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A finite difference method for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations. (English) Zbl 0907.65085

The authors consider the Djidjeli-form of the Korteweg-de Vries equation and discretize and linearize it using a finite difference approximation (the linearization taking place by substituting \(u(t_{n+1})u(t_n)\) for \(u^2(t_{n+1/2})\)). This process results into a second-order implicit linear difference scheme with a pentadiagonal matrix. The approach is extended to the Kadomtsev - Petviashvili equation. For both types of equations, the authors investigate the truncation error, linear stability and numerical dispersion as well as influence of boundary conditions.
Numerical experiments include single solitons and soliton interaction for line-type and lump-type solitons.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q51 Soliton equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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