Publications [#243358] of J. Thomas Beale
- Beale, JT, Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm,
Siam Journal on Numerical Analysis, vol. 47 no. 4
pp. 2476-2495, Society for Industrial & Applied Mathematics (SIAM), ISSN 0036-1429 [pdf], [doi]
(last updated on 2019/10/21)
We prove a regularity property of finite difference schemes for the heat or diffusion equation μ t = δμ 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 gai n 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(h 2) accuracy and first differences of uniform accuracy almost O(h 2). © 2009 Society for Industrial and Applied Mathematics.