source: palm/trunk/TUTORIAL/SOURCE/numerics_bc.tex @ 976

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added first LaTeX source code for the new tutorial

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1% $Id: numerics_bc.tex 915 2012-05-30 15:11:11Z maronga $
2\input{header_tmp.tex}
3%\input{header_lectures.tex}
4
5\usepackage[utf8]{inputenc}
6\usepackage{ngerman}
7\usepackage{pgf}
8\usetheme{Dresden}
9\usepackage{subfigure}
10\usepackage{units}
11\usepackage{multimedia}
12\usepackage{hyperref}
13\newcommand{\event}[1]{\newcommand{\eventname}{#1}}
14\usepackage{xmpmulti}
15\usepackage{tikz}
16\usetikzlibrary{shapes,arrows,positioning}
17\def\Tiny{\fontsize{4pt}{4pt}\selectfont}
18\usepackage{amsmath}
19\usepackage{amssymb}
20\usepackage{multicol}
21\usepackage{pdfcomment}
22
23\institute{Institut fÌr Meteorologie und Klimatologie, Leibniz UniversitÀt Hannover}
24\date{last update: \today}
25\event{PALM Seminar}
26\setbeamertemplate{navigation symbols}{}
27
28\setbeamertemplate{footline}
29  {
30    \begin{beamercolorbox}[rightskip=-0.1cm]&
31     {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}}
32    \end{beamercolorbox}
33    \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,
34      leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}
35      {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}
36    \end{beamercolorbox}
37    \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}
38    \end{beamercolorbox}
39  }
40%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.pdf}}
41
42\title[Numerics and Boundary Conditions]{Numerics and Boundary Conditions\\
43(used in PALM)
44}
45\author{Siegfried Raasch}
46
47\begin{document}
48
49% Folie 1
50\begin{frame}
51\titlepage
52\end{frame}
53
54% Folie 2
55\begin{frame}
56   \frametitle{Overview}
57   \scriptsize PALM is (almost) using simple, standard and fast numerical schemes:
58   \begin{itemize}
59      \scriptsize
60      \item<2-> \textbf{Spatial and temporal discretization by finite differences}\\
61      \item<3-> \textbf{Explicit timestep methods:}\\
62         - Euler\\
63         - Leapfrog\\
64         - \underline{Runge-Kutta}, second or \underline{third order}
65      \item<4-> \textbf{Advection method}\\
66         - Upstream\\
67         - Piacsek-Williams (second order central finite differences)\\
68         - Upstream-spline\\
69         - Bott-Chlond-scheme (monotone, positiv definit, for scalars only)\\
70         - \underline{5th-order scheme of Wicker and Skamarock}, (as used in WRF model)
71      \item<5-> \textbf{Poisson-equation for pressure}\\
72         - \underline{Direct FFT-method}\\
73         - Multigrid-method
74      \item<6-> \textbf{Lagrangian particle model included}
75      \item<7-> \textbf{Boundary conditions:}\\
76         - \underline{Cyclic} and non-cyclic horizontal boundary conditions\\
77         - Surface layer with Monin-Obukhov similarity\\
78         - Topography\\
79         - Turbulent inflow (for non-cyclic horizontal boundary conditions)
80         
81   \end{itemize}
82\end{frame}
83
84
85\section{Numerics}
86\subsection{Numerics}
87
88% Folie 3
89\begin{frame}
90   \frametitle{Numerical Grid}
91   \footnotesize
92   \vspace{2mm}
93   \includegraphics[width=\textwidth]{numerics_bc_figures/numerical_grid.png}
94   \begin{itemize}
95      \item<1->Equations are spatially discretized on an Arakawa-C grid.
96      \item<2->All scalar variables s (e.g. , $p^*$, $e$, $K_{\mathrm{m}}$,
97               $K_{\mathrm{h}}$) are defined at the cell centers.
98      \item<3->Velocity components ($u$, $v$, $w$) are shifted by half of the grid spacing.
99      \item<4->Spacings are equidistant, stretching along $z$ is possible.
