  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.
