required to be upper-bounded by a positive constant which depends on the flow velocity and the diffusion coefficient, but is independent of the mesh size h. We also give optimal error estimates for the IMEX LDG schemes, under the same temporal condition, if a monotone numerical flux is adopted for the convection. Numerical experiments are given to verify our main results.

show abstract

“…This scheme has provable stability and error estimates for linear equations and demonstrates good convergence to the viscosity solutions for nonlinear equations. Recently, some other direct solvers have also been proposed, such as the central DG scheme [18] and the local DG scheme [21]. This paper is based on our previous work [6], where a DG scheme for HJ equations arising from front propagation was considered using the direct solver in [12].…”

Section: Introductionmentioning

confidence: 99%

A discontinuous Galerkin scheme for front propagation with obstacles

2013

View full text Add to dashboard Cite

A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations

Cited by 74 publications

References 27 publications

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems

A discontinuous Galerkin scheme for front propagation with obstacles

Contact Info

Product

Resources

About