求解对流扩散方程的一种高精度紧致差分格式
A High-Order Compact Difference Scheme for Solving Convection-Diffusion Equations

在指数变换的基础上，将对流扩散方程变为扩散方程，消除了数值求解中较难处理的对流项，采用四阶紧致差分方法离散扩散方程的空间变量，采用扩展的$\frac{1}{3} $ -Simpson公式离散时间变量，格式的截断误差为O（τ⁴+h⁴）.理论分析证明该格式是无条件稳定的.通过数值算例验证了本文方法的有效性.
In this paper, based on exponential transform, the convection diffusion equation is transformed into a diffusion equation, thus eradicating the advection term, which is hard to treat in numerical solution. A high-order accurate implicit compact difference scheme is constructed for solving the one dimensional parabolic equation by the fourth-order pade' formula combined with time extension of the $\frac{1}{3} $-Simpson formulas. The truncation error of the scheme is O(τ⁴+h⁴). A theoretical analysis shows that the scheme is unconditionally stable. Numerical experiments verify the accuracy and reliability of the present scheme