对流扩散反应方程基于坐标变换的高阶紧致差分格式

A High-order Compact Difference Scheme Based on the Coordinate Transformation for the Convection Diffusion Reaction Equation

基于非均匀网格,提出了一种求解一维定常对流扩散反应方程的高精度紧致差分格式。首先采用坐标变换方法将原方程由物理空间的非均匀网格转换为计算空间的均匀网格,然后给出一阶导数和二阶导数在均匀网格上的中心差分逼近式,并结合变换后的方程,得到了定常对流扩散反应方程具有四阶精度的紧致差分格式。最后,通过数值算例验证了该方法的精确性和高分辨率的特点。数值实验结果表明,对于所研究问题,该方法较不进行坐标变换而直接在物理域上建立的非均匀网格上的高阶紧致格式具有更高精度。

A high accurate compact finite difference scheme based on the nonuniform grid is proposed to solve the one dimensional （1D） convection diffusion reaction equation. First, the original equation is transformed from the nonuniform grid of the physical space to the uniform grid of the computational space by using the coordinate transformation method; then the central difference approximate expressions for the 1st-order and 2nd-order derivatives are constructed on the uniform grids. After that, a fourth-order compact difference scheme for the 1D convection diffusion reaction equation is got by u- sing the original equation. At last, the numerical experiments are conducted to show the high accuracy and high resolution of the present method. And the computed results show that the present scheme is more accurate than the high-order compact transforma- tion-free method on the nonuniform grids.