Abstract
A method of lines approach to the numerical solution of nonlinear wave equations typified by the regularized long wave (RLW) is presented. The method developed uses a finite differences discretization to the space. Solution of the resulting system was obtained by applying fourth Runge-Kutta time discretization method. Using Von Neumann stability analysis, it is shown that the proposed method is marginally stable. To test the accuracy of the method some numerical experiments on test problems are presented. Test problems including solitary wave motion, two-solitary wave interaction, and the temporal evaluation of a Maxwellian initial pulse are studied. The accuracy of the present method is tested with <svg style="vertical-align:-3.3907pt;width:22.5875px;" id="M1" height="15.4" version="1.1" viewBox="0 0 22.5875 15.4" width="22.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g> <g transform="matrix(.012,-0,0,-.012,9.763,15.187)"><path id="x221E" d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196
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Highlights
Many researchers have introduced various methods to solve the regularized long wave (RLW) of the form ut + ux + εuux − μuxxt = 0, (1)where ε and μ are positive constants
A method of lines approach to the numerical solution of nonlinear wave equations typified by the regularized long wave (RLW) is presented
In order to show how good the numerical solutions are in comparison with exact ones, we will use the L2 and L∞ error norms defined by
Summary
Many researchers have introduced various methods to solve the regularized long wave (RLW) of the form ut + ux + εuux − μuxxt = 0, (1). An analytical solution for the RLW equation was found under the restricted initial and boundary condition [2], but this solution is not very useful; the availability of numerical method is essential. In previous work [3,4,5] various numerical studies have been reported based on the finite difference. A recent study in [6] has shown that the meshless kernelbased method of lines provides a highly accurate and efficient method for the modified regularized long wave equation. The method of lines (MOL) is applied to obtain a numerical solution for RLW with a constant grid of central finite differences and fourth Runge-Kutta approximation in time. The performance of the method was tested on two known model problems
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