Some high order schemes for parabolic Hamilton-Jacobi-Bellman equations

We consider high order numerical schemes for second order Hamilton-Jacobi-Bellman (HJB) equations. For high order approximation schemes (where “high” stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. We present a class of ``filtered'' schemes: a suitable local modification of the high order scheme is introduced by ``filtering'' it with a monotone one, so that it can be proven to converge and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests. For making the scheme working, it is fundamental to use a stable high order scheme. We focus on a second order BDF (Backward Differentiation Formula) scheme and prove its stability properties.