Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes,II
Journal of Computational Physics
High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations
SIAM Journal on Numerical Analysis
Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit
Nonlinear Analysis: Theory, Methods & Applications
A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations
SIAM Journal on Mathematical Analysis
A PDE-based fast local level set method
Journal of Computational Physics
Weighted ENO Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
High-Resolution Nonoscillatory Central Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
Journal of Computational Physics
Optimal Trajectories of Curvature Constrained Motion in the Hamilton---Jacobi Formulation
Journal of Scientific Computing
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We introduce two types of finite difference methods to compute the L-solution and the proper viscosity solution recently proposed by the second author for semi-discontinuous solutions to a class of Hamilton-Jacobi equations. By regarding the graph of the solution as the zero level curve of a continuous function in one dimension higher, we can treat the corresponding level set equation using the viscosity theory introduced by Crandall and Lions. However, we need to pay special attention both analytically and numerically to prevent the zero level curve from overturning so that it can be interpreted as the graph of a function. We demonstrate our Lax-Friedrichs type numerical methods for computing the L-solution using its original level set formulation. In addition, we couple our numerical methods with a singular diffusive term which is essential to computing solutions to a more general class of HJ equations that includes conservation laws. With this singular viscosity, our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws. These numerical methods are generalized to higher order accuracy using weighted ENO local Lax-Friedrichs methods as developed recently by Jiang and Peng. We verify that our numerical solutions approximate the proper viscosity solutions obtained by the second author in a recent Hokkaido University preprint. Finally, since the solution of scalar conservation law equations can be constructed using existing numerical techniques, we use it to verify that our numerical solution approximates the entropy solution.