Volume 13, Issue 1
The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Jinye Shen ,  Xuping Wang and Zhi-zhong Sun

10.4208/nmtma.OA-2019-0038

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 253-280.

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  • Abstract

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation  with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct  matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes  are constructed  for the  problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the  nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence  of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.


  • History

Published online: 2019-12

  • AMS Subject Headings

65M06, 65M12, 65M15

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