Re: Boundary conditions in implicit methods


In article <1177592226.766384.138610@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mark.t.douglas@xxxxxxxxx writes:
Hi everyone,

I am trying to apply an implicit method to solving a PDE but I am
having grief seeing how to apply continuous boundary conditions. I
have an excellent reference for the heat equation with fixed
boundaries, so I will work with that in this post, to avoid notation
soup.

To solve the heat equation with fixed boundaries implicitly is to
solve

/ \/ \
| 1+2a -a 0 ... 0 ||U_{-N+1}|
| -a 1+2a -a ... 0 ||. |
| 0 -a 1+2a ... 0 ||. | =
| . . . ... . ||. |
| . . . ... -a ||. |
\ 0 0 0 ... -a 1+2a /\U_{N-1} /



/ \ / \
|u_{-N+1}| | U_{-N}|
=| . | | 0 |
| . | +a| . |
| . | | . |
| . | | 0 |
\ u_{N-1}/ \ U_{N} /

for U. a is dt / (dx^2), u is the old value of U, and the subscripts
range from -N to N. This equation can be written in matrix form as M U
= u + b. The boundary conditions are contained in b; as the boundaries
are fixed U_{-N} and U_{N} are known for all times, and as we have
removed two unknowns we can knock two rows out of M. To find U one
simply uses ones favourite tridiagonal system solver.

My question is this. Continuous boundaries are boundaries such that
U_{N} = U_{N-1} and U_{-N} = U{-N+1} for all times. How do I change
you mean u here?
I read ths as "no boundary flux" or "von neumann homogeneous
boundary conditions". in this case you can add two fictitious points,
one at the left, one at the right.
add the discretization of the differential equation on the
boundary line (makes your matrix two rows/columns larger.
then eliminate formally the fictitious point:
gives the first row
1+2a, -2a, 0 ....... 0
and the last row
0,..0, -2a 1+2a .
then solve. unsymmetry is no problem here, the matrix keeps its
property "M".
hth
peter



the system to reflect the change in boundary conditions?

.

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