Smoothing Properties of Implicit Finite Difference Methods for a Diffusion Equation in Maximum Norm

We prove a regularity property of finite difference schemes for the heat or diffusion equation ut = ∆u in maximum norm with large time steps. For a class of time discretizations including L-stable single-step methods and the second-order backward difference formula, with the usual second-order Laplacian, we show that solutions of the scheme gain first spatial differences boundedly, and also second differences except for logarithmic factors, with respect to nonhomogeneous terms. A weaker property is shown for the Crank-Nicolson method. As a consequence we show that the numerical solution of a convection-diffusion equation with an interface can allow O(h) truncation error near the interface and still have a solution with uniform O(h2) accuracy and first differences of uniform accuracy almost O(h2).