FD approximation at internal boundary condition

[venta]

Hi,

I have to solve diffusion-advection PDE using finite difference method. The problem has two regions with different diffusion coefficients and velocities. At the interface between the two regions types of boundary condition :

1. No contact resistance
C1 = C2
- D1*dC1/dx + v1*C1 = - D2*dC2/dx + v2*C2

2. With surface resistance
- D1*dC1/dx + v1*C1 = - h (C2-C1)
- D1*dC1/dx + v1*C1 = - D2*dC2/dx + v2*C2

I am using fully-implicit in time and central-difference in space scheme and Tridiagonal Matrix Algorithm (Thomas's algorithm). However I found problem in doing FD approximaton at this internal interface. I would like to ask anybody who knows how to get the imaginary point for each regions for different types of BC.

Thank you in advance

 

[Jhero]

Hello,

Thank you for reaching out with your question. Solving diffusion-advection PDEs using finite difference methods can be challenging, especially when dealing with different regions and boundary conditions. I can offer some suggestions that may help you with your problem.

First, it is important to make sure that your finite difference scheme is accurate and stable. You mentioned using a fully-implicit in time and central-difference in space scheme, which is a good choice for this type of problem. However, I would recommend checking your discretization and boundary conditions to make sure they are implemented correctly.

In order to handle the interface between the two regions, you can use the concept of ghost points. These are points outside of your physical domain that act as imaginary points to help you approximate the solution at the interface. For example, if you have two regions with different diffusion coefficients and velocities, you can add ghost points to the side of the interface where the diffusion coefficient and velocity change. These ghost points will have the same boundary conditions as the points on the other side of the interface, allowing you to solve the PDE in a continuous manner.

In terms of the different types of boundary conditions, you will need to modify your finite difference scheme accordingly. For the no contact resistance condition, you can simply use central difference approximation at the interface, treating the interface as a regular internal point. For the surface resistance condition, you will need to use a ghost point and modify the finite difference scheme to account for the additional term -h(C2-C1).

I hope this helps and good luck with your problem! If you have any further questions, feel free to reach out.