Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations

Name: Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations
Author: Erbay, Hüsnü Ata, Erbay, Saadet, Erkip, A.

İsim	Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations
Yazar	Erbay, Hüsnü Ata, Erbay, Saadet, Erkip, A.
Basım Tarihi:	2018-09-13
Basım Yeri	- EDP Sciences
Konu	Nonlocal nonlinear wave equation, Discretization, Semi-discrete scheme, Improved Boussinesq equation, Convergence
Tür	Süreli Yayın
Dil	İngilizce
Dijital	Evet
Yazma	Hayır
Kütüphane:	Özyeğin Üniversitesi
Demirbaş Numarası	0764-583X
Kayıt Numarası	2a214d85-eb10-4c6d-9d41-3dede325d8d1
Lokasyon	Natural and Mathematical Sciences
Tarih	2018-09-13
Örnek Metin	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

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Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations

Yazar Erbay, Hüsnü Ata, Erbay, Saadet, Erkip, A.

Basım Tarihi 2018-09-13

Basım Yeri - EDP Sciences

Konu Nonlocal nonlinear wave equation, Discretization, Semi-discrete scheme, Improved Boussinesq equation, Convergence

Tür Süreli Yayın

Dil İngilizce

Dijital Evet

Yazma Hayır

Kütüphane Özyeğin Üniversitesi

Demirbaş Numarası 0764-583X

Kayıt Numarası 2a214d85-eb10-4c6d-9d41-3dede325d8d1

Lokasyon Natural and Mathematical Sciences

Tarih 2018-09-13

Örnek Metin 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