Two Kinds Of Difference Schemes For KdV Equation

In this paper, we consider two kinds of new difference schemes for KdV equation. One is Lax difference scheme, the other is Du Fort-Frankel difference scheme. At the same time, we also have detailed and complete introduction for this two kinds of difference schemes. The local truncation errors of Lax difference scheme and Du Fort-Frankel difference scheme are analysed in this paper. The local truncation errors of these two kinds of difference schemes are o(Ï„² + h²), both of which are two order accurate. However the local truncation error of difference scheme in reference document [1] made by Li Yi and Li Xun is o(Ï„ + h), whose precision is one order. We can clearly see that the precision of this two kinds of difference schemes in this paper are higher than that in paper made by Li Yi and Li Xun. Moreover Lax difference scheme and Du Fort-Frankel difference scheme correspond a system of linear equation with pentagonal diagonal matrix, which can be solved by double sweep method if boundary values are given. We also analyse consistency and stability of this two kinds of difference schemes. As to consistency, we have verified that Lax difference scheme and Du Fort-Frankel difference scheme both are consistent with the difference scheme U_t + UU_x + EU_xxx = 0. As to stability, Lax difference scheme and Du Fort-Frankel difference scheme are both absolute stability. While the difference scheme in reference document [1] made by Li Yi and Li Xun is conditional stability. So we can indicate that the stability of Lax difference scheme and Du Fort-Frankel difference scheme are better than that in paper made by Li Yi and Li Xun.The KdV equation U_t + UU_x+EU_xxx = 0 is used to describe wave spreading and mutual action of waves. KdV equation has extensive background in physics and application. In 1965, Zabusky and Kruskal had been put forward the problem of solving KdV equation. Then, in 1976, Greig and Morris discussed the equation of KdV too. But because of they didn't know how to deal with the nonlinear termin KdV equation, they just made a little theoretical analysis for the difference schemes which they constructed. In this paper, we not only discuss the theory of Lax difference scheme and Du Fort-Frankel difference scheme, but also make numerical experiment. The numerical experiment in this paper comes from reference document [1] made by Li Yi and Li Xun. We make a numerical table in order to compare with the numerical table in reference document [1]. By comparing, we indicate that Lax difference scheme and Du Fort-Frankel difference scheme in this paper are better responding the stability of wave. We also make three-dimensional figure and two-dimensional figure. The three-dimensional figure shows that this two schemes have long time stability. From two-dimensional figure, we can see the waves speedup at different time.