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Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm

Publication ,  Journal Article
Beale, JT
Published in: SIAM Journal on Numerical Analysis
July 20, 2009

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.

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Published In

SIAM Journal on Numerical Analysis

DOI

ISSN

0036-1429

Publication Date

July 20, 2009

Volume

47

Issue

4

Start / End Page

2476 / 2495

Related Subject Headings

  • Numerical & Computational Mathematics
  • 4903 Numerical and computational mathematics
  • 4901 Applied mathematics
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics
 

Citation

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Beale, J. T. (2009). Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm. SIAM Journal on Numerical Analysis, 47(4), 2476–2495. https://doi.org/10.1137/080731645
Beale, J. T. “Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm.” SIAM Journal on Numerical Analysis 47, no. 4 (July 20, 2009): 2476–95. https://doi.org/10.1137/080731645.
Beale JT. Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm. SIAM Journal on Numerical Analysis. 2009 Jul 20;47(4):2476–95.
Beale, J. T. “Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm.” SIAM Journal on Numerical Analysis, vol. 47, no. 4, July 2009, pp. 2476–95. Scopus, doi:10.1137/080731645.
Beale JT. Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm. SIAM Journal on Numerical Analysis. 2009 Jul 20;47(4):2476–2495.

Published In

SIAM Journal on Numerical Analysis

DOI

ISSN

0036-1429

Publication Date

July 20, 2009

Volume

47

Issue

4

Start / End Page

2476 / 2495

Related Subject Headings

  • Numerical & Computational Mathematics
  • 4903 Numerical and computational mathematics
  • 4901 Applied mathematics
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics