Error bounds for spatial discretization and waveform relaxation applied to parabolic functional differential equations

Abstract

The process of semi-discretization and waveform relaxation are applied to general nonlinear parabolic functional differential equations. Two new theorems are presented, which extend and improve some of the classical results. The first of these theorems gives an upper bound for the norm of the error of finite difference semi-discretization. This upper bound is sharper than the classical error bound. The second of these theorems gives an upper bound for the norm of the error, which is caused by both semi-discretization and waveform relaxation. The focus in the paper is on estimating this error directly without using the upper bound for the error, which is caused by the process of semidiscretization and the upper bound for the error, which is caused by the waveform relaxation method. Such estimating gives sharper error bound than the bound, which is obtained by estimating both errors separately.  2004 Elsevier Inc. All rights reserved.

A physics‐informed order‐of‐magnitude approach to handling dynamic iterations applied to models of physical systems: Theoretical framework

“…using the solver ode15s applied to (17) (where t n are grid points and x 𝑗,n are approximations calculated by ode15s) is about 2.31 • 10 −6 . For comparison, Figure 3A presents the errors (20) as functions of t n for fixed k = 8, where x (8) 1,n is defined by (24) and z (k) n , x (k) 2,n , x (k) 3,n are determined from (23). Here, k = 8 is chosen so that the errors (20) stay constant for the remaining iterates k ≥ 8 as presented in Figure 1B.…”

“…Apart from the benefits of parallelization, dynamic iterations turned out to be also useful for solving systems of delay differential equations [5,6] and more general functional-differential equations [7,8], where the functional argument is treated in a Picard-type of way. Thalamo-cortical systems written in terms of strongly coupled nonlinear Volterra integro-differential equations have also been solved by using dynamic iterations, which turned out to be quickly convergent [9].…”