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1% ================================================================
2% Chapter � Miscellaneous Topics
3% ================================================================
4\chapter{Miscellaneous Topics}
9$\ $\newline    % force a new ligne
11% ================================================================
12% Representation of Unresolved Straits
13% ================================================================
14\section{Representation of Unresolved Straits}
17In climate modeling, it often occurs that a crucial connections between water masses
18is broken as the grid mesh is too coarse to resolve narrow straits. For example, coarse
19grid spacing typically closes off the Mediterranean from the Atlantic at the Strait of
20Gibraltar. In this case, it is important for climate models to include the effects of salty
21water entering the Atlantic from the Mediterranean. Likewise, it is important for the
22Mediterranean to replenish its supply of water from the Atlantic to balance the net
23evaporation occurring over the Mediterranean region. This problem occurs even in
24eddy permitting simulations. For example, in ORCA 1/4\deg several straits of the Indonesian
25archipelago (Ombai, Lombok...) are much narrow than even a single ocean grid-point.
27We describe briefly here the three methods that can be used in \NEMO to handle
28such improperly resolved straits. The first two consist of opening the strait by hand
29while ensuring that the mass exchanges through the strait are not too large by
30either artificially reducing the surface of the strait grid-cells or, locally increasing
31the lateral friction. In the third one, the strait is closed but exchanges of mass,
32heat and salt across the land are allowed.
33Note that such modifications are so specific to a given configuration that no attempt
34has been made to set them in a generic way. However, examples of how
35they can be set up is given in the ORCA 2\deg and 0.5\deg configurations. For example,
36for details of implementation in ORCA2, search:
41IF( cp_cfg == "orca" .AND. jp_cfg == 2 )
45% -------------------------------------------------------------------------------------------------------------
46%       Hand made geometry changes
47% -------------------------------------------------------------------------------------------------------------
48\subsection{Hand made geometry changes}
51$\bullet$ reduced scale factor in the cross-strait direction to a value in better agreement
52with the true mean width of the strait. (Fig.~\ref{Fig_MISC_strait_hand}).
53This technique is sometime called "partially open face" or "partially closed cells".
54The key issue here is only to reduce the faces of $T$-cell ($i.e.$ change the value
55of the horizontal scale factors at $u$- or $v$-point) but not the volume of the $T$-cell.
56Indeed, reducing the volume of strait $T$-cell can easily produce a numerical
57instability at that grid point that would require a reduction of the model time step.
58The changes associated with strait management are done in \mdl{domhgr},
59just after the definition or reading of the horizontal scale factors.
61$\bullet$ increase of the viscous boundary layer thickness by local increase of the
62fmask value at the coast (Fig.~\ref{Fig_MISC_strait_hand}). This is done in
63\mdl{dommsk} together with the setting of the coastal value of fmask
64(see Section \ref{LBC_coast})
67\begin{figure}[!tbp]     \begin{center}
70\caption{   \label{Fig_MISC_strait_hand} 
71Example of the Gibraltar strait defined in a $1\deg \times 1\deg$ mesh.
72\textit{Top}: using partially open cells. The meridional scale factor at $v$-point
73is reduced on both sides of the strait to account for the real width of the strait
74(about 20 km). Note that the scale factors of the strait $T$-point remains unchanged.
75\textit{Bottom}: using viscous boundary layers. The four fmask parameters
76along the strait coastlines are set to a value larger than 4, $i.e.$ "strong" no-slip
77case (see Fig.\ref{Fig_LBC_shlat}) creating a large viscous boundary layer
78that allows a reduced transport through the strait.}
79\end{center}   \end{figure}
82% -------------------------------------------------------------------------------------------------------------
83% Cross Land Advection
84% -------------------------------------------------------------------------------------------------------------
85\subsection{Cross Land Advection (\mdl{tracla})}
91\colorbox{yellow}{Add a short description of CLA staff here or in lateral boundary condition chapter?}
92Options are defined through the  \ngn{namcla} namelist variables.
