Higher-order approximation of staggered-grid finite difference

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Higher-order approximation of staggered-grid finite difference

To approximate the 1st-order derivatives as accurate as possible, we express it in the following

$\begin{displaymath}\begin{split}\frac{\partial f}{\partial x}=&a_1\frac{f(x+\Del... ...x+5\Delta x/2)-f(x-5\Delta x/2)}{\Delta x}+\cdots\\ \end{split}\end{displaymath}$

(32)

where

. Substituting the

and

with (29) for $h=\Delta x/2,3\Delta x/2, \ldots$ results in

$\begin{displaymath}\begin{split}\frac{\partial f}{\partial x}=&c_1\left(\Delta x... ...4+\cdots)\frac{\partial^5 f}{\partial x^5}+\cdots\\ \end{split}\end{displaymath}$

(33)

Thus, taking first

terms means

$\begin{equation*}\left\{ \begin{split}a_1+a_2+a_3+\cdots+a_N&=1\\ a_1+3^2a_2+5^2... ...a_2+5^{2N-2}a_3+\cdots)+(2N-1)^{2N-2}a_N&=0\\ \end{split} \right.\end{equation*}$

(34)

In matrix form,

$\displaystyle \underbrace{ \begin{bmatrix}1 & 1 & \ldots & 1\\ 1^2 & 3^2 & \ldo... ...= \underbrace{ \begin{bmatrix}1\\ 0\\ \vdots\\ 0\\ \end{bmatrix} }_{\textbf{b}}$

(35)

The above matrix equation is Vandermonde-like system: $\textbf{V} \textbf{a}=\textbf{b}$ , $\textbf{a}=(a_1, a_2,\ldots, a_N)^T$ . The Vandermonde matrix

$\displaystyle \textbf{V} = \begin{bmatrix}1 & 1 & \ldots & 1\\ x_1 & x_2 & \ldo... ... & \ddots & \vdots\\ x_1^{N-1} & x_2^{N-1} & \ldots & x_N^{N-1}\\ \end{bmatrix}$

(36)

in which

, has analytic solutions. $\textbf{V} \textbf{a}=\textbf{b}$ can be solved using the specific algorithms, see Bjorck (1996). And we obtain

$\displaystyle \frac{\partial f}{\partial x}=\frac{1}{\Delta x}\sum_{i=1}^N c_i(f(x+i\Delta x/2)-f(x-i\Delta x/2)+O(\Delta x^{2N})$

(37)

The MATLAB code for solving the 2N-order finite difference coefficients is provided in the following.

$\begin{lstlisting} function c=staggered_fdcoeff(NJ)N=NJ/2; x=zeros(N,1); b=zer... ...1 b(i)=b(i)-b(i+1); end end for k=1:N c(k)=b(k)/(2*k-1); end \end{lstlisting}$