This procedure has been demonstrated to work very well in finite-element models.
In fact, these techniques were found to be far more efficient in the present finite-element
runs than for the finite-difference studies of Xu & Panchang (1993) and Panchang et al.
(1991).
Li (1994) has recently suggested modifications to the iterative schemes of
Panchang et al. (1991) for enhancing the convergence. Comparison between these two
schemes were given in Xu and Panchang (1995). Although these modifications resulted in
faster convergence, it was noticed that, unlike the basic procedures of Panchang et al.
(1991), they do not converge to a solution monotonically. Instead, the residual error
decreases in an oscillatory manner as the iterations proceed. Also, the gain in CPU time
varies from case to case and appears to be problem-specific, but it can save over 50% of
CPU time for some applications. This scheme has also been included in CGWAVE as an
alternative choice, and consequently, there are two types of solvers provided in
CGWAVE.
The solving algorithm is slightly different when non-linear mechanisms are
incorporated into the solution. In this case, the solution is first solved as if there were no
non-linear mechanisms. This solution is then used to prescribe initial conditions for the
non-linear mechanisms, and the system of equations are modified to incorporate the
resulting non-linear mechanisms. CGWAVE then solves using the modified equations.
This iterative method of solving, modifying the system of equations with non-linear
mechanisms, and solving again continues until the resulting solution no longer changes
between iterations.
The solving algorithm is also different when spectral wave conditions are used. In
this case, a solution is first obtained for each individual spectral component. The final
solution is obtained by linear superposition of the solutions for all spectral components.
This superposition is performed after a spectral run is completed, using a post processing
program.
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