94%The problem is resolved here by allowing the mixing of tracers and mass/volume between non-adjacent water columns at nominated regions within the model. Momentum is not mixed. The scheme conserves total tracer content, and total volume (the latter in $z*$- or $s*$-coordinate), and maintains compatibility between the tracer and mass/volume budgets. 
96% ================================================================
97% Closed seas
98% ================================================================
99\section{Closed seas (\mdl{closea})}
102\colorbox{yellow}{Add here a short description of the way closed seas are managed}
105% ================================================================
106% Sub-Domain Functionality (\textit{nizoom, njzoom}, namelist parameters)
107% ================================================================
108\section{Sub-Domain Functionality (\np{jpizoom}, \np{jpjzoom})}
111The sub-domain functionality, also improperly called the zoom option
112(improperly because it is not associated with a change in model resolution)
113is a quite simple function that allows a simulation over a sub-domain of an
114already defined configuration ($i.e.$ without defining a new mesh, initial
115state and forcings). This option can be useful for testing the user settings
116of surface boundary conditions, or the initial ocean state of a huge ocean
117model configuration while having a small computer memory requirement.
118It can also be used to easily test specific physics in a sub-domain (for example,
119see \citep{Madec_al_JPO96} for a test of the coupling used in the global ocean
120version of OPA between sea-ice and ocean model over the Arctic or Antarctic
121ocean, using a sub-domain). In the standard model, this option does not
122include any specific treatment for the ocean boundaries of the sub-domain:
123they are considered as artificial vertical walls. Nevertheless, it is quite easy
124to add a restoring term toward a climatology in the vicinity of such boundaries
125(see \S\ref{TRA_dmp}).
127In order to easily define a sub-domain over which the computation can be
128performed, the dimension of all input arrays (ocean mesh, bathymetry,
129forcing, initial state, ...) are defined as \np{jpidta}, \np{jpjdta} and \np{jpkdta} 
130( in \ngn{namcfg} namelist), while the computational domain is defined through
131\np{jpiglo}, \np{jpjglo} and \jp{jpk} (\ngn{namcfg} namelist). When running the
132model over the whole domain, the user sets \np{jpiglo}=\np{jpidta} \np{jpjglo}=\np{jpjdta} 
133and \jp{jpk}=\jp{jpkdta}. When running the model over a sub-domain, the user
134has to provide the size of the sub-domain, (\np{jpiglo}, \np{jpjglo}, \np{jpkglo}),
135and the indices of the south western corner as \np{jpizoom} and \np{jpjzoom} in
136the  \ngn{namcfg} namelist (Fig.~\ref{Fig_LBC_zoom}).
138Note that a third set of dimensions exist, \jp{jpi}, \jp{jpj} and \jp{jpk} which is
139actually used to perform the computation. It is set by default to \jp{jpi}=\np{jpjglo} 
140and \jp{jpj}=\np{jpjglo}, except for massively parallel computing where the
141computational domain is laid out on local processor memories following a 2D
142horizontal splitting. % (see {\S}IV.2-c) ref to the section to be updated
145\begin{figure}[!ht]    \begin{center}
147\caption{   \label{Fig_LBC_zoom}
148Position of a model domain compared to the data input domain when the zoom functionality is used.}
149\end{center}   \end{figure}
153% ================================================================
154% Accelerating the Convergence
155% ================================================================
156\section{Accelerating the Convergence (\np{nn\_acc} = 1)}
162Searching an equilibrium state with an global ocean model requires a very long time
163integration period (a few thousand years for a global model). Due to the size of
164the time step required for numerical stability (less than a few hours),
165this usually requires a large elapsed time. In order to overcome this problem,
166\citet{Bryan1984} introduces a technique that is intended to accelerate
167the spin up to equilibrium. It uses a larger time step in
168the tracer evolution equations than in the momentum evolution
169equations. It does not affect the equilibrium solution but modifies the
170trajectory to reach it.
172Options are defined through the  \ngn{namdom} namelist variables.
