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

We study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton-Jacobi-Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use ofwell-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the L-2 norm for linear and semi-linear equations, and in the H-1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in L-2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Holder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms.