扩散方程一种无条件稳定的保正并行有限差分方法

A POSITIVITY-PRESERVING PARALLEL DIFFERENCE SCHEME FOR DIFFUSION EQUATION WITH UNCONDITIONAL STABILITY

本文针对扩散方程提出了一种保正的并行差分格式,并且这个格式为无条件稳定的.我们在每个时间层将计算区域分成许多个子区域以便于实施并行计算.格式构造中首先我们使用前两个时间层的计算结果在分区界面处通过一种非线性的保正外插来预估子区域界面值.然后在每个子区域内部使用经典的全隐格式进行计算.最后在界面处使用全隐格式进行校正(本质上这一步计算是显式计算).我们给出了一维与二维情形下的保正并行差分格式,并相应的给出了无条件稳定性证明.数值实验显示此并行格式具有二阶数值精度,而且无条件稳定性与保正性也均在数值实验中得到验证.

In this work,we propose a positivity-preserving parallel difference scheme for solving diffusion equation,and the scheme is unconditionally stable.On each time level,the domain is decomposed into many sub-domains for parallelism.Firstly we give a nonlinear positivitypreserving extrapolation method to predict the values on sub-domain interfaces using the values on previous two time levels.Then the fully implicit scheme is used with the predicted values for sub-domain problems.Finally the values on inner interface are updated by the fully implicit scheme,which is in fact an explicit calculation with the use of the sub-domain values just obtained.In this work,the schemes are given for both one and two dimension problem,and the unconditional stability is proved.Numerical tests demonstrate that the parallel scheme has second-order accuracy.Moreover,positivity and unconditional stability are also tested in the numerical experiments.