173The acceleration of convergence option is used when \np{nn\_acc}=1. In that case,
174$\rdt=rn\_rdt$ is the time step of dynamics while $\widetilde{\rdt}=rdttra$ is the
175tracer time-step. the former is set from the \np{rn\_rdt} namelist parameter while the latter
176is computed using a hyperbolic tangent profile and the following namelist parameters :
177\np{rn\_rdtmin}, \np{rn\_rdtmax} and \np{rn\_rdth}. Those three parameters correspond
178to the surface value the deep ocean value and the depth at which the transition occurs, respectively.
179The set of prognostic equations to solve becomes:
180\begin{equation} \label{Eq_acc}
182\frac{\partial \textbf{U}_h }{\partial t} 
183   &\equiv \frac{\textbf{U}_h ^{t+1}-\textbf{U}_h^{t-1} }{2\rdt} = \ldots \\ 
184\frac{\partial T}{\partial t} &\equiv \frac{T^{t+1}-T^{t-1}}{2 \widetilde{\rdt}} = \ldots \\ 
185\frac{\partial S}{\partial t} &\equiv \frac{S^{t+1} -S^{t-1}}{2 \widetilde{\rdt}} = \ldots \\ 
189\citet{Bryan1984} has examined the consequences of this distorted physics.
190Free waves have a slower phase speed, their meridional structure is slightly
191modified, and the growth rate of baroclinically unstable waves is reduced
192but with a wider range of instability. This technique is efficient for
193searching for an equilibrium state in coarse resolution models. However its
194application is not suitable for many oceanic problems: it cannot be used for
195transient or time evolving problems (in particular, it is very questionable
196to use this technique when there is a seasonal cycle in the forcing fields),
197and it cannot be used in high-resolution models where baroclinically
198unstable processes are important. Moreover, the vertical variation of
199$\widetilde{ \rdt}$ implies that the heat and salt contents are no longer
200conserved due to the vertical coupling of the ocean level through both
201advection and diffusion. Therefore \np{rn\_rdtmin} = \np{rn\_rdtmax} should be
202a more clever choice.
205% ================================================================
206% Accuracy and Reproducibility
207% ================================================================
208\section{Accuracy and Reproducibility (\mdl{lib\_fortran})}
211\subsection{Issues with intrinsinc SIGN function (\key{nosignedzero})}
214The SIGN(A, B) is the \textsc {Fortran} intrinsic function delivers the magnitude
215of A with the sign of B. For example, SIGN(-3.0,2.0) has the value 3.0.
216The problematic case is when the second argument is zero, because, on platforms
217that support IEEE arithmetic, zero is actually a signed number.
218There is a positive zero and a negative zero.
220In \textsc{Fortran}~90, the processor was required always to deliver a positive result for SIGN(A, B)
221if B was zero. Nevertheless, in \textsc{Fortran}~95, the processor is allowed to do the correct thing
222and deliver ABS(A) when B is a positive zero and -ABS(A) when B is a negative zero.
223This change in the specification becomes apparent only when B is of type real, and is zero,
224and the processor is capable of distinguishing between positive and negative zero,
225and B is negative real zero. Then SIGN delivers a negative result where, under \textsc{Fortran}~90
226rules,  it used to return a positive result.
227This change may be especially sensitive for the ice model, so we overwrite the intrinsinc
228function with our own function simply performing :   \\
229\verb?   IF( B >= 0.e0 ) THEN   ;   SIGN(A,B) = ABS(A)  ?    \\
230\verb?   ELSE                   ;   SIGN(A,B) =-ABS(A)     ?  \\
231\verb?   ENDIF    ? \\
232This feature can be found in \mdl{lib\_fortran} module and is effective when \key{nosignedzero}
233is defined. We use a CPP key as the overwritting of a intrinsic function can present
234performance issues with some computers/compilers.
237\subsection{MPP reproducibility}
240The numerical reproducibility of simulations on distributed memory parallel computers
241is a critical issue. In particular, within NEMO global summation of distributed arrays
242is most susceptible to rounding errors, and their propagation and accumulation cause
243uncertainty in final simulation reproducibility on different numbers of processors.
