Non-stationary wave relaxation methods for general linear systems of Volterra equations: convergence and parallel GPU implementation

In the present paper, a parallel-in-time discretization of linear systems of Volterra equations of type u¯(t)=u¯0+∫0tK(t-s)u¯(s)ds+f¯(t),0<t≤T,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \bar{u}(t) = \bar{u}_0 + \int _0^t \textbf{K}(t-s) \bar{u}(s)\,\text{ d }s + \bar{f}(t),\qquad 0<t\le T, \end{aligned}$$\end{document}is addressed. Related to the analytical solution, a general enough functional setting is firstly stated. Related to the numerical solution, a parallel numerical scheme based on the Non-Stationary Wave Relaxation (NSWR) method for the time discretization is proposed, and its convergence is studied as well. A CUDA parallel implementation of the method is carried out in order to exploit Graphics Processing Units (GPUs), which are nowadays widely employed for reducing the computational time of several general purpose applications. The performance of these methods is compared to some sequential implementation. It is revealed throughout several experiments of special interest in practical applications the good performance of the parallel approach.