Crank-Nicoslon scheme and nonlinear boundary conditions

[AiRAVATA]

Hey guys, I'm having a conceptual problem implementing the Crank-Nicolson scheme to a PDE with nonlinear boundary conditions.

The problem is the following:

$$u_t + u_{xxxx} = 0,$$

$$u(0,t) = 1,\quad u_x(0,t) = 0, \quad u_{xx}(1,t) = 0,$$

$$u_t(1,t) - u_{xxx}(1,t) = f\bigl(u(1,t)\bigr).$$

Taking m points in space and denoting $u(x_j,t^i) = u_j^i$, I have, for the linear part:

$$u_{-1}^i = u_0^i = 1, \quad u_{m+1}^i - 2 u_m^i + u_{m-1}^i = 0,$$

and

$$u^{i+1}_{j-2} - 4 u^{i+1}_{j-1} + \Bigl(6 + \frac{2 k^4}{h}\Bigr) u^{i+1}_{j} - 4 u^{i+1}_{j+1} + u^{i+1}_{j+2} = -u^{i}_{j-2} + 4 u^{i}_{j-1} - \Bigl(6 - \frac{2 k^4}{h}\Bigr) u^{i}_{j} + 4 u^{i}_{j+1} - u^{i}_{j+2}, \quad 0 < j < m,$$

and for the nonlinear part,

$$u^{i+1}_{m-2} - 2 u^{i+1}_{m-1} + \Bigl(\frac{4 k^3}{h} - 2k^3 f'(u_m^i)\Bigr) u_m^{i+1} + 2 u^{i+1}_{m+1} - u_{m+2}^{i+1} = -u^{i}_{m-2} + 2 u^{i}_{m-1} + \Bigl(\frac{4 k^3}{h} - 2k^3 f'(u_m^i)\Bigr) u_m^i - 2 u^{i}_{m+1} + u_{m+2}^{i} + 4k^3f(u_m^i),$$

where k is the step in space and h is the step in time.

Now, I have m+1 equations with m+2 unknowns so, where is my missing equation?

Should I add an artificial "free boundary" condition to the right of the point m, or I should extend the PDE to the right boundary (allowing j = m)?

I'm doing something wrong?

---EDIT---

I suppose that taking $u_{xxx}(1^-,t) = u_{xxx}(1,t)$ will close the sistem correctly by modifying the last equation.

 

[jvicens]

Hello there,

Thank you for sharing your problem with us. It seems like you are on the right track with your implementation of the Crank-Nicolson scheme. However, there are a few things that I would like to point out.

Firstly, you are correct in assuming that you need one additional equation to close the system. This is because you have m+1 equations for m+2 unknowns, as you mentioned. There are two possible approaches to address this issue - adding an artificial "free boundary" condition or extending the PDE to the right boundary.

Adding an artificial "free boundary" condition would mean that you are imposing a condition on u_{m+1}^{i+1}, which is not a physical point in your domain. This is a commonly used technique in numerical methods, and it usually involves extrapolating the solution from the last known point. However, this approach may introduce some errors in your solution.

On the other hand, extending the PDE to the right boundary would mean that you are allowing j = m in your equations. This would give you an additional equation for u_{m+1}^{i+1}, and it would not require any extrapolation. This approach is more accurate but may require some modifications to your equations.

Lastly, I would like to mention that taking u_{xxx}(1^-,t) = u_{xxx}(1,t) will not close the system correctly. This is because u_{xxx}(1^-,t) and u_{xxx}(1,t) are not the same point in the domain. Instead, you can use the fact that u_{xx}(1,t) = 0 to obtain an additional equation for u_{m+1}^{i+1}.

I hope this helps you in implementing the Crank-Nicolson scheme correctly. If you have any further questions, please don't hesitate to ask.