244To avoid so, based on \citet{He_Ding_JSC01} review of different technics,
245we use a so called self-compensated summation method. The idea is to estimate
246the roundoff error, store it in a buffer, and then add it back in the next addition.
248Suppose we need to calculate $b = a_1 + a_2 + a_3$. The following algorithm
249will allow to split the sum in two ($sum_1 = a_{1} + a_{2}$ and $b = sum_2 = sum_1 + a_3$)
250with exactly the same rounding errors as the sum performed all at once.
252   sum_1 \ \  &= a_1 + a_2 \\
253   error_1     &= a_2 + ( a_1 - sum_1 ) \\
254   sum_2 \ \  &= sum_1 + a_3 + error_1 \\
255   error_2     &= a_3 + error_1 + ( sum_1 - sum_2 ) \\
256   b \qquad \ &= sum_2 \\
258This feature can be found in \mdl{lib\_fortran} module and is effective when \key{mpp\_rep}.
259In that case, all calls to glob\_sum function (summation over the entire basin excluding
260duplicated rows and columns due to cyclic or north fold boundary condition as well as
261overlap MPP areas).
262Note this implementation may be sensitive to the optimization level.
264\subsection{MPP scalability}
267The default method of communicating values across the north-fold in distributed memory applications
268(\key{mpp\_mpi}) uses a \textsc{MPI\_ALLGATHER} function to exchange values from each processing
269region in the northern row with every other processing region in the northern row. This enables a
270global width array containing the top 4 rows to be collated on every northern row processor and then
271folded with a simple algorithm. Although conceptually simple, this "All to All" communication will
272hamper performance scalability for large numbers of northern row processors. From version 3.4
273onwards an alternative method is available which only performs direct "Peer to Peer" communications
274between each processor and its immediate "neighbours" across the fold line. This is achieved by
275using the default \textsc{MPI\_ALLGATHER} method during initialisation to help identify the "active"
276neighbours. Stored lists of these neighbours are then used in all subsequent north-fold exchanges to
277restrict exchanges to those between associated regions. The collated global width array for each
278region is thus only partially filled but is guaranteed to be set at all the locations actually
279required by each individual for the fold operation. This alternative method should give identical
280results to the default \textsc{ALLGATHER} method and is recommended for large values of \np{jpni}.
281The new method is activated by setting \np{ln\_nnogather} to be true ({\bf nammpp}). The
282reproducibility of results using the two methods should be confirmed for each new, non-reference
285% ================================================================
286% Model optimisation, Control Print and Benchmark
287% ================================================================
288\section{Model Optimisation, Control Print and Benchmark}
294 \gmcomment{why not make these bullets into subsections?}
295Options are defined through the  \ngn{namctl} namelist variables.
297$\bullet$ Vector optimisation:
299\key{vectopt\_loop} enables the internal loops to collapse. This is very
300a very efficient way to increase the length of vector calculations and thus
301to speed up the model on vector computers.
303% Add here also one word on NPROMA technique that has been found useless, since compiler have made significant progress during the last decade.
305% Add also one word on NEC specific optimisation (Novercheck option for example)
307$\bullet$ Control print %: describe here 4 things:
3091- \np{ln\_ctl} : compute and print the trends averaged over the interior domain
310in all TRA, DYN, LDF and ZDF modules. This option is very helpful when
311diagnosing the origin of an undesired change in model results.
3132- also \np{ln\_ctl} but using the nictl and njctl namelist parameters to check
314the source of differences between mono and multi processor runs.
3163- \key{esopa} (to be rename key\_nemo) : which is another option for model
317management. When defined, this key forces the activation of all options and
318CPP keys. For example, all tracer and momentum advection schemes are called!
319Therefore the model results have no physical meaning.
320However, this option forces both the compiler and the model to run through
321all the \textsc{Fortran} lines of the model. This allows the user to check for obvious
322compilation or execution errors with all CPP options, and errors in namelist options.
