Erbay, Hüsnü A. and Erbay, Saadet and Erkip, Albert (2018) Convergence of a semidiscrete numerical method for a class of nonlocal nonlinear wave equations. ESAIM: Mathematical Modelling and Numerical Analysis, 52 (3). pp. 803826. ISSN 0764583X (Print) 12903841 (Online)
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Official URL: http://dx.doi.org/10.1051/m2an/2018035
Abstract
In this article, we prove the convergence of a semidiscrete numerical method applied to a general class of nonlocal nonlinear wave equations where the nonlocality is introduced through the convolution operator in space. The most important characteristic of the numerical method is that it is directly applied to the nonlocal equation by introducing the discrete convolution operator. Starting from the continuous Cauchy problem defined on the real line, we first construct the discrete Cauchy problem on a uniform grid of the real line. Thus the semidiscretization in space of the continuous problem gives rise to an infinite system of ordinary differential equations in time. We show that the initialvalue problem for this system is wellposed. We prove that solutions of the discrete problem converge uniformly to those of the continuous one as the mesh size goes to zero and that they are secondorder convergent in space. We then consider a truncation of the infinite domain to a finite one. We prove that the solution of the truncated problem approximates the solution of the continuous problem when the truncated domain is sufficiently large. Finally, we present some numerical experiments that confirm numerically both the expected convergence rate of the semidiscrete scheme and the ability of the method to capture finitetime blowup of solutions for various convolution kernels.
Item Type:  Article 

Uncontrolled Keywords:  Nonlocal nonlinear wave equation; discretization; semidiscrete scheme; improved Boussinesq equation; convergence 
Subjects:  Q Science > QA Mathematics > QA299.6433 Analysis 
Divisions:  Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences 
Depositing User:  Albert Erkip 
Date Deposited:  07 Jan 2019 11:41 
Last Modified:  31 Jul 2023 11:55 
URI:  https://research.sabanciuniv.edu/id/eprint/36718 
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Convergence of a semidiscrete numerical method for a class of nonlocal nonlinear wave equations. (deposited 01 Aug 2018 11:35)
 Convergence of a semidiscrete numerical method for a class of nonlocal nonlinear wave equations. (deposited 07 Jan 2019 11:41) [Currently Displayed]