Improved contour integral methods for parabolic PDEs. (English) Zbl 1186.65125

Weideman, J. A. C.

IMA J. Numer. Anal. 30, No. 1, 334-350 (2010).

Summary: 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 J. A. C. Weideman and L. N. Trefethen [Math. Comput. 76, No. 259, 1341–1356 (2007; Zbl 1113.65119)] 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.

MSC:

65M20	Method of lines for initial value and initial-boundary value problems involving PDEs
35K20	Initial-boundary value problems for second-order parabolic equations
65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs

Keywords:

matrix exponential; Laplace transform; numerical contour integration; semidiscretization; Runge-Kutta method; Chebyshev spectral collocation method; finite difference method; numerical examples; Dunford-Cauchy integral formula; roundoff error; convection-diffusion equation

Citations:

Zbl 1113.65119

Software:

Eigtool