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On stability of some finite difference schemes for the Korteweg-de Vries equation. (English) Zbl 1337.65136

Summary: Numerical stability in regard to some difference schemes for the Korteweg-de Vries (K-dV) equation is discussed. A stability criterion for the leap-frog explicit scheme which has been used by Zabusky and Kruskal for solving the initial value problem is proposed. It is shown by the energy method that this criterion implies the stability of a linearized difference equation closely related to the scheme concerned. Further, an unconditionally stable implicit scheme is proposed. Some numerical comparisons between the two schemes are given. At the same time, a numerical comparison with analytical solutions of the K-dV equation is also given. These results agree well with each other.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
76M20 Finite difference methods applied to problems in fluid mechanics
35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
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