三维热传导方程的高精度有限差分方法

High Order Finite Difference Method for Solving the Threedimensional Heat Equation

提出了数值求解三维热传导方程的一个四阶精度的有限差分格式,首先对三个空间方向上的二阶导数项,采用四次样条函数来近似,从而得到半离散的常微分方程.然后利用常微分方程的解析解表达式,时间矩阵利用Padé近似,得到时间和空间均为四阶精度的差分格式.最后利用方法计算了两个数值算例,并与文献中结果进行了对比,从而验证了高精度格式的性能.

A high order finite difference scheme for solving the three-dimensional heat equation is proposed based on the quartic spline functions for the spatial derivative and （2,2） Pade approximation for the temporal derivative. The scheme is fourth order accuracy in both time and space. The performance of unconditional stability is proved in theory. The accuracy and effectiveness of the present method is validated by two numerical experiments and by comparisons with the computed results in the literature.