Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Journal of Computational Physics
High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations
SIAM Journal on Numerical Analysis
On a cell entropy inequality for discontinuous Galerkin methods
Mathematics of Computation
High-resolution conservative algorithms for advection in incompressible flow
SIAM Journal on Numerical Analysis
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
A Discontinuous Galerkin Finite Element Method for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Analysis of the discontinuous Galerkin method for Hamilton—Jacobi equations
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
Weighted ENO Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
A Local Discontinuous Galerkin Method for KdV Type Equations
SIAM Journal on Numerical Analysis
High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes
SIAM Journal on Scientific Computing
A second order discontinuous Galerkin fast sweeping method for Eikonal equations
Journal of Computational Physics
A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations
Journal of Scientific Computing
Journal of Scientific Computing
Journal of Computational Physics
A local discontinuous Galerkin method for directly solving Hamilton-Jacobi equations
Journal of Computational Physics
Some Improvements for the Fast Sweeping Method
SIAM Journal on Scientific Computing
A boundary-only meshless method for numerical solution of the Eikonal equation
Computational Mechanics
Optimal Control of the Classical Two-Phase Stefan Problem in Level Set Formulation
SIAM Journal on Scientific Computing
A Discontinuous Galerkin Solver for Front Propagation
SIAM Journal on Scientific Computing
Uniformly Accurate Discontinuous Galerkin Fast Sweeping Methods for Eikonal Equations
SIAM Journal on Scientific Computing
Computers & Mathematics with Applications
The Chebyshev spectral viscosity method for the time dependent Eikonal equation
Mathematical and Computer Modelling: An International Journal
Optimal Trajectories of Curvature Constrained Motion in the Hamilton---Jacobi Formulation
Journal of Scientific Computing
A uniformly second order fast sweeping method for eikonal equations
Journal of Computational Physics
Alternating evolution discontinuous Galerkin methods for Hamilton-Jacobi equations
Journal of Computational Physics
Hi-index | 31.47 |
In this paper, we propose a new discontinuous Galerkin finite element method to solve the Hamilton-Jacobi equations. Unlike the discontinuous Galerkin method of [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666-690.] which applies the discontinuous Galerkin framework on the conservation law system satisfied by the derivatives of the solution, the method in this paper applies directly to the solution of the Hamilton-Jacobi equations. For the linear case, this method is equivalent to the traditional discontinuous Galerkin method for conservation laws with source terms. Thus, stability and error estimates are straightforward. For the nonlinear convex Hamiltonians, numerical experiments demonstrate that the method is stable and provides the optimal (k+1)th order of accuracy for smooth solutions when using piecewise kth degree polynomials. Singularities in derivatives can also be resolved sharply if the entropy condition is not violated. Special treatment is needed for the entropy violating cases. Both one and two-dimensional numerical results are provided to demonstrate the good qualities of the scheme.