Stability and convergence of second order backward differentiation schemes for parabolic Hamilton-Jacobi-Bellman equations
It provides rigorous stability and convergence results for a non-monotone second-order scheme, addressing a gap in numerical methods for HJB equations.
The paper analyzes a second-order BDF scheme for parabolic HJB equations, proving stability in a general nonlinear setting and convergence under additional regularity assumptions, with numerical validation.
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.