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Two Kinds of Difference Schemes for KdV Equation

Author: WangShuang
Tutor: ABuDuReXiTi·ABuDuWaiLi
School: Xinjiang University
Course: Computational Mathematics
Keywords: KdV equation Stability Solitary wave
CLC: O241.84
Type: Master's thesis
Year: 2005
Downloads: 135
Quote: 2
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Abstract

In this paper, Solving the KdV equation U _t UU _x EU _xxx = 0 the two new difference scheme, called: LaX difference scheme and Du Fort-Frankel difference scheme. This paper describes the structure of these two difference schemes analyzed LaX difference scheme and Du Fort-Frankel difference scheme truncation error, the order of these two difference schemes are o (T ^{2 h < sup> 2 ), is the second-order accuracy, and References [1] in Li Yi and Li Xun done, \), is an order accuracy, apparently constructed herein difference scheme precision than Li Yi and Li Xun article made higher precision. Besides these two differential formats iterative matrix are five pairs of diagonal matrix. This paper also analyzes the LaX difference scheme and Du Fort-Frankel difference scheme for compatibility and stability, in terms of compatibility, the two differential formats and are compatible with the original differential equation approximation; both in terms of stability kind of difference schemes are absolutely stable, while Li Yi and Li Kaoru do differential format is conditionally stable, so this construct two new difference scheme is superior in terms of stability Li Yi and Li Xun in the reference literature [1] to do a differential format. KdV equation U _t UU _x EU _xxx = 0 is used to describe wave propagation and interaction, has a wide range of physical background. As early as 1965 Zabusky and Kruskal on the question of how to solve the problems KdV equation, 1976 Greig and Morris also discussed this issue, but because they do not know how to handle nonlinear terms in the equation, they are just doing their format some theoretical analysis. This paper LaX difference schemes and Du Fort-Frankel difference scheme in addition to doing the above theoretical analysis also did some numerical experiments. In this paper, Li Yi and Li Xun in reference [1] to do the numerical examples, for example, do a numerical experiment, and references [1] compared the numerical results, the paper better reflect wave stability. The article also made based on the numerical results of the three-dimensional wave propagation from the chart, we can clearly see a more intuitive structure LaX herein difference scheme and Du Fort-Frankel difference scheme is stable for a long time. In order to better describe the wave propagation behavior, the paper also made based on the numerical results of wave propagation in two-dimensional map, from the two-dimensional figure, we can see that in the case of different time catching waves.}

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Numerical Analysis > The numerical solution of differential equations, integral equations > Stability theory of differential equations
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