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Numerical inverse Laplace transform for convection-diffusion equations

Authors: Nicola Guglielmi, María López-Fernández and Giancarlo Nino
Journal: Math. Comp. 89 (2020), 1161-1191
MSC (2010): Primary 65L05, 65R10, 65J10, 65M20, 91-08
Published electronically: January 6, 2020
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Abstract: In this paper a novel contour integral method is proposed for linear convection-diffusion equations. The method is based on the inversion of the Laplace transform and makes use of a contour given by an elliptic arc joined symmetrically to two half-lines. The trapezoidal rule is the chosen integration method for the numerical inversion of the Laplace transform, due to its well-known fast convergence properties when applied to analytic functions. Error estimates are provided as well as careful indications about the choice of several involved parameters. The method selects the elliptic arc in the integration contour by an algorithmic strategy based on the computation of pseudospectral level sets of the discretized differential operator. In this sense the method is general and can be applied to any linear convection-diffusion equation without knowing any a priori information about its pseudospectral geometry. Numerical experiments performed on the Black-Scholes ($ 1D$) and Heston ($ 2D$) equations show that the method is competitive with other contour integral methods available in the literature.

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Additional Information

Nicola Guglielmi
Affiliation: Gran Sasso Science Institute, via Crispi 7, L’Aquila, Italy

María López-Fernández
Affiliation: Departamento de Análisis Matemático, Estadística e I.O., Matemática Aplicada, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos s/n, 29080, Málaga, Spain; and Department of Mathematics Guido Castelnuovo, Sapienza University of Rome, Italy

Giancarlo Nino
Affiliation: Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211 Genève, Switzerland

Keywords: Contour integral methods, pseudospectra, Laplace transform, numerical inversion of Laplace transform, trapezoidal rule, quadrature for analytic functions.
Received by editor(s): September 5, 2018
Received by editor(s) in revised form: July 2, 2019, and August 21, 2019
Published electronically: January 6, 2020
Additional Notes: The first and second authors acknowledge INdAM GNCS for their financial support.
The second author acknowledges partial support of the Spanish grant MTM2016-75465.
Article copyright: © Copyright 2020 American Mathematical Society