100   \end{itemize}
101
102   \tikzstyle{plain} = [rectangle, draw, fill=white!20, text width=0.33\textwidth, font=\small]
103   \begin{tikzpicture}[remember picture, overlay]
104      \node at (current page.north west){%
105      \begin{tikzpicture}[overlay]
106         \node[plain, draw,anchor=west] at (88mm,-55mm) {
107         \noindent \scriptsize general definition (cylic):\\
108         $\Psi$(0:nz+1,-1:ny+1,-1:nx+1)\\
109         $\Psi$(:,-1,:) $=\Psi$(:,ny,:)\\
110         $\Psi$(:,ny+1,:) $=\Psi$(:,0,:)
111
112         };
113      \end{tikzpicture}
114      };
115   \end{tikzpicture}
116\end{frame}
117
118% Folie 4
119\begin{frame}
120   \frametitle{Timestep Methods (I)}
121   \footnotesize
122   \begin{itemize}
123      \item<1->\textbf{Euler}\\
124      \vspace{1mm}
125      $\dfrac{\partial \psi(t)}{\partial t} = F (\psi(t)) \rightarrow 
126      \dfrac{\psi(t + \Delta t) - \psi(t)}{\Delta t}
127      \approx F (\psi(t))$ \hspace{8mm} 
128      \onslide<2-> $u\dfrac{\Delta t}{\Delta x}=C<1$\\
129      \begin{flushright}
130         for stability
131      \end{flushright}
132
133      \onslide<1->$\psi (t+\Delta t) = \psi(t) 
134      + \Delta t \cdot F(\psi(t)) \hspace{28mm} \mathcal{O}(\Delta t)$\\
135      (used for SGS-TKE in special cases)
136         
137      \vspace{3mm}
138      \item<3-> \textbf{Leapfrog}\\
139      $\psi(t + \Delta t) = \psi(t - \Delta t)+2 \Delta t \cdot F(\psi(t))$ \hspace{16mm} $
140      \mathcal{O}(\Delta t^2)$ \hspace{3mm} $C \le 0.1$\\
141      Time-splitting requires a weak time filter (Asselin filter)
142         
143      \vspace{3mm}
144      \item<4-> \textbf{Runge-Kutta, third-order}\\
145      $k_1=F(\psi(t))$\\
146      \vspace{1mm}
147      $k_2=F \left( \psi(t) + \frac{1}{3} \Delta t \cdot k_1 \right)$\\
148      \vspace{1mm}
149      $k_3=F \left( \psi(t) - \frac{3}{16} \Delta t \cdot k_1 
150      + \frac{15}{16} \Delta t \cdot k_2 \right)$\\
151      \vspace{1mm}
152      $\psi(t + \Delta t) = \psi(t) + \frac{1}{30}\Delta t (5 k_1 + 9 k_2 + 16 k_3)$ 
153      \hspace{12mm} $\mathcal{O}(\Delta t^2)$ \hspace{3mm} $C \le 0.9$\\
154   \end{itemize}
155\end{frame}
156
157% Folie 5
158\begin{frame}
159   \frametitle{Timestep Methods (II)}
160   \footnotesize
161   \onslide<1->In the PALM code, the different timestep schemes are treated by one
162   equation using switches:
163   $\psi (t + \Delta t ) = (1 - c_1) \cdot \psi (t - \Delta t ) + c_1 \cdot \psi (t) 
164   + \Delta t \cdot \left[ c_2 \cdot F (\psi (t) ) + c_3 \cdot F (\psi (t - 
165   \Delta t ) ) \right]$
166   \vspace{1mm}
167
168   \onslide<2->
169   \begin{centering}
170      \begin{table}
171         \begin{tabular}{cccc}
172            \bf{Scheme} & \bf{c$_1$} & \bf{c$_2$} & \bf{c$_3$}\\
173            Euler & 1 & 1 & 0\\
174            Leapfrog & 0 & 2 & 0\\
175            RK (1st step) & 1 & 1/3 & 0\\
176            RK (2nd step) & 1 & 15/16 & -25/48\\
177            RK (3rd step) & 1 & 8/15 & 1/15\\
178         \end{tabular}
179      \end{table}
180   \end{centering}
181
182   \onslide<3->
183   \begin{align*}
184      \psi (t - \Delta t) &= \psi (t) \hspace{15mm} \textbf{after each RK substep}\\
185      \psi (t) &= \psi (t + \Delta t)
186   \end{align*}
187\end{frame}
188
189% Folie 6
190\begin{frame}
191   \frametitle{Advection Methods (I)}
192   \small
193   \begin{itemize}
194      \item<1-> Piacsek Williams C3 (1970, J. Comput. Phy., 6, 392).
195      \item<2-> Scheme of 2nd order accuracy.
196      \item<3-> Conserves integrals of linear and quadratic quantities.
