Arakawa's C grid

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

I'm stuck with how to define $u$

given

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

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

How do I calculate $u$?

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

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

Originally Posted by davefulton
Hi all,

I'm stuck with how to define $u$

given

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

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

How do I calculate $u$?

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

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

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

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

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

Thank you for your help