Abstract
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 of well-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 Hölder 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.
Highlights
This paper provides stability and convergence results for a type of implicit finite difference scheme for the approximation of nonlinear parabolic equations using backward differentiation formulae (BDF)
The scheme we propose is constructed by using a second order BDF approximation for the first derivatives in both time and space, combined with a standard three-point central finite difference for the second spatial derivative in one dimension
We have proved the well-posedness and stability in L2 and H 1 norms of a second order BDF scheme for HJB equations with enough regularity of the coefficients
Summary
This paper provides stability and convergence results for a type of implicit finite difference scheme for the approximation of nonlinear parabolic equations using backward differentiation formulae (BDF). We assume throughout the whole paper the well-posedness of the problem, namely the existence and uniqueness of a solution in the viscosity sense Under such weak assumptions, convergence of numerical schemes can only be guaranteed if they satisfy certain monotonicity properties, in addition to the more standard consistency and stability conditions for linear equations [2]. The scheme is second order consistent by construction For this scheme, under the assumption of uniform parabolicity, we establish new stability results in the H 1-norm for fully nonlinear HJB and Isaacs equations, and in the L2-norm for the semilinear case (see Theorems 4 and 5, respectively). We will make individual assumptions for each result as we go along
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