and φ represents the variables (η, *u*α, *v*α). The finite-difference operator defini-

tions in Equation 27 corresponding to the *x *and *y *components of the momentum

equation are centered at (*i*+1/2,*j*) and (*i,j*+1/2), respectively.

The finite difference formulation of the continuity equation (Equation 21)

yields an algebraic equation that is explicitly solved for η at all grid points. The

formulation for the *x *and *y *momentum equations (Equations 22 and 23) have

been decoupled by placing the *v*xt, *v*yt, and *v*xyt terms on the right-hand side of the

This reduces the momentum equations to tridiagonal equations for *u*α and *v*α

along lines in the *x *and *y *direction, respectively. Tridiagonal matrices are much

easier to store and solve than the large sparse matrix equation that would be

obtained if the equations were not decoupled. The major disadvantage of this

approach, however, is that the iterative step takes longer to converge for shorter-

period waves propagating at large angles to the grid where the higher-order

derivative terms (*u*xxt, *v*yyt).

The numerical scheme is stable provided that the Courant number, *C*R, is less

than 1, i.e.,

2 2 1

1

=

<

1

(28)

∆*x*

∆*y *

where *C *is the phase speed based on the average zero-crossing period of the

incident waves. It is, however, recommended that the Courant number be kept

within the range 0.5 to 0.7 since nonlinear wave-wave interactions, wave break-

ing, and the presence of reflected waves can affect the stability criterion of the

numerical model.

To solve the governing equations, appropriate boundary conditions have to

be imposed at the boundaries of the computational domain. This requires specifi-

cation of waves propagating into the domain and the absorption of waves propa-

gating out of the domain. The equations have also been modified to simulate

wave interaction with fully/partially reflecting structures within the computa-

tional domain. The types of boundaries considered in BOUSS-2D include:

15

Chapter 3 Numerical Solution

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