Numerics

         |    BSBT
PROP    <                     | Sec |
         |    GSE  [waveage] <  MIn  >
         |                    |  HR |
         |                    | DAy |

Command to choose:

• the BSBT scheme (stationary and nonstationary) instead of the default S&L scheme (in case of nonstationary cases) or the default SORDUP scheme (in case of stationary cases) or
• the alleviation of the garden-sprinkler effect (GSE).

BSBT	the BSBT scheme will be used in the computations.
GSE	garden-sprinkler effect is to be counteracted in the S&L propagation scheme
	(default for nonstationary regular grid computations) or in the propagation
	scheme for unstructured grids by adding a diffusion term to the basic equation.
	This may affect the numerical stability of SWAN (see Scientific/Technical
	documentation).
[waveage]	the time interval used to determine the diffusion which counteracts the so-called
	garden-sprinkler effect. The default value of [waveage] is zero, i.e. no added
	diffusion. The value of [waveage] should correspond to the travel time of
	the waves over the computational region.

Notes:

• All schemes (BSBT, SORDUP and S&L) can be used in combination with curvilinear grids. With the higher order schemes (S&L and SORDUP) it is important to use a gradually varying grid, otherwise there may be a severe loss of accuracy. If sharp transitions in the grid cannot be avoided it is safer to use the BSBT scheme.
• In the computation with unstructured meshes, a lowest order upwind scheme will be employed. This scheme is very robust but rather diffusive. This may only be significant for the case when swell waves propagate over relative large distances (in the order of thousands of kilometers) within the model domain. However and most fortunately, in such a case, this will alleviate the garden-sprinkler effect.
• Alleviating the garden-sprinkler effect by adding some diffusion makes the SWAN computation conditionally stable. You can either use i) a smaller time step, ii) a lower value of [waveage], iii) better resolution in the directional space, or iv) worse resolution in the geographic space, in order of preference, to make the model stable when necessary.

NUMeric ( STOPC [dabs] [drel] [curvat] [npnts]                      &

                    | -> STAT [mxitst] [alfa] |
                   <                           >  [limiter]       ) &
                    |    NONSTAT [mxitns]     |

          ( DIRimpl [cdd]                                         ) &

          ( SIGIMpl [css] [eps2] [outp] [niter]                   ) &

          ( CTheta [cfl]                                          ) &

          ( CSigma [cfl]                                          ) &

          ( SETUP [eps2] [outp] [niter]                           )

With this optional command the user can influence some of the numerical properties of SWAN.