197      \item<4-> Requires incompressibility $\rightarrow$ flux form of advection term.
198                \onslide<4-> \includegraphics[width=0.8\textwidth]{numerics_bc_figures/advection_methods.png}
199   \end{itemize}
200   $$\left.\frac{\partial (u \psi)}{\partial x}\right\vert_i = \frac{1}{2 \Delta x}
201   \left( u_{i+\frac{1}{2}} \psi_{i+1} - u_{i-\frac{1}{2}} \psi_{i-1} \right)$$
202   \begin{itemize}
203      \item<5-> In case of momentum advection (e.g. $\psi=u$), $u_{i-1}$ and
204                $u_{i+1}$ have to be obtained by linear interpolation.
205      \item<5-> May cause $2 \Delta x$ wiggles in case of sharp gradients.
206   \end{itemize}
207\end{frame}
208
209% Folie 7
210\begin{frame}
211   \frametitle{Advection Methods (II)}
212   \begin{itemize}
213      \item<1-> \small Upstream-spline\\ \scriptsize
214         \onslide<1-> - Requires Euler timestep\\
215         \onslide<2-> - Nonlocal scheme: produces heavy load on communication network\\
216         \onslide<3-> - Numerically unstable under stable stratification
217      \item<4-> \small Bott-Chlond\\ \scriptsize
218         \onslide<4-> - Chlond (1994)\\
219         \onslide<5-> - Monotone, positive definit. Can only be used for scalars\\
220         \onslide<6-> - Conserves sharp gradients\\
221         \onslide<7-> - Numerically expensive\\
222         \onslide<8-> - Not optimized for use on cache-based machines.
223      \item<9-> \small Default: Wicker and Skamarock scheme (5th order)\\ \scriptsize
224         \onslide<9-> - Much better accuracy than Piacsek Williams\\
225         \onslide<10-> - Much simpler algorithm than Bott-Chlond\\
226         \onslide<11-> - Requires additional ghost layers\\
227         \onslide<12-> - Adds additional numerical dissipation
228         
229   \end{itemize}
230\end{frame}
231
232% Folie 8
233\begin{frame}
234   \frametitle{Advection Methods – Wicker/Skamarock (I)}
235   \footnotesize
236   \begin{itemize}
237      \item Wicker and Skamarock (2002, Mon. Wea. Rev. 130, 2088 – 2097).
238      \item Based on flux form of advection term
239      \item Difference of fluxes at the edge of the grid cell is used to
240      discretise advection term
241   \end{itemize}
242
243   \begin{columns}[T]
244      \begin{column}{0.55\textwidth}
245         \hspace{8mm}\includegraphics[width=0.8\textwidth]{numerics_bc_figures/numerical_grid_small.png}
246      \end{column}
247      \begin{column}{0.45\textwidth}
248         $\frac{ \partial \psi}{\partial t} = - \nabla (u_i \psi) \approx 
249         - \frac{F_{i+\frac{1}{2}} - F_{i-\frac{1}{2}}}{\Delta x}$
250      \end{column}
251   \end{columns}
252
253   \tikzstyle{plain} = [rectangle, text width=0.1\textwidth, font=\small]
254   \begin{tikzpicture}[remember picture, overlay]
255      \node at (current page.north west){%
256
257      \begin{tikzpicture}[overlay]
258         \node[plain, anchor=west] at (2mm,-68mm) {
259         \tikz 
260         {
261         \draw[blue, -latex', line width=5pt] (1,0) -- (2,0);
262         }
263
264         $F_{i-\frac{1}{2}}$
265         };
266      \end{tikzpicture}
267
268      \begin{tikzpicture}[overlay]
269         \node[plain, anchor=west] at (62mm,-68mm) {
270         \tikz 
271         {
272         \draw[blue, -latex', line width=5pt] (,0) -- (2,0);
273         }
274         $F_{i+\frac{1}{2}}$
275         };
276      \end{tikzpicture}
277
278      };
279   \end{tikzpicture}
