# 4th order Padé scheme formula derivation

I am trying to derive the formula of the 4th order Padé scheme that passes through the points $x_i$, $x_{i-1}$ and $x_{i+1}$

$$\Big(\frac{\partial\phi}{\partial x} \Big)_i = -\frac{1}{4}\Big(\frac{\partial\phi}{\partial x} \Big)_{i+1}-\frac{1}{4}\Big(\frac{\partial\phi}{\partial x} \Big)_{i-1} + \frac{3}{4}\frac{\phi_{i+1}-\phi_{i-1}}{\Delta x}$$

To achieve this I am thinking of solving it using linear algebra, Gauss elimination, to find the coefficients $a_i$

$$A=\pmatrix{ 1 & x_{i-1} & x_{i-1}^2 & x_{i-1}^3 &x_{i-1}^4 \\ 1 & x_{i} & x_{i}^2 & x_{i}^3 &x_{i}^4 \\ 1 & x_{i+1} & x_{i+1}^2 & x_{i+1}^3 &x_{i+1}^4 \\ } ~~~~~~ x=\pmatrix{ a_0\\a_1\\a_2\\a_3\\a_4 } ~~~~~~ y=\pmatrix{ \phi_{i-1} \\ \phi_{i} \\ \phi_{i+1} }$$

once I have the coefficients I can use them to find the derivatives in $x_i$, $x_{i-1}$ and $x_{i+1}$ and try to get the formula. I think this is a lengthy procedure, is this a good approach to derive the formula? Is there a better way (faster) to do it?

• Is this a homework problem? – David Ketcheson Apr 2 '13 at 22:48
• Yes it is some sort of homework. I am not asking for the prove here though, I'm just asking if my approach is a good one or if there is a better way to do it. – BRabbit27 Apr 3 '13 at 7:26

## 2 Answers

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 coefficients on both sides to be equal. Here we have, $$2\alpha + 1 = a$$

• $f_n$ coefficient could only appear on the right hand side and its coefficient must be set equal to zero. In this case, this does not give us any equations.

• Start with coefficients of the lowest powers of $\Delta x$ and set them to be equal, until you get enough equations, e.g., $$a = 2 \dfrac{3!}{2!} \alpha$$

I would think a Taylor table would be the way to go. Moin's book "Fundamentals of Engineering Numerical Analysis" has good easy to follow examples.

• Ok I'll take a look to that book. Just to be sure, the procedure I was thinking of, was it a true procedure to obtain the Padé's formula? I mean you tell me a Taylor table is the way to go but it doesn't mean my long approach was wrong, right? – BRabbit27 Apr 5 '13 at 23:05
• I don't think it would work, Taylor series is the fundamental tool in deriving these schemes. – Isopycnal Oscillation Apr 6 '13 at 3:26