Finite difference method for the diffusion-advection equation

[LogarithmLuke]

Homework Statement

Consider the diffusion-advection equation given by $\mu \Delta u + \mathbf{v} \cdot \nabla u = f \ \text{in} \ \Omega$ with some appropiate boundary conditions. Here the velocity field $\mathbf{v} : \Omega \rightarrow \mathbb{R}^2$ and the source function $f: \Omega \rightarrow \mathbb{R}$ may depend on the position. Here $\Delta$ is the Laplace operator, and $\nabla$ is the gradient.

Solve the problem on a unit square with Dirichlet boundary conditions. Set up a finite difference scheme using central differences for both the first and second derivatives of $u$. Implement the scheme. Write your program so it can solve the problem for different choices of $\mathbf{v}$

Relevant Equations

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So for my scheme I obtained $\frac{\mu}{h^2} U_{p}+(\frac{v_{1}}{2 h}-\frac{\mu}{h^2})U_{E}+(\frac{v_{2}}{2 h} - \frac{\mu}{h^2})U_{N} - (\frac{v_{1}}{2 h}+\frac{\mu}{h^2})U_{W} - (\frac{v_{2}}{2 h} + \frac{\mu}{h^2})U_{N} + \tau = f$ however I am not sure this is correct. I am quite new to the finite difference method, and honestly find it quite difficult. I am also not sure how to go about setting up the A matrix and progressing onward to solve the rest of the problem. Any help would be greatly appreciated.

 

[Chestermiller]

You have N, W, and E. What happened to S?

I assume that this is a 2D problem, correct? You should be using subscript indexes of I and J on U, not N, S, E, and W.

 

[LogarithmLuke]

Yes this is a 2D problem. The last $U_N$ should have been a $U_S$. Ok so using subscript indicies i and j instead I obtain the following scheme:

$4\frac{\mu}{h^2} U_{i,j}+(\frac{v_{1}}{2 h}-\frac{\mu}{h^2})U_{i+1,j}+(\frac{v_{2}}{2 h} -\frac{\mu}{h^2})U_{i,j+1} - (\frac{v_{1}}{2 h}+\frac{\mu}{h^2})U_{i-1,j} - (\frac{v_{2}}{2 h} + \frac{\mu}{h^2})U_{i,j-1} + \tau = f$ where $\tau$ is an error term.

Does this seem correct? I realized I had forgotten a factor 4 when I recalculated it.

 

[Chestermiller]

I think you have the signs wrong on the diffusion terms. I haven't checked the advection terms yet. Are you assume that the two velocity components are independent of x and y? The f term should be subscripted i,j unless f is constant.

 

[LogarithmLuke]

Im not familiar with what the diffusion terms and advection terms of the equation are. I did not think about whether the velocity components were independent of x and y, I simply dotted the components with the nabla vector for u, where I replaced the partial derivatives with central differences.

 

[Chestermiller]

The signs of the terms involving $\mu$ are wrong.

 

[LogarithmLuke]

So this is the correct scheme? $-4\frac{\mu}{h^2} U_{i,j}+(\frac{v_{1}}{2 h}+\frac{\mu}{h^2})U_{i+1,j}+(\frac{v_{2}}{2 h} +\frac{\mu}{h^2})U_{i,j+1} - (\frac{v_{1}}{2 h}-\frac{\mu}{h^2})U_{i-1,j} - (\frac{v_{2}}{2 h} - \frac{\mu}{h^2})U_{i,j-1} + \tau = f_{ij}$ How would I transition further to set up the pentadiagonal(?) matrix A to solve the problem?

 

[Chestermiller]

So this is the correct scheme? $-4\frac{\mu}{h^2} U_{i,j}+(\frac{v_{1}}{2 h}+\frac{\mu}{h^2})U_{i+1,j}+(\frac{v_{2}}{2 h} +\frac{\mu}{h^2})U_{i,j+1} - (\frac{v_{1}}{2 h}-\frac{\mu}{h^2})U_{i-1,j} - (\frac{v_{2}}{2 h} - \frac{\mu}{h^2})U_{i,j-1} + \tau = f_{ij}$

. Looks OK.

How would I transition further to set up the pentadiagonal(?) matrix A to solve the problem?

You define a solution vector $y_k$, and then associate the U values with this vector in order, row-by-row or column-by-column.