Math @ Duke

Publications [#243358] of J. Thomas Beale
Papers Published
 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
(2009),
pp. 24762495, Society for Industrial & Applied Mathematics (SIAM), ISSN 00361429 [pdf], [doi]
(last updated on 2019/01/20)
Abstract: 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 Lstable singlestep methods and the secondorder backward difference formula, with the usual secondorder 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 CrankNicolson method. As a consequence we show that the numerical solution of a convectiondiffusion 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.


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