Stability and convergence of second order backward differentiation schemes for parabolic Hamilton-Jacobi-Bellman equations

TL;DR

This paper analyzes the stability and convergence of a second order implicit BDF scheme for parabolic HJB equations, addressing challenges due to non-monotonicity and extending results to multi-dimensional and Isaacs equations.

Contribution

It provides rigorous stability and convergence analysis for a second order BDF scheme applied to nonlinear parabolic HJB equations, including multi-dimensional cases.

Findings

Established stability results in nonlinear settings

Proved convergence under regularity assumptions

Validated the scheme with numerical tests

Abstract

We study a second order BDF (Backward Differentiation Formula) scheme for the numerical approximation of parabolic HJB (Hamilton-Jacobi-Bellman) equations. The scheme under consideration is implicit, non-monotone, and second order accurate in time and space. The lack of monotonicity prevents the use of well-known convergence results for solutions in the viscosity sense. In this work, we establish rigorous stability results in a general nonlinear setting as well as convergence results for some particular cases with additional regularity assumptions. While most results are presented for one-dimensional, linear parabolic and non-linear HJB equations, some results are also extended to multiple dimensions and to Isaacs equations. Numerical tests are included to validate the method.