Pseudospectral roaming contour integral methods for convection-diffusion equations

TL;DR

This paper introduces a new class of contour integral methods for solving linear convection-diffusion PDEs, optimizing numerical inversion of Laplace transforms with spectral level sets, and compares different integration contours for improved computational efficiency.

Contribution

It develops a new fast pseudospectral roaming method, compares various contour profiles, and enhances solution approximation over multiple time windows without extra computational cost.

Findings

Comparison of three integration profiles (parabolic, hyperbolic, elliptic)

Introduction of a fast pseudospectral roaming method

Optimization of time window selection for solution approximation

Abstract

We generalize ideas in the recent literature and develop new ones in order to propose a general class of contour integral methods for linear convection-diffusion PDEs and in particular for those arising in finance. These methods aim to provide a numerical approximation of the solution by computing its inverse Laplace transform. The choice of the integration contour is determined by the computation of a few suitably weighted pseudo-spectral level sets of the leading operator of the equation. Parabolic and hyperbolic profiles proposed in the literature are investigated and compared to the elliptic contour originally proposed by Guglielmi, L\'opez-Fern\'andez and Nino. In summary, the article (i) provides a comparison among three different integration profiles; (ii) proposes a new fast pseudospectral roaming method; (iii) optimizes the selection of time windows on which one may arbitrarily approximate the solution by no extra computational cost with respect to the case of a fixed time instant; (iv) focuses extensively on computational aspects and it is the reference of the MATLAB code https://github.com/MattiaManucci/Contour_Integral_Methods.git, where all algorithms described here are implemented.