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This is the procedure to derive formulations for Compact finite difference schemes.

First, assume the stencil and the general structure of the scheme. For the classical Padé scheme you have:

$$\alpha f'_{n+1} + f'_{n} + \alpha f'_{n-1} = a \dfrac{f_{n+1} - f_{n-1}}{2 \Delta x} \qquad(1)$$

You write the Taylor series for the derivatives and the functions, e.g.,

$$f'_{n+1}=f'_{n} + \Delta x f''_n + \dfrac{(\Delta x)^2}{2!} f^{(3)}_n + \cdots$$

You then replace the derivatives and the functions in Eq. (1) with their Taylor series. The system of equations to solve consists of the following equations:

• Set $$f'_{n}$$'s and $$f_{n}$$'s coefficient equal to 1.
• Start with coefficients of the lowest powers of $$\Delta x$$ and set them to be equal to zero.