Author
Erbay, Hüsnü Ata, Erbay, Saadet, Erkip, A.
Publication Date
2018-09-13
Publication Place
-
EDP Sciences
Subject
Nonlocal nonlinear wave equation, Discretization, Semi-discrete scheme, Improved Boussinesq equation, Convergence
Type
Periodical
Language
English
Digital
Yes
Manuscript
No
Library
Özyeğin University
Library Asset ID
0764-583X
Record ID
2a214d85-eb10-4c6d-9d41-3dede325d8d1
Library Location
Natural and Mathematical Sciences
Date
2018-09-13
Sample Text
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.
DOI
10.1051/m2an/2018035
Cilt
52
