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Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation. (English) Zbl 1131.65073

We develop a finite-volume scheme for the Korteweg de Vries equation which conserves both the momentum and energy. The main ingredient of the method is a numerical device we developed in recent years that enables us to construct a numerical method for a partial differential equation that also simulates its related equations. In the method, numerical approximations to both the momentum and energy are conservatively computed.

The operator splitting approach is adopted in constructing the method in which the conservation and dispersion parts of the equation are alternatively solved; our numerical device is applied in solving the conservation part of the equation. The feasibility and stability of the method is discussed, which involves an important property of the method, the so-called Jensen condition. The truncation error of the method is analyzed, which shows that the method is second-order accurate.

Finally, several numerical examples, including the Zabusky-Kruskal’s example, are presented to show the good stability property of the method for long-time numerical integration.

MSC:
65M06Finite difference methods (IVP of PDE)
35Q53KdV-like (Korteweg-de Vries) equations
65M12Stability and convergence of numerical methods (IVP of PDE)