280\end{frame}
281
282% Folie 9
283\begin{frame}
284   \frametitle{Advection Methods – Wicker/Skamarock (II)}
285   
286
287
288   \textbf{Finite difference approxiamtion of 6$^{\text{th}}$ order}
289   \begin{tikzpicture}[scale=2]
290      \tikzstyle{ann} = [draw=none,fill=none,right]
291      \matrix[nodes={draw, thick, fill=blue!20}, row sep=0.3cm,column sep=0.5cm]{
292      \node[rectangle, rounded corners]{
293      $F^{\text{6th}}_{i-\frac{1}{2}} = \frac{1}{60} u_{i-\frac{1}{2}} \left( 37 
294      (\Psi_i + \Psi_{i-1}) - 8 (\Psi_{i+1} + \Psi_{i-2}) + (\Psi_{i+2}
295      + \Psi_{i-3}) \right)$ 
296       };\\
297       };
298   \end{tikzpicture}
299   
300   \vspace{5mm}
301   
302   \textbf{Artificially added numerical dissipation term}
303   \begin{tikzpicture}[scale=2]
304      \tikzstyle{ann} = [draw=none,fill=none,right]
305      \matrix[nodes={draw, thick, fill=blue!40}, row sep=0.3cm,column sep=0.5cm]{
306      \node[rectangle, rounded corners]{
307      $\frac{-1}{60} \left| u_{i-\frac{1}{2}} \right| \left( 10 (\Psi_i - 
308      \Psi_{i-1}) - 5 (\Psi_{i+1} - \Psi_{i-2}) + (\Psi_{i+2} - \Psi_{i-3}) \right)$ 
309       };\\
310       };
311   \end{tikzpicture}
312\end{frame}
313
314% Folie 10
315\begin{frame}
316   \frametitle{Advection Methods – Wicker/Skamarock (III)}
317   
318   \begin{tikzpicture}[scale=2]
319      \tikzstyle{ann} = [draw=none,fill=none,right]
320      \matrix[nodes={draw, thick, fill=blue!20}, row sep=0.3cm,column sep=0.5cm]{
321      \node[rectangle, rounded corners]{
322      $F^{\text{6th}}_{i-\frac{1}{2}} = \frac{1}{60} u_{i-\frac{1}{2}}
323      \left( 37 (\Psi_i + \Psi_{i-1}) - 8 (\Psi_{i+1} + \Psi_{i-2}) + (\Psi_{i+2} + 
324      \Psi_{i-3}) \right)$ 
325       };\\
326       };
327   \end{tikzpicture}
328   
329   \begin{columns}[T]
330      \begin{column}{0.7\textwidth}
331         \includegraphics[width=1\textwidth]{numerics_bc_figures/numerical_oscillations.png}
332      \end{column}
333      \begin{column}{0.3\textwidth}
334         Centered Finite Differences produces numerical oscillations (''wiggles'')
335         near sharp gradients.
336      \end{column}
337   \end{columns}
338\end{frame}
339
340% Folie 11
341\begin{frame}
342   \frametitle{Advection Methods – Wicker/Skamarock (IV)}
343   \footnotesize
344   
345   \begin{columns}[T]
346      \begin{column}{0.1\textwidth}
347            \begin{tikzpicture}[scale=2]
348      \tikzstyle{ann} = [draw=none,fill=none,right]
349      \matrix[nodes={draw, thick, fill=blue!40}, row sep=0.3cm,column sep=0.5cm]{
350      \node[draw=none,fill=none]{
351      $F^{\text{5th}}_{i-\frac{1}{2}} = F^{\text{6th}}_{i-\frac{1}{2}}$
352      };\\
353       };
354   \end{tikzpicture}
355      \end{column}
356      \begin{column}{0.8\textwidth}
357            \begin{tikzpicture}[scale=2]
358      \tikzstyle{ann} = [draw=none,fill=none,right]
359      \matrix[nodes={draw, thick, fill=blue!40}, row sep=0.3cm,column sep=0.5cm]{
360      \node[rectangle, rounded corners]{
361      $- \frac{1}{60} \left| u_{i-\frac{1}{2}} \right| \left( 10 (\Psi_i - \Psi_{i-1}) - 
362      5 (\Psi_{i+1} - \Psi_{i-2}) + (\Psi_{i+2} - \Psi_{i-3}) \right)$ 
363       };\\
364       };
365   \end{tikzpicture}
366      \end{column}
367   \end{columns}
368   
369   \begin{columns}[T]
370      \begin{column}{0.7\textwidth}
371         \includegraphics[width=1\textwidth]{numerics_bc_figures/numerical_oscillations_2.png}
