一种求解抛物型偏微分方程的时空高阶方法

A Kind of Space-time High Order Numerical Methods for Parabolic Partial Differential Equations

在对微分系统进行数值求解时,研究者们总希望能够在尽可能短的时间内达到尽可能高的计算精度.考虑一类线性抛物型偏微分方程,首先用最优的二次样条配置法求解此方程,可以得到一个刚性常微分方程系统;再采用一种高阶隐式时间积分方法求解此常微分方程系统.这种混合方法对空间网格尺寸和时间步长均为四阶收敛.通过分析这种混合方法在相邻时间步之间的迭代矩阵的谱半径,可以看出这种方法是稳定的,而且可以避免振荡现象的发生.通过数值算例可以看出,新方法的计算效率明显高于现有的一些高效数值方法,即新方法可以在保持计算精度的前提下大大缩短计算时间,节省计算资源.

It is always expected that high accuracy can be obtained within as short computational time as possible, when solving a system of differential equations. In this paper we consider a kind of linear parabolic partial differential equations（PDEs）. First, opti- mal quadratic spline collocation method is employed for the system, which leads to a stiff system of ordinary differential equations （ODEs）. Then we use a class of high order implicit time integrations for the system of ODEs. The resulting errors at the mesh points of both the space partition and the time partition are fourth order. The hybrid method is unconditionally stable, and immune to spurious oscillations. Numerical experiments are carried out to show that the new method behaves much better than some other e^cient methods. It means that it can save a lot of computational cost for achieving a desired accuracy.