Iteration process and stopping criteria

Generally, the velocities $c_x$ and $c_y$ have different signs in the geographical domain and hence, more steps are needed. Energy propagates along a wave ray and this cannot be done in 1 iteration as soon as this ray is curved. Wave rays can be curved in coastal waters due to depth changes and ambient current. This enhances the number of iterations as it must cover the propagation of energy density along the whole characteristic curve. The key issue is the maximum number of iterations needed to cover energy propagation across the model domain.


Any wave ray can be divided into a finite number of pieces so that each piece can be covered effectively by one of the four sweeps. This number is related to the directional change of the wave ray. Since these pieces have to be captured sequentially the total number of iterations needed is proportional to the number of pieces. In turn, this number depends on the size of the model domain and the way the waves propagate through the domain.


If the source terms are included then in every wet grid point a balance between refraction, wind input, white capping, wave breaking, etc. within the spectral space is evaluated. On top of this various wave components are marching in the model domain from the open boundaries. It depends on the strength of the source terms on the one hand and propagation throughout the domain and the domain size on the other hand how many iterations are actually needed. As a matter of fact, the total number of iterations depends rather on the local change in wave propagation e.g. in shallow waters, tidal inlets, and around shoals, irrespective of the ordering and sweeping.


Due to refraction and nonlinear wave energy transfer, interactions occur between the directional quadrants. To properly take these interactions into account and the fact that we employ the Gauss-Seidel technique and linearization of the source term (3.32), the quadrant sweeping and the solution of system (3.35) need to be repeated until some convergence criteria are met. The iteration process runs from $s = 1$ to $s = S$ and is terminated if the maximum number of iterations $S$ (usually 50) is reached or the following criteria for the significant wave height $H_{m0}$ and mean relative wave period $T_{m01}$, as given by

  $$H_{m0} = 4\sqrt{m_0}\, , \quad T_{m01} = 2\pi \frac{m_0}{m_1}\, ,... ...nt_{0}^{\infty} \int_{0}^{2\pi} \sigma^j E(\sigma,\theta) d\sigma d\theta \, ,$$ (3.36)

are both satisfied in at least 98% of all wet grid points $(i,j)$:

  $$\frac{\vert\Delta H^{s}_{m0}(i,j)\vert}{H^{s-1}_{m0}(i,j)} < {\v... ...x{or} \quad \vert\Delta H^{s}_{m0}(i,j)\vert < {\varepsilon}^{\rm a}_{\rm H}$$ (3.37)

and

  $$\frac{\vert\Delta T^{s}_{m01}(i,j)\vert}{T^{s-1}_{m01}(i,j)} < {... ...\quad \vert\Delta T^{s}_{m01}(i,j)\vert < {\varepsilon}^{\rm a}_{\rm T} \, .$$ (3.38)

Here, $\Delta Q^s \equiv Q^s - Q^{s-1}$, with $Q$ some quantity. The default values of this limiting criteria are: ${\varepsilon}^{\rm r}_{\rm H} = {\varepsilon}^{\rm r}_{\rm T} = 0.02$, ${\varepsilon}^{\rm a}_{\rm H} = 0.02$ m and ${\varepsilon}^{\rm a}_{\rm T} = 0.2$ s. The rationale behind the use of the integral wave parameters $H_{m0}$ and $T_{m01}$ in the stopping criteria is that these are the output variables typically of interest. The iterative solution procedure is accelerated by calculating a reasonable first guess of the wave field based on second-generation source terms of Holthuijsen and De Boer (1988).


In general, the iterative method should be stopped if the approximate solution is accurate enough. A good termination criterion is very important, because if the criterion is too weak the solution obtained may be useless, whereas if the criterion is too severe the iteration process may never stop or may cost too much work. Experiences with SWAN have shown that the above criteria (3.37) and (3.38) are often not strict enough to obtain accurate results after termination of the iterative procedure. It was found that the iteration process can converge so slowly that at a certain iteration $s$ the difference between the successive iterates, $H^s_{m0} - H^{s-1}_{m0}$, can be small enough to meet the convergence criteria, causing the iteration process to stop, even though the converged solution has not yet been found. In particular, this happens when convergence is non-monotonic such that the process is terminated at local maxima or minima that may not coincide with the converged solution.