372      \end{column}
373      \begin{column}{0.3\textwidth}
374         \vspace{3mm}
375         \underline{Advantage}\\
376         Numerical Dissipation damps small scale oscillations.\\
377         \vspace{3mm}
378         \underline{Disadvantage}\\
379         In a turbulent flow numerical dissipation removes energy from small scales.
380
381      \end{column}
382   \end{columns}
383\end{frame}
384
385% Folie 12
386\begin{frame}
387   \frametitle{Advection Methods – Wicker/Skamarock (V)}
388   
389    \begin{columns}[T]
390      \begin{column}{0.6\textwidth}
391         \includegraphics[width=1\textwidth]{numerics_bc_figures/numerical_properties.png}
392      \end{column}
393      \begin{column}{0.4\textwidth}
394         \includegraphics[width=1\textwidth]{numerics_bc_figures/pw_ws.png}
395      \end{column}
396   \end{columns}
397   
398   \begin{itemize}
399      \item Better resolution of larger scales $(> 8\,\Delta x)$ and hence less
400      numerical energy transfer from larger to smaller scales compared to lower
401      order schemes.
402      \item Less spectral energy at smaller scales.
403   \end{itemize}
404   
405\end{frame}
406
407% Folie 13
408\begin{frame}
409   \frametitle{Pressure Solver (I)}
410   \footnotesize
411   \begin{itemize}
412      \item<1-> Governing equations of PALM require incompressibility
413      \item<2-> Incompressibility is reached by a predictor-corrector method\\
414            \scriptsize
415            1. Momentum equations are solved without the pressure term giving a
416            provisional velocity field which is not free of divergence.\\
417            $\overline{u}^{t+\Delta t}_{i_{\mathrm{prov}}} = \overline{u}^t_i +
418            \Delta t \left( - \frac{\partial}{\partial x_k} \overline{u}^t_k
419            \overline{u}^t_i - (\varepsilon_{ijk} f_j \overline{u}^t_k
420            - \varepsilon_{i3k} f_3 u_{\mathrm{g}_k}) 
421            + g \frac{\overline{\theta^*}^t}{\theta_0} \delta_{i3}
422            - \frac{\partial}{\partial x_k} \overline{u'_k u'_i}^t \right)$\\
423            \vspace{3mm}
424            \onslide<3-> 2. Assign all remaining divergences to the (perturbation)
425            pressure $p^*$ so that the new corrected velocity field is the sum of the
426            provisional, divergent field and the perturbation pressure term.\\
427            $\overline{u}^{t+\Delta t}_{i} = 
428            \overline{u}^{t+\Delta t}_{i_{\mathrm{prov}}} - 
429            \frac{\Delta t}{\rho_0} \frac{\partial \overline{p^*}^t}{\partial x_i}$\\
430            \vspace{3mm}
431            \onslide<4-> 3. The divergence operator is applied to this equation.
432            Demanding a corrected velocity field free of divergence, this leads to a
433            Poisson equation for the perturbation pressure.\\
434            $\frac{\partial^2 \overline{p^*}^t}{\partial x_i^2} = \frac{\rho_0}
435            {\Delta t} \frac{\partial \overline{u}_{i_{\mathrm{prov}}}^{t + \Delta t}}
436            {\partial x_i}$\\
437            \vspace{3mm}
438            \onslide<5-> 4. After solving the Poisson equation, the final velocity field
439            is \\
440            calculated as given in step 2.\\
441
442   \end{itemize}
443\end{frame}
444
445% Folie 14
446\begin{frame}
447   \frametitle{Pressure Solver (II)}
448   \small
449   
450   \begin{itemize}
451      \item FFT-solver\\
452      \onslide<1-> 1. Discretization of the Poisson-equation by central differences\\
453      \onslide<2-> 2. 2D discrete FFT in both horizontal directions\\
454      \onslide<3-> 3. Solving the resulting tridiagonal set of linear equations\\
455      \onslide<4-> 4. Inverse 2D discrete FFT in both horizontal directions leading
456      to the perturbation pressure
457
458      \begin{itemize}
459         \item<5-> Very fast and accurate, $\mathcal{O}(n \log n)$, $n$:
460         number of gridpoints
461         \item<6-> CPU requirement $<$ 50\% of total CPU time
462         \item<7-> Due to non-locality of the FFT, transpositions are required
463         on parallel computers
464         \item<8-> Usage requires periodic boundary conditions and uniform grids
465         along $x$ and $y$
466      \end{itemize}
467   \end{itemize}
468\end{frame}
469
470% Folie 15
471\begin{frame}
472   \frametitle{Pressure Solver (III)}
473   \scriptsize   
474   \begin{columns}
475      \begin{column}{1.03\textwidth}
476         \begin{itemize}
477            \item<1-> Multigrid-method\\
