On the method of Neumann series for highly oscillatory equations
type: Item
instance of: scholarly article
title: On the method of Neumann series for highly oscillatory equations
zbMATH Open document ID: 1071.65099
DOI: 10.1023/B:BITN.0000046810.25353.95
published in: BIT
publication date: 2005-04-07T00:00:00Z
review text: The purpose of the author is to describe and analyse techniques for the numerical solution of highly oscillatory ordinary differential equations by exploiting a Neumann expansion. In the first part he presents the theoretical notions which are used in the sequel. Thus, given the formal equation \[ Y'= A(t)Y,\quad t\geq 0,\quad Y(0)=Y_0,\tag{1} \] where \(A\) might be a matrix, a differential operator or an integral operator, its solution is given by the series \[ Y(y)= (I- K)^{-1}Y_0= \sum^\infty_{m=0} K^m Y_0,\tag{2} \] where \[ K^0X= X(t),\quad K^{m+1}X= \int^t_0 A(x)K^m(x)\,dx,\quad m\geq 0.\tag{3} \] The Neumann expansions is known in physics as Dyson series and in matrix analysis as the Peano series. The second part computes the size of \(\| K\|\), where \(\|\cdot\|\) is the Euclidian norm and explores how it is attenuated by rapid oscillation. The third part describes the practical implementation of the Neumann series for the solution of a linear ordinary differential equation \[ y'= A(t)y,\quad t\geq 0,\quad y(0)= y_0\in\mathbb{R}^N\tag{4} \] having a fast oscillating solution. The author preconditions the solution of (4) locally by the known solution of a linear system with constant coefficients and solves \[ x'= B(t)x,\quad t\geq 0,\quad x(0)= y_n\in \mathbb{R}^N,\quad y_n= y(t_n),\quad n\geq 0, \] where \(B(t)= e^{-t\overline A}[A(t_n+ t)-\overline A] e^{t\overline A}\) with \(\overline A= A(t_n+{1\over 2} h)\), \(h= t_{n+1}- t_n\). Thus truncating (2) he gets \[ \begin{multlined} x_{n+1}= y_n+ \int^h_0 B(x)\,dxy_n+ \int^h_0\int^x_0 B(x_1)B(x_2)\,dx_2 dx_1y_n+\cdots\\ +\int^h_0 \int^{x_1}_0\cdots \int^{x_{r-1}}_{01} B(x_1) B(x_2)\cdots B(x_r)\,dx_r\cdots dx_1 y_n\end{multlined} \] which needs the replacement of multivariate integrals by quadrature. He describes how to accomplish this to high accuracy with a modest computational effort. The fourth section contains the application to the wave equation with a forcing term.
reviewed by: Romulus Militaru
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 2153688
zbMATH Keywords: high oscillation
zbMATH Keywords: numerical quadrature
zbMATH Keywords: Neumann series
zbMATH Keywords: wave equation
zbMATH Keywords: Dyson series
zbMATH Keywords: Peano series
author: Arieh Iserles
MaRDI profile type: Publication
full work available at URL: b:bitn.0000046810.25353.95
OpenAlex ID: W2050353205
description: scientific article; zbMATH DE number 2153688