In this article, we prove the convergence of a semi-discrete 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 semi-discretization in space of the continuous problem gives rise to an infinite system of ordinary differential equations in time. We show that the initial-value problem for this system is well-posed. 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 second-order 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 semi-discrete scheme and the ability of the method to capture finite-time blow-up of solutions for various convolution kernels.
Mots-clés : Nonlocal nonlinear wave equation, discretization, semi-discrete scheme, improved Boussinesq equation, convergence
@article{M2AN_2018__52_3_803_0,
author = {Erbay, H.A. and Erbay, S. and Erkip, A.},
title = {Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {803--826},
publisher = {EDP-Sciences},
volume = {52},
number = {3},
year = {2018},
doi = {10.1051/m2an/2018035},
mrnumber = {3865550},
zbl = {1414.35224},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018035/}
}
TY - JOUR AU - Erbay, H.A. AU - Erbay, S. AU - Erkip, A. TI - Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 803 EP - 826 VL - 52 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018035/ DO - 10.1051/m2an/2018035 LA - en ID - M2AN_2018__52_3_803_0 ER -
%0 Journal Article %A Erbay, H.A. %A Erbay, S. %A Erkip, A. %T Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 803-826 %V 52 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018035/ %R 10.1051/m2an/2018035 %G en %F M2AN_2018__52_3_803_0
Erbay, H.A.; Erbay, S.; Erkip, A. Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 803-826. doi : 10.1051/m2an/2018035. http://www.numdam.org/articles/10.1051/m2an/2018035/
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