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

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.