3244- last digit comparison (\np{nn\_bit\_cmp}). In an MPP simulation, the computation of
325a sum over the whole domain is performed as the summation over all processors of
326each of their sums over their interior domains. This double sum never gives exactly
327the same result as a single sum over the whole domain, due to truncation differences.
328The "bit comparison" option has been introduced in order to be able to check that
329mono-processor and multi-processor runs give exactly the same results.
330%THIS is to be updated with the mpp_sum_glo  introduced in v3.3
331% nn_bit_cmp  today only check that the nn_cla = 0 (no cross land advection)
333$\bullet$  Benchmark (\np{nn\_bench}). This option defines a benchmark run based on
334a GYRE configuration (see \S\ref{CFG_gyre}) in which the resolution remains the same
335whatever the domain size. This allows a very large model domain to be used, just by
336changing the domain size (\jp{jpiglo}, \jp{jpjglo}) and without adjusting either the time-step
337or the physical parameterisations.
340% ================================================================
341% Elliptic solvers (SOL)
342% ================================================================
343\section{Elliptic solvers (SOL)}
349When the filtered sea surface height option is used, the surface pressure gradient is
350computed in \mdl{dynspg\_flt}. The force added in the momentum equation is solved implicitely.
351It is thus solution of an elliptic equation \eqref{Eq_PE_flt} for which two solvers are available:
352a Successive-Over-Relaxation scheme (SOR) and a preconditioned conjugate gradient
353scheme(PCG) \citep{Madec_al_OM88, Madec_PhD90}. The solver is selected trough the
354the value of \np{nn\_solv}   \ngn{namsol} namelist variable.
356The PCG is a very efficient method for solving elliptic equations on vector computers.
357It is a fast and rather easy method to use; which are attractive features for a large
358number of ocean situations (variable bottom topography, complex coastal geometry,
359variable grid spacing, open or cyclic boundaries, etc ...). It does not require
360a search for an optimal parameter as in the SOR method. However, the SOR has
361been retained because it is a linear solver, which is a very useful property when
362using the adjoint model of \NEMO.
364At each time step, the time derivative of the sea surface height at time step $t+1$ 
365(or equivalently the divergence of the \textit{after} barotropic transport) that appears
366in the filtering forced is the solution of the elliptic equation obtained from the horizontal
367divergence of the vertical summation of \eqref{Eq_PE_flt}.
368Introducing the following coefficients:
369\begin{equation}  \label{Eq_sol_matrix}
371&c_{i,j}^{NS}  &&= {2 \rdt }^2 \; \frac{H_v (i,j) \; e_{1v} (i,j)}{e_{2v}(i,j)}              \\
372&c_{i,j}^{EW} &&= {2 \rdt }^2 \; \frac{H_u (i,j) \; e_{2u} (i,j)}{e_{1u}(i,j)}            \\
373&b_{i,j} &&= \delta_i \left[ e_{2u}M_u \right] - \delta_j \left[ e_{1v}M_v \right]\ ,   \\
376the resulting five-point finite difference equation is given by:
377\begin{equation}  \label{Eq_solmat}
379       c_{i+1,j}^{NS} D_{i+1,j}  + \;  c_{i,j+1}^{EW} D_{i,j+1}   
380  +   c_{i,j}    ^{NS} D_{i-1,j}   + \;  c_{i,j}    ^{EW} D_{i,j-1}                                          &    \\
381  -    \left(c_{i+1,j}^{NS} + c_{i,j+1}^{EW} + c_{i,j}^{NS} + c_{i,j}^{EW} \right)   D_{i,j}  &=  b_{i,j}
384\eqref{Eq_solmat} is a linear symmetric system of equations. All the elements of
385the corresponding matrix \textbf{A} vanish except those of five diagonals. With
386the natural ordering of the grid points (i.e. from west to east and from
387south to north), the structure of \textbf{A} is block-tridiagonal with
388tridiagonal or diagonal blocks. \textbf{A} is a positive-definite symmetric
389matrix of size $(jpi \cdot jpj)^2$, and \textbf{B}, the right hand side of
390\eqref{Eq_solmat}, is a vector.