478            \begin{itemize}\scriptsize 
479               \item Iterative solver\\
480               \scriptsize
481               basic idea: Poisson equation is transformed to a fixed point problem:\\
482               $\vec{p}^{k+1} = T \cdot \vec{p}^k + \vec{c}^k$\\
483               \vspace{1mm}
484               \onslide<2-> starting from a first guess, the solution will be
485               improved by repeated execution of the fixed point problem:\\
486               $\begin{array}{rcl}
487               \vec{p}^{1} &=&T \cdot \vec{p}^0 + \vec{c}^0\\
488               \vec{p}^{2} &=&T \cdot \vec{p}^1 + \vec{c}^1
489               \vspace{-2mm}\\
490               
491               &\vdots&
492               \vspace{-1.5mm}\\
493               
494               \vec{p}^{k} &=&T \cdot \vec{p}^{k-1} + \vec{c}^{k-1}\\
495               \vec{p}^{k+1} &=&T \cdot \vec{p}^k + \vec{c}^k\\
496               \end{array}$\\
497               \vspace{1mm}
498               \onslide<3-> Depending on the structure of the matrix $T$ and vector
499               $c$ different iterative solvers can be defined, e.g.: Jacobi-scheme
500               (here on 2D-uniform grid):\\
501               $p^{k+1}_{i,j} = \frac{1}{4} \cdot \left( p^k_{i-1,j} + p^k_{i+1,j}
502               + p^k_{i,j-1} + p^k_{i,j+1} - \Delta x^2 f(i,j,k) \right)$\\
503               \scriptsize \vspace{2mm}
504               \item<4-> With each iteration step $k$ the improved solution converges towards
505               the exact solution.
506               \item<5-> Iterative schemes are 'local schemes' $\rightarrow$ information
507               is needed \\ only from neighboring grid-points.
508               \item<6-> Very low convergence: $\mathcal{O}(n^2)$.
509           \end{itemize}
510         \end{itemize}
511      \end{column}
512   \end{columns}
513\end{frame}
514   
515% Folie 16
516\begin{frame}
517   \frametitle{Pressure Solver (IV)} 
518   \begin{itemize}
519      \item<1-> Multigrid-method\\
520      \begin{itemize}
521         \item Due to their locality, iterative solvers show a frequency-dependent
522            reduction of the residual: low frequencies are reduced slower than high
523            frequencies.
524         \item<2-> The main idea of the multigrid method is to reduce errors of different
525         frequencies on grids with different grid spacing:
526         \begin{itemize}
527            \item errors of high frequency are reduced on fine grids
528            \item errors of low frequency are reduced on coarse grids.
529         \end{itemize}
530      \end{itemize}
531   \end{itemize}
532   \onslide<2->
533   \begin{figure}[htp]
534      \centering
535      \includegraphics[scale=0.35]{numerics_bc_figures/errors.png}
536   \end{figure}
537\end{frame}
538
539% Folie 17
540\begin{frame}
541   \frametitle{Pressure Solver (V)} 
542   
543   \begin{columns}[T]
544      \begin{column}{0.65\textwidth}
545         \begin{itemize}
546            \item<1-> Multigrid-method\\
547            \begin{itemize}
548               \footnotesize
549               \item On each grid-level an approximate solution of the fixed point
550                  equation is obtained performing a few iterations.
551               \item<2-> The solution is transmitted to the next coarser grid-level
552                  where it is used as the first guess to solve the fixed point problem.
553               \item<3-> This procedure is performed up to the coarsest grid-level
554                  containing two grid-points in each direction.
555               \item<4-> From the coarsest grid-level the procedure is passed in
556                  backward order to get the final solution.
557               \item<5-> For large grids faster than FFT method.
558               \item<6-> V- and W-cycles are implemented.
559            \end{itemize}
560      \end{itemize}
561      \end{column}
562      \begin{column}{0.5\textwidth}
563         \onslide<2->
564         \includegraphics[width=1\textwidth]{numerics_bc_figures/multigrid.png}
565      \end{column}
566   \end{columns}
567\end{frame}
568
569
570
571\section{Boundary Conditions}
572\subsection{Boundary Conditions}
573
574% Folie 18
575\begin{frame}
576   \frametitle{Boundary Conditions (I)}
577   
578   \begin{itemize}
579      \item<1-> Lateral $(xy)$ boundary conditions:\\
580      \begin{itemize}
581         \item Cyclic by default, allowing undisturbed evolution / advection of turbulence.
