# Arakawa's C grid

• Jan 31st 2010, 10:13 AM
davefulton
Arakawa's C grid
Hi all,

I'm stuck with how to define $\displaystyle u$

given

$\displaystyle \frac {\partial u}{\partial x}\approx \frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x}$

say I'm incrementing $\displaystyle \Delta x$ and $\displaystyle \Delta y$ at some defined value. I should have a 2D grid with $\displaystyle x_{i}$ and $\displaystyle y_{j}$ down either side and the central values will be $\displaystyle u_{i,j}$.

How do I calculate $\displaystyle u$?

Is it simply $\displaystyle u_{i,j}=\frac{\partial x_{i}}{\partial t_{n}}+\frac{\partial y_{j}}{\partial t_{n}}$

where $\displaystyle t_{n}$ is defined as $\displaystyle n\Delta t$
• Jan 31st 2010, 01:07 PM
HallsofIvy
Quote:

Originally Posted by davefulton
Hi all,

I'm stuck with how to define $\displaystyle u$

given

$\displaystyle \frac {\partial u}{\partial x}\approx \frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x}$

say I'm incrementing $\displaystyle \Delta x$ and $\displaystyle \Delta y$ at some defined value. I should have a 2D grid with $\displaystyle x_{i}$ and $\displaystyle y_{j}$ down either side and the central values will be $\displaystyle u_{i,j}$.

How do I calculate $\displaystyle u$?

Is it simply $\displaystyle u_{i,j}=\frac{\partial x_{i}}{\partial t_{n}}+\frac{\partial y_{j}}{\partial t_{n}}$

where $\displaystyle t_{n}$ is defined as $\displaystyle n\Delta t$

I'm not sure what you want here. u is simply some function that is defined at each node of grid. $\displaystyle u_{i,j}$ is the value of that function at the i,j-node. It is NOT a derivative, it is the function itself. $\displaystyle \frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x}$ is an approximation to $\displaystyle \frac {\partial u}{\partial x}$ on that grid and, similarly, $\displaystyle \frac{u_{i,j+1}-u_{i,j-1}}{2\Delta y}$ is an approximation to $\displaystyle \frac {\partial u}{\partial y}$.
• Jan 31st 2010, 02:38 PM
davefulton
$\displaystyle u$ is a velocity component. As is $\displaystyle v$. I'm using to solve Euler's equations.

$\displaystyle \frac{\partial u}{\partial x}\approx \frac{u_{i+1,j}-u_{i,j}}{\Delta x}$

$\displaystyle \frac{\partial v}{\partial y}\approx \frac{v_{i,j+1}-v_{i,j}}{\Delta y}$