392Note that in the linear free surface case, the depth that appears in \eqref{Eq_sol_matrix}
393does not vary with time, and thus the matrix can be computed once for all. In non-linear free surface
394(\key{vvl} defined) the matrix have to be updated at each time step.
396% -------------------------------------------------------------------------------------------------------------
397%       Successive Over Relaxation
398% -------------------------------------------------------------------------------------------------------------
399\subsection{Successive Over Relaxation (\np{nn\_solv}=2, \mdl{solsor})}
402Let us introduce the four cardinal coefficients:
404a_{i,j}^S &= c_{i,j    }^{NS}/d_{i,j}     &\qquad  a_{i,j}^W &= c_{i,j}^{EW}/d_{i,j}       \\
405a_{i,j}^E &= c_{i,j+1}^{EW}/d_{i,j}    &\qquad   a_{i,j}^N &= c_{i+1,j}^{NS}/d_{i,j}   
407where $d_{i,j} = c_{i,j}^{NS}+ c_{i+1,j}^{NS} + c_{i,j}^{EW} + c_{i,j+1}^{EW}$ 
408(i.e. the diagonal of the matrix). \eqref{Eq_solmat} can be rewritten as:
409\begin{equation}  \label{Eq_solmat_p}
411a_{i,j}^{N}  D_{i+1,j} +\,a_{i,j}^{E}  D_{i,j+1} +\, a_{i,j}^{S}  D_{i-1,j} +\,a_{i,j}^{W} D_{i,j-1}  -  D_{i,j} = \tilde{b}_{i,j}
414with $\tilde b_{i,j} = b_{i,j}/d_{i,j}$. \eqref{Eq_solmat_p} is the equation actually solved
415with the SOR method. This method used is an iterative one. Its algorithm can be
416summarised as follows (see \citet{Haltiner1980} for a further discussion):
418initialisation (evaluate a first guess from previous time step computations)
420D_{i,j}^0 = 2 \, D_{i,j}^t - D_{i,j}^{t-1}
422iteration $n$, from $n=0$ until convergence, do :
423\begin{equation} \label{Eq_sor_algo}
425R_{i,j}^n  = &a_{i,j}^{N} D_{i+1,j}^n       +\,a_{i,j}^{E}  D_{i,j+1} ^n         
426         +\, a_{i,j}^{S}  D_{i-1,j} ^{n+1}+\,a_{i,j}^{W} D_{i,j-1} ^{n+1}
427                 -  D_{i,j}^n - \tilde{b}_{i,j}                                           \\
428D_{i,j} ^{n+1}  = &D_{i,j} ^{n}   + \omega \;R_{i,j}^n     
431where \textit{$\omega $ }satisfies $1\leq \omega \leq 2$. An optimal value exists for
432\textit{$\omega$} which significantly accelerates the convergence, but it has to be
433adjusted empirically for each model domain (except for a uniform grid where an
434analytical expression for \textit{$\omega$} can be found \citep{Richtmyer1967}).
435The value of $\omega$ is set using \np{rn\_sor}, a \textbf{namelist} parameter.
436The convergence test is of the form:
438\delta = \frac{\sum\limits_{i,j}{R_{i,j}^n}{R_{i,j}^n}}
439                    {\sum\limits_{i,j}{ \tilde{b}_{i,j}^n}{\tilde{b}_{i,j}^n}} \leq \epsilon
441where $\epsilon$ is the absolute precision that is required. It is recommended
442that a value smaller or equal to $10^{-6}$ is used for $\epsilon$ since larger
443values may lead to numerically induced basin scale barotropic oscillations.
444The precision is specified by setting \np{rn\_eps} (\textbf{namelist} parameter).
445In addition, two other tests are used to halt the iterative algorithm. They involve
446the number of iterations and the modulus of the right hand side. If the former
447exceeds a specified value, \np{nn\_max} (\textbf{namelist} parameter),
448or the latter is greater than $10^{15}$, the whole model computation is stopped
449and the last computed time step fields are saved in a NetCDF file.
