解三维抛物型方程的一个高精度显式差分格式

A High-order Accurate Explicit Difference Scheme for Solving Three-dimensional Parabolic Equations

提出了求解三维抛物型方程的一个高精度显式差分格式.首先,推导了一个特殊节点处一阶偏导数(■u)/(■/t)的一个差分近似表达式,利用待定系数法构造了一个显式差分格式,通过选取适当的参数使格式的截断误差在空间层上达到了四阶精度和在时间层上达到了三阶精度.然后,利用Fourier分析法证明了当r<1/6时,差分格式是稳定的.最后,通过数值试验比较了差分格式的解与精确解的区别,结果说明了差分格式的有效性.

An explicit difference schemes with high accuracy for solving three-dimensionalparabolic equations is given. First, a difference approximation expression of the first orderpartial derivative was deduced at a special node; an explicit difference scheme is constructed bythe method of undetermined coefficients, and appropriate parameters are chosen to endow thetruncation error of scheme is fourth-order accurate in space and third-order accurate in time.In turn, the new difference scheme is proved to be stable if r 〈 with the Fourier analysismethod. Finally, the numerical experiment shows the numerical solutions of difference schemesand the exact solutions are matched and the difference scheme is effective.