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Numerical solution for the KdV equation based on similarity reductions. (English) Zbl 1168.65376
Summary: The present paper is devoted to the development of a new scheme to solve the KdV equation locally on sub-domains using similarity reductions for partial differential equations. Each sub-domain is divided into three-grid points. The ordinary differential equation deduced from the similarity reduction can be linearized, integrated analytically and then used to approximate the flux vector in the KdV equation. The arbitrary constants in the analytical solution of the similarity equation can be determined in terms of the dependent variables at the grid points in each sub-domain. This approach eliminates the difficulties associated with boundary conditions for the similarity reductions over the whole solution domain. Numerical results are obtained for two test problems to show the behavior of the solution of the problems. The computed results are compared with other numerical results.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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