450In both cases, this usually indicates that there is something wrong in the model
451configuration (an error in the mesh, the initial state, the input forcing,
452or the magnitude of the time step or of the mixing coefficients). A typical value of
453$nn\_max$ is a few hundred when $\epsilon = 10^{-6}$, increasing to a few
454thousand when $\epsilon = 10^{-12}$.
455The vectorization of the SOR algorithm is not straightforward. The scheme
456contains two linear recurrences on $i$ and $j$. This inhibits the vectorisation.
457\eqref{Eq_sor_algo} can be been rewritten as:
461= &a_{i,j}^{N}  D_{i+1,j}^n +\,a_{i,j}^{E}  D_{i,j+1} ^n
462 +\,a_{i,j}^{S}  D_{i-1,j} ^{n}+\,_{i,j}^{W} D_{i,j-1} ^{n} -  D_{i,j}^n - \tilde{b}_{i,j}      \\
463R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{S}\; R_{i,j-1}^n                                             \\
464R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{W}\; R_{i-1,j}^n
467This technique slightly increases the number of iteration required to reach the convergence,
468but this is largely compensated by the gain obtained by the suppression of the recurrences.
470Another technique have been chosen, the so-called red-black SOR. It consist in solving successively
471\eqref{Eq_sor_algo} for odd and even grid points. It also slightly reduced the convergence rate
472but allows the vectorisation. In addition, and this is the reason why it has been chosen, it is able to handle the north fold boundary condition used in ORCA configuration ($i.e.$ tri-polar global ocean mesh).
474The SOR method is very flexible and can be used under a wide range of conditions,
475including irregular boundaries, interior boundary points, etc. Proofs of convergence, etc.
476may be found in the standard numerical methods texts for partial differential equations.
478% -------------------------------------------------------------------------------------------------------------
479%       Preconditioned Conjugate Gradient
480% -------------------------------------------------------------------------------------------------------------
481\subsection{Preconditioned Conjugate Gradient  (\np{nn\_solv}=1, \mdl{solpcg}) }
484\textbf{A} is a definite positive symmetric matrix, thus solving the linear
485system \eqref{Eq_solmat} is equivalent to the minimisation of a quadratic
488\textbf{Ax} = \textbf{b} \leftrightarrow \textbf{x} =\text{inf}_{y} \,\phi (\textbf{y})
489\quad , \qquad
490\phi (\textbf{y}) = 1/2 \langle \textbf{Ay},\textbf{y}\rangle - \langle \textbf{b},\textbf{y} \rangle 
492where $\langle , \rangle$ is the canonical dot product. The idea of the
493conjugate gradient method is to search for the solution in the following
494iterative way: assuming that $\textbf{x}^n$ has been obtained, $\textbf{x}^{n+1}$ 
495is found from $\textbf {x}^{n+1}={\textbf {x}}^n+\alpha^n{\textbf {d}}^n$ which satisfies:
497{\textbf{ x}}^{n+1}=\text{inf} _{{\textbf{ y}}={\textbf{ x}}^n+\alpha^n \,{\textbf{ d}}^n} \,\phi ({\textbf{ y}})\;\;\Leftrightarrow \;\;\frac{d\phi }{d\alpha}=0
499and expressing $\phi (\textbf{y})$ as a function of \textit{$\alpha $}, we obtain the
500value that minimises the functional:
502\alpha ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle  / \langle {\textbf{ A d}^n, \textbf{d}^n} \rangle
504where $\textbf{r}^n = \textbf{b}-\textbf{A x}^n = \textbf{A} (\textbf{x}-\textbf{x}^n)$ 
505is the error at rank $n$. The descent vector $\textbf{d}^n$ s chosen to be dependent
506on the error: $\textbf{d}^n = \textbf{r}^n + \beta^n \,\textbf{d}^{n-1}$. $\beta ^n$ 
507is searched such that the descent vectors form an orthogonal basis for the dot
508product linked to \textbf{A}. Expressing the condition
509$\langle \textbf{A d}^n, \textbf{d}^{n-1} \rangle = 0$ the value of $\beta ^n$ is found:
510 $\beta ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle  / \langle {\textbf{r}^{n-1}, \textbf{r}^{n-1}} \rangle$.
