Convergent Difference Schemes for Degenerate Elliptic and Parabolic Equations: Hamilton--Jacobi Equations and Free Boundary Problems

Authors:
Adam M. Oberman
Affiliations:
-
Venue:
SIAM Journal on Numerical Analysis
Year:
2006

Citing 0
Cited 8

The Flexible, Extensible and Efficient Toolbox of Level Set Methods

Journal of Scientific Computing
Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation

Journal of Computational Physics
Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge-Ampère Equation in Dimensions Two and Higher

SIAM Journal on Numerical Analysis
Optimal Trajectories of Curvature Constrained Motion in the Hamilton---Jacobi Formulation

Journal of Scientific Computing
Convergence of a semi-discretization scheme for the Hamilton-Jacobi equation: A new approach with the adjoint method

Applied Numerical Mathematics
A semi-Lagrangian scheme for the game p-Laplacian via p-averaging

Applied Numerical Mathematics
Numerical solution of the Optimal Transportation problem using the Monge-Ampère equation

Journal of Computational Physics
Finite element methods for fully nonlinear second order PDEs based on a discrete Hessian with applications to the Monge-Ampère equation

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	0.01

Visualization

Abstract

Convergent numerical schemes for degenerate elliptic partial differential equations are constructed and implemented. Simple conditions are identified which ensure that nonlinear finite difference schemes are monotone and nonexpansive in the maximum norm. Explicit schemes endowed with an explicit CFL condition are built for time-dependent equations and are used to solve stationary equations iteratively. Explicit and implicit formulations of monotonicity for first- and second-order equations are unified. Bounds on orders of accuracy are established. An example of a scheme which is stable, but nonmonotone and nonconvergent, is presented. Schemes for Hamilton--Jacobi equations, obstacle problems, one-phase free boundary problems, and stochastic games are built and computational results are presented.