Alternative boundary conditions - Thomas-algorithm

[Huibert]

Alternative boundary conditions -- Thomas-algorithm

Hello,

I have to solve a diffusion equation:
MatrixL * Csim(:,i+1) = MatrixR * Csim(:,i) + BoundaryConditions
where Csim = concentration, j = location, i = time.

Boundary conditions are of type Dirichlet (Csim = 5 at j = 1, Csim = 0 at j = end). So I used:

MatrixL(1,:) = (1 0 0 . . 0) 
MatrixR(1,:) = (0 0 0 . . 0) 
BoundaryConditions(1) = (5)

so
Matrices, first line: (1 0 . 0) * Csim(:,i+1) = (0 0 . 0) * Csim(:,i) + 5
Matrices, last line: (. . 0 1) * Csim(:,i+1) = (. . 0 1) * Csim(:,i) + 0

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To solve this problem with the Thomas Algorithm, I have to write the equation as

MatrixL2*Csim(:,i+1)  = Csim(:,i)

So MatrixL2 = inv(MatrixR)*MatrixL

However, it is not possible to calculate inv(MatrixR) when MatrixR(1,:)= MatrixR(end,:)= 0

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So I tried to describe the boundary conditions in the folloing way :

Matrices, first line: (1 0 0 . .) * Csim(:,i+1) = (1 0 0 . .) * Csim(:,i)
Matrices, last line: (. . 0 0 1) * Csim(:,i+1) = (. . 0 0 1) * Csim(:,i)

But that cat won't jump. So could you please help me to find what's wrong with this?

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Because the problem is a littlebit complicated to explain, a longer description is in the attachment.

 

[KataKoniK]

Thank you for sharing your problem and asking for assistance. I am a scientist who specializes in numerical methods for solving differential equations, and I have some suggestions for alternative boundary conditions to use with the Thomas Algorithm in your diffusion equation problem.

Firstly, I would like to point out that the Thomas Algorithm is specifically designed for tridiagonal systems, where only the main diagonal and the two adjacent diagonals have non-zero elements. In your case, your system is not strictly tridiagonal due to the Dirichlet boundary conditions at j=1 and j=end. Therefore, the Thomas Algorithm may not be the most efficient or accurate method to use in this case.

One alternative boundary condition that you could try is to use a ghost node approach. This involves extending your domain by one node on each end, and treating the boundary nodes as ghost nodes that have the same concentration as the adjacent interior node. This way, your system becomes tridiagonal and the Thomas Algorithm can be used effectively. However, this may not be the most physically accurate representation of your problem, so it is important to carefully consider the implications of using this approach.

Another option is to use a different numerical method altogether, such as the Crank-Nicolson method, which is better suited for non-tridiagonal systems and can handle a wider range of boundary conditions. It may also be worth exploring if there are any specific methods or techniques that have been developed for solving diffusion equations with Dirichlet boundary conditions.

In any case, it is important to carefully assess the accuracy and stability of your chosen method, and to consider the physical implications of your boundary conditions. I hope this helps and good luck with your problem!