5 ITERATIVE CG SOLUTION
METHOD
CGWAVE provides the choice of choosing the wavelength dependent resolution
to solve the governing equation. An in-depth investigation was performed to determine
the sensitivity of the solution as a function of wavelength. Based on our analysis, it is
recommended that ten or more points per wavelength to be used. In regions where the
change in topography or wave-amplitude is rapid, a finer grid is necessary. To solve the
elliptic mild-slope equation without approximating the physics (as in parabolic models),
the problem has to be solved simultaneously over the entire domain. This leads to a very
large linear system of equations (e.g. Equation 69). Direct methods (e.g. Gaussian
elimination) are often inapplicable due to extremely large storage requirement of the
matrix [A]. Iterative methods, on the other hand, require memory for only the non-zero
elements in [A]. Since [A] is highly sparse, iterative methods can significantly enhance the
ability of the elliptic type mild-slope equation models to handle large domain problems.
However, most iterative procedures require [A] to be diagonally-dominant or symmetric
and positive-definite. Unfortunately, the coefficient matrix [A] is not diagonally-dominant,
nor symmetric and positive definite. Panchang et al. (1991) recommended that the
following Gauss transformation is applied to Equation 69 :
[ ]A ]{η}= [ ] f }
A[ $
A {
*
*
(142)
where A* is the complex conjugate transpose of [A]. Xu (1995) has shown that the new
coefficient matrix [A*][A] is Hermitian and positive-definite. Therefore, the modified
conjugate-gradient method, which is often several orders of magnitude faster than many
other schemes, including the traditional conjugate gradient scheme, is guaranteed to
converge when applied to Equation 125. The algorithm is implemented in CGWAVE as
42
Previous Page