STOPC	With this option the user can influence the criterion for terminating the
	iterative procedure in the SWAN computations (both stationary and
	nonstationary). The criterion make use of the second derivative, or curvature,
	of the iteration curve of the significant wave height. As the solution of a
	simulation approaches full convergence, the curvature of the iteration curve will
	tend to zero. SWAN stops the process if the relative change in $H_s$ from one
	iteration to the next is less than [drel] and the curvature of the iteration
	curve of $H_s$ normalized with $H_s$ is less than [curvat] or the absolute change in
	$H_s$ from one iteration to the next is less than [dabs]. Both conditions need to
	be fulfilled in more than fraction [npnts]% of all wet grid points.
	With respect to the QC modelling, another stopping criteria will be employed.
	Namely, SWAN stops the iteration process if the absolute change in $H_s$ from one
	iterate to another is less than [dabs] $\times H_{\rm inc}$, where $H_{\rm inc}$ is the representative
	incident wave height, or the relative change in $H_s$ from one to the next iteration
	is less than [drel]. These criteria must be fulfilled in more than [npnts]% of
	all active, well-defined points.
[dabs]	Default: [dabs] = 0.005 [m] or [dabs] = 0.05 [$-$] in case of QC model.
[drel]	Default: [drel] = 0.01 [$-$]
[curvat]	Default: [curvat] = 0.005 [$-$]
	Note that [curvat] is not used in the QC model.
[npnts]	Default: [npnts] = 99.5 [$-$]
STAT	indicates the use of parameters in a stationary computation.
[mxitst]	the maximum number of iterations for stationary computations.
	The computation stops when this number is exceeded.
	Default: [mxitst] = 50.
	Note that [mxitst] can be set to 0 if one wants to check the input to the
	model without making computations.
[alfa]	proportionality constant used in the frequency-dependent under-relaxation
	technique. Based on experiences, a suggestion for this parameter is [alfa] = 0.01.
	In case of diffraction computations, the use of this parameter is recommended.
	Default: [alfa] = 0.00.
	NOT MEANINGFUL FOR NONSTATIONARY COMPUTATIONS.
NONSTAT	indicates the use of parameters in a nonstationary computation.
[mxitns]	the maximum number of iterations per time step for nonstationary computations.
	The computation moves to the next time step when this number is exceeded.
	Default: [mxitns] = 1.
	Note that [mxitns] can be set to 0 if one wants to check the input to the
	model without making computations.
[limiter]	determines, in both stationary and nonstationary runs, the maximum change per
	iteration of the energy density per spectral ($\sigma$,$\theta$)-bin, given in
	terms of a fraction of the omni-directional Phillips level (see Scientific/
	Technical documentation).
	Default: [limiter] = 0.1.
DIRIMPL	this option is used to influence the numerical scheme for refraction.
[cdd]	A value of [cdd]=0 corresponds to a central scheme and has the largest
	accuracy (diffusion $\approx$ 0) but the computation may more easily generate
	spurious fluctuations. A value of [cdd]=1. corresponds to a first order
	upwind scheme and it is more diffusive and therefore preferable if (strong)
	gradients in depth or current are present.
	Default: [cdd] = 0.5.
SIGIMPL	controls the accuracy of computing the frequency shifting and the stopping criterion
	and amount of output for the SIP solver (used in the computations in the presence
	of currents or time varying depth).
CTHETA	this option prevents an excessive directional turning at a single grid point or vertex
	due to a very coarse bathymetry or current locally. This option limits the directional
	turning rate $c_{\theta}$ based on the CFL restriction. (See Eq. 3.41 of Scientific/
	Technical documentation). See also the final remark in Section 2.6.3.
	Note that if this command is not specified, then the limiter is not activated.
[cfl]	upper limit for the CFL restriction for $c_{\theta}$. A suggestion for this parameter is
	[cfl] = 0.9.
	Default: [cfl] = 0.9 (when command CTHETA is activated).
CSIGMA	this option prevents an excessive frequency shifting at a single grid point or vertex
	due to a very coarse bathymetry or current locally. This option limits the frequency
	shifting rate $c_{\sigma}$ based on the CFL restriction. See also the final remark in Section
	2.6.3. Note that if this command is not specified, then the limiter is not activated.
[cfl]	upper limit for the CFL restriction for $c_{\sigma}$. A suggestion for this parameter is
	[cfl] = 0.9.
	Default: [cfl] = 0.9 (when command CSIGMA is activated).
SETUP	controls the stopping criterion and amount of output for the SOR solver in the
	computation of the wave-induced set-up.
[css]	A value of [css]=0 corresponds to a central scheme and has the largest
	accuracy (diffusion $\approx$ 0) but the computation may more easily generate
	spurious fluctuations. A value of [css]=1. corresponds to a first order upwind
	scheme and it is more diffusive and therefore preferable if (strong) gradients in
	depth or current are present.
	Default: [css] = 0.5.
[eps2]	Relative stopping criterion to terminate the linear solver (SIP or SOR). The
	criterion for the SIP solver is based on	$\Vert A {\vec{N}}_k - \vec{b} \Vert _2 \leq$ [eps2] $\Vert \vec{b} \Vert _2$
	where $A$ is a matrix, $\vec{N}$ is the action density vector, $\vec{b}$ is the right hand vector
	and $k$ is the iteration number.
	The criterion for the SOR solver is based on $\Vert {\overline{\eta}}_{k+1} - {\overline{\eta}}_{k} \Vert _{\infty} \leq$ [eps2] where
	$\overline{\eta}$ is the set-up.
	Default: [eps2] = 1.e-4 in case of SIP and [eps2] = 1.e-6 in case of SOR.
[outp]	output for the iterative solver:
	0 = no output
	1 = additional information about the iteration process is written to the PRINT file
	2 = gives a maximal amount of output concerning the iteration process
	3 = summary of the iteration process
	Default: [outp] = 0.
[niter]	maximum number of iterations for the linear solver.
	Default: [niter] = 20 in case of SIP and [niter] = 1000 in case of SOR.