Furthermore, it became apparent that, unlike $H_{m0}$, the quantity $T_{m01}$ is not an effective measure of convergence. It was found that the relative error in $T_{m01}$, i.e. $\vert T^s_{m01} - T^{s-1}_{m01}\vert/T^{s-1}_{m01}$, does not monotonically decrease near convergence, but keeps oscillating during the iteration process. This behaviour is due to small variations in the spectrum at high frequencies, to which $T_{m01}$ is sensitive. This behaviour is problematic when any form of stricter stopping criterion is developed based on $T_{m01}$. Therefore, in the improved termination criterion proposed, $T_{m01}$ has been abandoned as a convergence measure and only $H_{m0}$, which displays more monotonic behaviour near convergence, is retained.


Stiffness and nonlinearity of the action balance equation are found to yield less rapid and less monotone convergence. Ferziger and Perić (1999) explain the slow convergence in terms of the eigenvalue or spectral radius of the iteration process generating the sequence $\{{\phi}^0, {\phi}^1, {\phi}^2,\cdots\}$. They show that the actual solution error is given by

  $${\phi}^{\infty} - {\phi}^s \approx \frac{{\phi}^{s+1} - {\phi}^s}{1-\rho} \, ,$$ (3.39)

where ${\phi}^{\infty}$ denotes the steady-state solution and $\rho$ is the spectral radius indicating the rate of convergence. The smaller $\rho$, the faster convergence. This result shows that the solution error is larger than the difference between successive iterates. Furthermore, the closer $\rho$ is to 1, the larger the ratio of solution error to the difference between successive iterates. In other words, the lower the rate of convergence of the iteration process, the smaller this difference from one iteration to the next must be to guarantee convergence. The stopping criterion of SWAN could be improved by making the maximum allowable relative increment in $H_{m0}$ a function of its spectral radius instead of imposing a fixed allowable increment. By decreasing the allowable relative increment as convergence is neared, it would be possible to delay run termination until a more advanced stage of convergence. Such a stopping criterion was used by, e.g. Zijlema and Wesseling (1998). This criterion is adequate if the iteration process converges in a well-behaved manner and $\rho < 1$ for all iterations. However, due to nonlinear energy transfer in spectral space SWAN typically does not display such smooth behaviour. Therefore, this criterion may be less suited for SWAN.


An alternative way to evaluate the level of convergence is to consider the second derivative or curvature of the curve traced by the series of iterates (iteration curve). Since the curvature of the iteration curve must tend towards zero as convergence is reached, terminating the iteration process when a certain minimum curvature has been reached would be a robust break-off procedure. The curvature of the iteration curve of $H_{m0}$ may be expressed in the discrete sense as

  $$\Delta (\Delta \tilde{H}_{m0}^s)^s = \tilde{H}_{m0}^s - 2\tilde{H}_{m0}^{s-1} + \tilde{H}_{m0}^{s-2} \, ,$$ (3.40)

where $\tilde{H}_{m0}^s$ is some measure of the significant wave height at iteration level $s$. To eliminate the effect of small amplitude oscillations on the curvature measure, we define $\tilde{H}_{m0}^s \equiv (H_{m0}^s + H_{m0}^{s-1})/2$. The resulting curvature-based termination criterion at grid point $(i,j)$ is then

  $$\frac{\vert H_{m0}^s(i,j) - ( H_{m0}^{s-1}(i,j) + H_{m0}^{s-2}(i,... ...vert }{2 H_{m0}^s(i,j)}< \varepsilon_{\rm C} \, , \,\,\, s = 3, 4, \cdots \, ,$$ (3.41)

where $\varepsilon_{\rm C}$ is a given maximum allowable curvature. The curvature measure is made non-dimensional through normalization with $H_{m0}^s$. Condition (3.41) must be satisfied in at least 99% of all wet grid points before the iterative process stops. This curvature requirement is considered to be the primary criterion. However, the curvature passes through zero between local maxima and minima and, at convergence, the solution may oscillate between two constant levels due to the action density limiter (see Section 3.7.2), whereas the average curvature is zero. As safeguard against such a situation, the weaker criterion (3.37) is retained in addition to the stricter criterion (3.41).


Since version 41.01, the curvature-based stopping criteria, Eqs. (3.41) and (3.37) are the default, whereas the previous employed stopping criteria, Eqs. (3.37) and (3.38), are obsolete.