Abstract
In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well.
Highlights
Modified Kortweg-de Vries (MKdV) equation admitted soliton solution which can be defined as a solitary wave solution, highly stable and retains its identity, upon interaction
MKdV equation has a limited number of numerical studies in the literature
Raslan and Baghdady [4] [5], derived finite difference method and finite element method using collocation method with septic spline for solving the MKdV equation, they show that both schemes are unconditionally stable in the linearized sense
Summary
This article is concerned with, the numerical solution of non-linear MKdV equation [1]. Raslan and Baghdady [4] [5], derived finite difference method and finite element method using collocation method with septic spline for solving the MKdV equation, they show that both schemes are unconditionally stable in the linearized sense. We will derive four numerical schemes for solving the MKdV equation are presented; based on the Padé approximation of fourth order accuracy in space, together with Crank-Nicolson scheme of second order accuracy in the time direction.
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