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.

我们研究了线性抛物方程和非线性Hamilton-Jacobi-Bellman（HJB）方程数值近似的二阶向后微分公式（BDF）方案。BDF方案缺乏单调性，因此无法在粘度意义上将众所周知的收敛结果用于解决方案。我们首先考虑一维均匀抛物线方程并证明其摄动的稳定性，线性和半线性方程的\（L ^ 2 \）范数，完全非线性方程的\（H ^ 1 \）范数HJB和Isaacs类型。然后将这些结果扩展到二维半线性方程和可能具有退化性的线性方程。根据这些稳定性结果，我们得出\（L ^ 2 \）中的误差估计均匀抛物线形半线性HJB方程经典解的范数，其阶次取决于它们的Hölder正则性，而在光滑情况下则可恢复完全二阶。对Eikonal方程和受控扩散方程的数值测试说明了该方案在不同规范下的实际准确性。