Chapter 4: Basic Aspects of Discretization

Finite Difference Method

用 \(u_{i,j}\) 表示 \((i, j)\) 点的速度分量

\[ \begin{equation} u_{i+1, j} = u_{i, j} + \left. \frac{\partial u}{\partial x} \right|_{i, j} \Delta x + \left. \frac{\partial^2 u}{\partial x^2} \right|_{i, j} \frac{\Delta x^2}{2} + \left. \frac{\partial^3 u}{\partial x^3} \right|_{i, j} \frac{\Delta x^3}{6} + \cdots \end{equation} \]

一阶精度向前差分：

\[ \begin{equation} \left. \frac{\partial u}{\partial x} \right|_{i, j} = \frac{u_{i+1, j} - u_{i, j}}{\Delta x} + \mathcal{O}(\Delta x) \end{equation} \]

二阶精度中心差分：

\[ \begin{equation} \left. \frac{\partial u}{\partial x} \right|_{i, j} = \frac{u_{i+1, j} - u_{i-1, j}}{2 \Delta x} + \mathcal{O}(\Delta x^2) \end{equation} \]

精度更高，但需要更多信息