I have run the different schemes with varying diffusion coefficient D and
advection velocitiy u, and tested the relaxation parameters w=1.0 (no relaxation)
w =1.1, 1.5, 0.9 and 0.5. I have used the linear FEM scheme (tuneint =0.333),
mesch points 64, timestep 0.5 and I hav e only used the Gaussian.
It seems like as long as D and u are approximaely the same size, the
unrelaxed schemes are the best. When u is much smaller than D, the slightly
over relaxed scheme is a little bit better, but the difference is small.
When u is larger than D, the underrelaxted schemes converge a little faster
than the unrelaxed ones, and when u is sufficiently large, only the underrelaxed
schemes work at all ( SSOR D=1, u=5). For some values of D and u, underrelaxation
cuts the number of iterations in half (SSOR D=0.05 u=2, D=1.5, u=16 ).
According to Numerical Recepies only over relaxation schould ever increase
the convergence rate.
Somewhat surprisingly, the Jacobi scheme converges as fast, or almost
as fast as the other unrelaxed schemes. For some combinations of u and
D , BGS require more iterations than FGS (D=1, u=1, w=1). SSOR typically
requires almost half as many iterations than the other schemes, which is
natural, since in SSOR one makes first a FGS and then a BGS iteration.
FGS is tough sometimes as fast as SSOR.
It is very difficult to compare the schemes, since the number of iterations
required to solve this problem are so few. The differences might show for
more slowly converging problems. The only difference in th BGS and FGS,
is which half of A is used for the updated part of x. Since A is not symmetrical
(if u is not zero), the difference in required iterations are due to the
increasing assymetry of A. When u is larger the elements in the under-diagonal
part of A becomes smaller, and the elements in the upper-diagonal part of
A becomes larger. A becomes less diagonally dominant and the number of
iterations required increases. According to litterature, the effect of
relaxation on convergence is only beneficial after ~N iterations, and this
is never seen here, so the stabilising effect of the under relaxation when
u is much larger than D, must be due to it preventing the first iterations
from giving a too large error.
Here
is the A- matrix (equation 73, Af(t+dt)=Bf(t))