How Can I Implement Non-linear Boundary Conditions in a 1-D PDE System?

[santhoo24]

Hi all,

I am Santosh, a grad student at FSU. I am trying to solve a system of 1-D pde's using finite difference scheme.
Here's a brief description of my boundary conditions:

Let my variables be S, T, V, & W, and k1, k2, k3, k4, k5, a, b, and c are constants. At x = 1,dA/dX = Ra,

where A = S, T, V, W, and Ra = k1/{k2 + (k3/T^a) + (k4/S^b) + (k5/V^c)}

Previously, I had a no-flux boundary conditions, which I could solve using the BTCS technique. Now, my boundary conditions are a non-linear function of the variables. I have a a term similar to Ra as a source term. I solve the Laplacian part first and use the solution to solve the source term by forward Euler technique.

But with the non-linear term in the BC, I am confused as to how I can implement the BC. I was wondering if someone can help me in this regard. Any suggestion s including a different technique are much appreciated.

Thanks.
Warm regards,
DSK.

 

[tanujkush]

you need to post your problem statement (i.e. the PDE you are solving) and the boundary condition more clearly

 

[santhoo24]

The following is the description of the problem. I am trying to solve the system of pde's in 1-D. My system of equations consist of four variables as I mentioned in my previous post, namely, S, T, V, and W. The following are the system of equations:
dA/dt = d^2/dX^2 + RA
Where, d is the partial, A is one of S, T, V, and W, and
RA= k1/{k2 + (k3/T^a) + (k4/S^b) + (k5/V^c)},
k1, k2, k3, k4, k5, a, b, and c are constants.

The initial conditions are A(X,0) = constant
BC: At X = 0, DA*dA/dX = 0 for A = S, T, V, and W

At X = 1, DS*dS/dX = C*RA, where C is a constant
DT*dT/dX = -C*RA
DV*dV/dX = -C*RA
DW*dW/dX = N*(W0 - W) - C*RA, where N and W0 are constants.

I am trying to solve for steady state solution. Previously, I had no flux at both the boundaries. I used to solve the system of equations using time-split method, solving the Laplacian part by BTCS finite difference scheme and then using this intermediate solution to solve the source term part by forward Euler method. This approach worked perfectly.
The way I implemented the BC was to solve for v(M) as a function of v(M-1) and then substituting for v(M) in v(M-1). But now, I can not use this approach as I am unable to solve for v(M) as a function of v(M-1). Here v is the discrete form of the variables and not one of the variables, V.

I am looking forward to any suggestions that can assist me in solving the problem. Thank you for your time for helping me out.

Warm regards,
Santosh.

 

[santhoo24]

Hello all,

I was expecting a suggestion from anyone that would help me solve my problem.

Regards,
DSK.