污染扩散方程高精度紧致格式及其稳定性分析

针对污染扩散方程提出了时间任意阶精度的显式格式,并对该格式的稳定性和精度进行了分析,理论结果表明:一阶精度的计算格式是传统的显格式,其稳定条件为:s≤1/2(s=D.Δt/Δx2,D为扩散系数,Δt为时间步长,Δx为空间步长),随着保留精度阶数的增加,稳定性范围也会随之增大;当保留无穷阶精度时,格式是无条件稳定的。这也就从一个侧面揭示了稳定性与时间精度之间的关系,为高性能数值计算格式的构思提供了可以借鉴的原则。数值算例的结果表明,本文格式具有一定的实用性。

High-order compact scheme and its stability analysis for the pollutant diffusion equation

An explicit numerical method which has arbitrary order of accuracy in time was derived in this paper,including the analysis of the stability and the accuracy of the pollutant diffusion equation.The conventional explicit Forward Time Central Space scheme(FTCS) is the specific form with first order accuracy of the proposed method.The theoretical analysis demonstrated that the stable region would expand along with the increase of the order of accuracy and the proposed explicit method was unconditionally stable when remaining the infinite order of accuracy.Finally,the validity of the proposed method was tested by a numerical example and it is shown that the numerical results agree quite well with the anterior theoretical analysis.And it also reveals the relationship between the stability and the accuracy,and provides the advisory principle for the development of the high-performance computational schemes.