【作者】 王廷春；

【导师】 张鲁明；

【作者基本信息】 南京航空航天大学 ， 计算数学， 2005， 硕士

【摘要】 本文将前人对正则长波方程的数值求解的部分结果进行了简单总结,并基于方程本身的的守恒律为出发点,提出了四个新的守恒差分格式,对格式的二阶精度进行了证明,并运用能量分析[46-48]的方法对格式的收敛性及稳定性进行了分析研究,并通过数值试验与前人的部分研究成果进行了比较,比较表明,本文格式精度的明显好于文[43,44]格式,特别是取适当参数时,精度有较大幅度提高,且由于格式是线性的,从而保持了计算量小的特点,显示出新格式的优越性。另文章对 Burgers 方程首先提出一种新的差分格式,其具有二阶精度,利用该格式结合Saul′yev型非对称格式,文章构造了 Burgers 方程的一种便于并行计算的交替分段隐格式,分段格式的截断误差理论上有所增加,不再是O(τ~2 + h2),但由于其在交替分段过程中,相邻两层的误差有所抵消,因此分段格式的精度没有明显降低,甚至个别点可能有所提高。而且格式线性化绝对稳定并有效地避免了数值振荡。数值实验表明本文格式具有很好的适用性。更多还原

【Abstract】 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.更多还原