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IMA Journal of Numerical Analysis Advance Access published online on April 17, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn074
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Improved contour integral methods for parabolic PDEs

J. A. C. Weideman{dagger}

Applied Mathematics, Stellenbosch University, Stellenbosch 7600,South Africa

{dagger} Email: weideman{at}sun.ac.za

Received on 1 May 2008. Accepted for publication 12 September 2008.


  Abstract

One way of computing the matrix exponential that arises in semidiscrete parabolic partial differential equations is via the Dunford–Cauchy integral formula. The integral is approximated by the trapezoidal or midpoint rules on a Hankel contour defined by a suitable change of variables. In a recent paper by Weideman & Trefethen (2007, Math. Comput., 76, 1341–1356) two widely used contours were analysed. Estimates for the optimal parameters that define these contours were proposed. In this paper this analysis is extended in two directions. First, the effect of roundoff error is now included in the error model. Second, we extend the results to the case of a model convection–diffusion equation, where a large convective term causes the matrix to be highly non-normal.

Key Words: matrix exponential; Laplace transform; numerical contour integration; convection–diffusion PDE


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