New Finite Difference Schemes For Two Nonlinear Equations

In this paper, we generalized some results about the Regularized Long- wave(RLW) equation which people have got, and gave four finite difference schemecontaining parameter η ≥ 0 for it based on its conserving theorems. All of themhave error of O (τ~2 + h~2)(Where τ is time step and h is space step.). They haveadvantages that there are discrete energy which are conserved. Their convergencesand stabilities of difference solutions were proved. Numerical experiment resultsdemonstrate that the precision of the new schemes with suitable η ≥ 0 are betterthan those schemes existed, and the new schemes are particularly attractive whenlong time solutions are sought. At the same time, a new two level implicit scheme which has a truncation errorof O(τ~2 + h~2) is presented for solving Burgers Equation. An alternating segmentimplicit (ASI) method is proposed and its unconditional linear stability is proved. TheASI method is suitable for parallel computing and avoids numerical oscillation.Though the error of ASI method is not O (τ~2 + h~2) , it is set off even better betweenthe two adjacent level, thus the precision of the ASI method is not debased obviously,even more better at some dot. A numerical example shows the method has goodapplicability and high accuracy.