Volume 52, Number 3, May–June 2018
|803 - 826
|13 September 2018
Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations
Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University,
2 Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey
* Corresponding author: firstname.lastname@example.org
Accepted: 9 May 2018
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.
Mathematics Subject Classification: 35Q74 / 65M12 / 65Z05 / 74S30
Key words: Nonlocal nonlinear wave equation / discretization / semi-discrete scheme / improved Boussinesq equation / convergence
© EDP Sciences, SMAI 2018
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