511 As a result, the errors $ \textbf{r}^n$ form an orthogonal
512base for the canonic dot product while the descent vectors $\textbf{d}^n$ form
513an orthogonal base for the dot product linked to \textbf{A}. The resulting
514algorithm is thus the following one:
516initialisation :
519\textbf{x}^0 &= D_{i,j}^0   = 2 D_{i,j}^t - D_{i,j}^{t-1}       \quad, \text{the initial guess }     \\
520\textbf{r}^0 &= \textbf{d}^0 = \textbf{b} - \textbf{A x}^0       \\
521\gamma_0 &= \langle{ \textbf{r}^0 , \textbf{r}^0} \rangle
525iteration $n,$ from $n=0$ until convergence, do :
528\text{z}^n& = \textbf{A d}^n \\
529\alpha_n &= \gamma_n /  \langle{ \textbf{z}^n , \textbf{d}^n} \rangle \\
530\textbf{x}^{n+1} &= \textbf{x}^n + \alpha_n \,\textbf{d}^n \\
531\textbf{r}^{n+1} &= \textbf{r}^n - \alpha_n \,\textbf{z}^n \\
532\gamma_{n+1} &= \langle{ \textbf{r}^{n+1} , \textbf{r}^{n+1}} \rangle \\
533\beta_{n+1} &= \gamma_{n+1}/\gamma_{n}  \\
534\textbf{d}^{n+1} &= \textbf{r}^{n+1} + \beta_{n+1}\; \textbf{d}^{n}\\
539The convergence test is:
541\delta = \gamma_{n}\; / \langle{ \textbf{b} , \textbf{b}} \rangle \leq \epsilon
543where $\epsilon $ is the absolute precision that is required. As for the SOR algorithm,
544the whole model computation is stopped when the number of iterations, \np{nn\_max}, or
545the modulus of the right hand side of the convergence equation exceeds a
546specified value (see \S\ref{MISC_solsor} for a further discussion). The required
547precision and the maximum number of iterations allowed are specified by setting
548\np{rn\_eps} and \np{nn\_max} (\textbf{namelist} parameters).
550It can be demonstrated that the above algorithm is optimal, provides the exact
551solution in a number of iterations equal to the size of the matrix, and that
552the convergence rate is faster as the matrix is closer to the identity matrix,
553$i.e.$ its eigenvalues are closer to 1. Therefore, it is more efficient to solve
554a better conditioned system which has the same solution. For that purpose,
555we introduce a preconditioning matrix \textbf{Q} which is an approximation
556of \textbf{A} but much easier to invert than \textbf{A}, and solve the system:
557\begin{equation} \label{Eq_pmat}
558\textbf{Q}^{-1} \textbf{A x} = \textbf{Q}^{-1} \textbf{b}
561The same algorithm can be used to solve \eqref{Eq_pmat} if instead of the
562canonical dot product the following one is used:
563${\langle{ \textbf{a} , \textbf{b}} \rangle}_Q = \langle{ \textbf{a} , \textbf{Q b}} \rangle$, and
564if $\textbf{\~{b}} = \textbf{Q}^{-1}\;\textbf{b}$ and $\textbf{\~{A}} = \textbf{Q}^{-1}\;\textbf{A}$ 
565are substituted to \textbf{b} and \textbf{A} \citep{Madec_al_OM88}.
566In \NEMO, \textbf{Q} is chosen as the diagonal of \textbf{ A}, i.e. the simplest form for
567\textbf{Q} so that it can be easily inverted. In this case, the discrete formulation of
568\eqref{Eq_pmat} is in fact given by \eqref{Eq_solmat_p} and thus the matrix and
569right hand side are computed independently from the solver used.
571% ================================================================
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