Numerical boundary conditions for wide approximation finite difference

[amalak]

Hi,

I have to use a wide 5 point stencil to solve a problem to fourth order accuracy. In particular, the one I'm using is:

u'' = -f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x - 2h) / 12h²

or when discretized

u'' = -U_j-2 + 16U_j-1 -30U_j + 16U_j+1 -U_j+2 / 12h²

In addition to dirichlet boundary conditions (which are not troubling me to implement), I have to implement numerical boundary conditions for

U_-1 and U_N+1

The problem I'm encountering is I'm not sure what to try for these numerical boundary conditions (as in, I haven't a clue as to what may work). I have the scheme set up without those conditions, but that's not what I want. The only time I know U_-1 and U_N+1 come up are with Neumann boundary conditions, which I don't have.

Any help or pointers would be immensely appreciated, thank you.

 

[AlephZero]

Your Dirichlet boundary conditions are at $u_0$ and $u_N$, right?

I would create an unsymmetrical fourth order approximation for $u''_1$ in terms of $u_0, u_1, u_2, u_3, u_4$ and simiilarly for $u''_{N-1}$.

Then, you don't need $u_{-1}$ and $u_{N+1}$.

For example see section 11.3.4 of http://www.rsmas.miami.edu/personal/miskandarani/Courses/MSC321/lectfiniteDifference.pdf

 

[amalak]

Thank you very much for your response. That method seems to make more sense, I think, but I've been instructed to use U_-1 and U_N+1, but thank you again.

 

[AlephZero]

OK, so they probably want your numerical solution satisfy the differential equation at the boundary points $u_0$ and $u_N$, not just to satisfy the Dirichlet boundary conditions at the boundary points.

So, make finite difference approximations for the derivatives at $u_0$ using $u_{-1}, u_0,. u_1, \dots$ and plug them into the differential equation, and similarly at $u_N$. That will give you two more equations, so you have enough equations to solve for $u_{-1}, u_0,\dots, u_N, u_{N+1}$.