Linear stationary system of 2 partial DEs, numerical implementation

[motorhue]

Hello.
I have a fluid in a rectangular basin, driven by stokes drift at the southern wall. The problem is formulated as follows:

$$\begin{align*} -U_y+V_x &= g_1(x,y) \\ U_x+V_y &= g_2(x,y) \\ U(x=0)=U(x=M)&=V(y=0)=V(y=N) = 0 \end{align*}$$

Here, g_1 and g_2 are known functions of the horisontal coordinates x and y. M and N are east and north boundary, subscript denotes derivatives.

My question is how can I implement this numerically? If I reduce to one variable I get a problem implementing the boundary conditions on the other variable. I believe it should be doable with a rather straightforward MATLAB routine?

 

[jvicens]

Hello,

Thank you for sharing your problem with us. Implementing this numerically can be done using a variety of methods, but I would recommend using finite difference methods. Here is a possible approach:

1. Discretize the domain into a grid with M+1 points in the x-direction and N+1 points in the y-direction. This will give you (M+1) x (N+1) points in total.

2. Define the grid spacing in the x and y-directions as dx and dy, respectively.

3. Use the finite difference approximation to discretize the equations at each grid point. For example, at the point (i,j) on the grid, the first equation becomes:

\begin{align*}
-\frac{U_{j+1,i}-U_{j,i}}{dy} + \frac{V_{j,i+1}-V_{j,i}}{dx} &= g_1(i*dx,j*dy)
\end{align*}

Similarly, the second equation becomes:

\begin{align*}
\frac{U_{j,i+1}-U_{j,i}}{dx} + \frac{V_{j+1,i}-V_{j,i}}{dy} &= g_2(i*dx,j*dy)
\end{align*}

4. Use these discretized equations to form a system of linear equations. This will give you a system of (M+1) x (N+1) equations with (M+1) x (N+1) unknowns (U and V at each grid point).

5. Apply the boundary conditions by setting U=0 and V=0 at the boundaries (i.e. for i=0, M and j=0, N).

6. Solve the resulting system of linear equations using a standard linear algebra solver, such as the backslash operator in MATLAB.

This should give you a numerical solution for U and V at each grid point. You can then plot the results to visualize the flow in the basin.

I hope this helps. Best of luck with your implementation!