Windowing Waveform Relaxation of Initial Value Problems

Abstract

We present a windowing technique of waveform relaxation for dynamic systems. An effective estimation on window length is derived by an iterative error expression provided here. Relaxation processes can be speeded up if one takes the windowing technique in advance. Numerical experiments are given to further illustrate the theoretical analysis.

A windowing waveform relaxation method for time-fractional differential equations

“…Further, for a given convergence rate ε(ε > 0) and an iteration number k s , the suggested length of window of ε-convergence for WR is estimated by Remark 3.1. When α = 1, Theorem 3.1 becomes Theorem 1 in [21]. This is because ϒ (k,i) (t) ≡ 0 and…”

“…In the following, we describe the basic settings of the original windowing WR method (cf. [21]), aiming to solve (2.1). First, the time domain is decomposed into N elements:…”

“…Basic idea of the windowing WR is that the interval of integration is split into a series of subintervals, called windows, with iteration taking place on each window successively. During the last years, there has been an explosion of research work on the analysis of the windowing WR method (for example, see [21,22]). However, there is currently no development of windowing WR method for fractional differential equations.…”

Error analysis of waveform relaxation method for semi-linear partial differential equations

“…This procedure is called windowing. Result on convergence of windowing for ODEs is given in [9], the windowing technique is employed to accelerate the convergence of the WR process in [10]. Here as an extension of these results, the convergence of windowing is proven for a large class of PDEs.…”

“…Right: the result with fourth order Runge-Kutta method (blue), the theoretical estimation (red) according to(10) and the classical estimation (15) (black) are plotted. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)…”