Optimizing Talbot's Contours for the Inversion of the Laplace Transform

Talbot’s method for the numerical inversion of the Laplace transform consists of numerically integrating the Bromwich integral on a special contour by means of the trapezoidal or midpoint rules. In this paper we address the issue of parameter selection in the method, for the particular situation when parabolic PDEs are solved. In the process the well-known subgeometric convergence rate $O(\exp(-c \sqrt{N}))$ of this method is improved to the geometric rate $O(\exp(-c N))$, with $N$ the number of nodes in the integration rule. The value of the maximum decay rate $c$ is explicitly determined. Numerical results for two versions of the heat equation are presented. With the choice of parameters derived here, the rule of thumb is that to achieve an accuracy of $10^{-\ell}$ at any given time, the associated elliptic problem has to be solved no more than $\ell$ times.

[1]  Rémi Vaillancourt,et al.  Applied Integral Transforms , 1993 .

[2]  FuSen F. Lin,et al.  Numerical inversion of laplace transforms by the trapezoidal-type methods , 2003 .

[3]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[4]  V. Thomée,et al.  Time discretization of an evolution equation via Laplace transforms , 2004 .

[5]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[6]  A. Fokas,et al.  Complex Variables: Introduction and Applications , 1997 .

[7]  C. Palencia,et al.  On the numerical inversion of the Laplace transform of certain holomorphic mappings , 2004 .

[8]  A. Talbot.  The Accurate Numerical Inversion of Laplace Transforms , 1979 .

[9]  Eduardo Cuesta,et al.  A Numerical Method for an Integro-Differential Equation with Memory in Banach Spaces: Qualitative Properties , 2003, SIAM J. Numer. Anal..

[10]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[11]  A. Turing.  A method for the calculation of the zeta-function Universal Turing Machine , 2011 .

[12]  Dongwoo Sheen,et al.  A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature , 2000, Math. Comput..

[13]  N. S. Barnett,et al.  Private communication , 1969 .

[14]  E. T. Goodwin.  The evaluation of integrals of the form , 1949, Mathematical Proceedings of the Cambridge Philosophical Society.

[15]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[16]  J. A. C. Weideman,et al.  Numerical Integration of Periodic Functions: A Few Examples , 2002, Am. Math. Mon..

[17]  Dongwoo Sheen,et al.  A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature , 2003 .

[18]  Dean G. Duffy,et al.  On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications , 1993, TOMS.

[19]  S. Orszag,et al.  Advanced Mathematical Methods For Scientists And Engineers , 1979 .

[20]  Lloyd N. Trefethen,et al.  Parabolic and hyperbolic contours for computing the Bromwich integral , 2007, Math. Comput..

[21]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .

[22]  Ivan P. Gavrilyuk,et al.  Exponentially Convergent Parallel Discretization Methods for the First Order Evolution Equations , 2001 .

[23]  Cesar Palencia,et al.  A spectral order method for inverting sectorial Laplace transforms , 2005 .

[24]  Satish C. Reddy,et al.  A MATLAB differentiation matrix suite , 2000, TOMS.