PARALLEL IN TIME ALGORITHM WITH SPECTRAL-SUBDOMAIN ENHANCEMENT FOR VOLTERRA INTEGRAL EQUATIONS

• Published in: SIAM journal on numerical analysis Vol. 51; no. 3; pp. 1735 - 1756
• Main Authors: LI, XIANJUAN, TANG, TAO, XU, CHUANJU
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
• Subjects: Accuracy; Algorithms; Applied mathematics; Approximation; Computational efficiency; Convergence; Cost estimates; Decomposition; Degrees of polynomials; Differential equations; Error rates; Integral equations; Mathematical domains; Mathematical models; Mathematical problems; Methods; Polynomials; Spectra; Spectral methods; Textual collocation; Volterra integral equations
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/120876241

This paper proposes a parallel in time (also called time parareal) method to solve Volterra integral equations of the second kind. The parallel in time approach follows the spirit of the domain decomposition that consists of breaking the domain of computation into subdomains and solving iteratively the subproblems in a parallel way. To obtain a high order of accuracy, a spectral collocation accuracy enhancement in subdomains will be employed. Our main contributions in this work are twofold: (i) A time parareal method is designed for the integral equations, which to our knowledge is the first of its kind. The new method is an iterative process combining a coarse prediction in the whole domain with fine corrections in subdomains by using spectral approximation, leading to an algorithm of very high accuracy. (ii) A rigorous convergence analysis of the overall method is provided. The numerical experiment confirms that the overall computational cost is considerably reduced while the desired spectral rate of convergence can be obtained.