582            \begin{columns}[T]
583               \begin{column}{0.2\textwidth}
584                  %leer
585               \end{column}
586               \begin{column}{0.5\textwidth}
587                  \includegraphics[width=1\textwidth]{numerics_bc_figures/lateral_bc.png}\\
588                  \vspace{2mm}
589               \end{column}
590               \begin{column}{0.4\textwidth}
591                  $\begin{array}{rcl}
592                     \Psi(-1) &=& \Psi(n)\\
593                     \Psi(n+1) &=& \Psi(0)\\
594                  \end{array}$
595               \end{column}
596            \end{columns}
597   
598         \item<2-> Dirichlet (inflow) and radiation (outflow) conditions are allowed along
599            either $x$- or $y$-direction.
600
601         \item<3-> In case of a Dirichlet condition, the inflow is laminar (by default) and
602            the domain has to be extended to allow for the development of a turbulent
603            state, if neccessary.
604         \item<4->  Non-cyclic lateral conditions require the use of the multigrid-method
605            for solving the Poisson-equation.
606           
607      \end{itemize}
608   \end{itemize}
609\end{frame}
610
611% Folie 19
612\begin{frame}
613   \frametitle{Boundary Conditions (II)}
614   
615   \begin{columns}[T]
616      \begin{column}{0.7\textwidth}
617         \scriptsize
618         \begin{itemize}
619            \item<1-> Surface boundary condition:
620            \begin{itemize}
621               \scriptsize
622               \item<1-> Monin-Obukhov-similarity is used by default, i.e. a Prandtl-layer is
623                  assumed between the surface and the first grid layer.\\
624                  $\frac{\partial \overline{u}}{\partial z} = \frac{u_{*}}{\kappa z}
625                  \Phi_{\mathrm{m}}$; \hspace{3mm} $u_{*} = \sqrt{- \overline{w' u'_0}}
626                  = \sqrt{\frac{\tau_0}{\overline{\rho}}}$\\
627                  $\frac{\partial \overline{\theta}}{\partial z} = \frac{\vartheta_{*}}
628                  {\kappa z} \Phi_{\mathrm{h}}$; \hspace{3mm} $\vartheta_{*} = 
629                  \frac{\overline{w' \theta'_0}}{u_{*}}$\\
630               \vspace{2mm}
631               \item<2-> Integration between $z=z_0$ (roughness height) and $z=z_{\mathrm{p}}$
632                  (top of Prandtl-layer, $k=1$) gives the only unknowns $u_{*}$ and
633                  $\theta_{*}$ which then define the surface momentum and heat flux,
634                  used as the real boundary conditions.\\
635               \vspace{2mm}
636               \item<3-> $\Phi_{\mathrm{m}}$, $\Phi_{\mathrm{h}}$: Dyer-Businger functions\\
637               \onslide<4->$\Phi_{\mathrm{m}} = \left\{ \begin{array}{cc}
638                     1+5\,\mathrm{Rif} & \text{stable}\\
639                     1 & \text{neutral}\\
640                     (1-16\,\mathrm{Rif})^{-1/4} & \text{unstable}\\
641                  \end{array} \right. $
642
643            \end{itemize}
644         \end{itemize}
645      \end{column}
646      \begin{column}{0.3\textwidth}
647         \onslide<1->
648         \includegraphics[width=1\textwidth]{numerics_bc_figures/surface_bc.png}\\
649         \vspace{-2mm}
650         Prandtl-layer\\
651         \vspace{8mm}
652         \onslide<5->
653         $\mathrm{Rif} = \frac{\frac{g}{\tilde{\theta}}
654         \overline{w' \theta'_0}}{\overline{w' u'}
655         \frac{\partial \overline{u}}{\partial z}}$\\
656         \scriptsize Richardson flux number
657      \end{column}
658   \end{columns}
659\end{frame}
660
661% Folie 20
662\begin{frame}
663   \frametitle{Boundary Conditions (III)}
664
665   \begin{itemize}
666      \item<1-> Surface boundary condition:
667      \begin{itemize}
668         \footnotesize
669         \item<1-> Monin-Obukhov-similarity is only valid for a
670            horizontal surface with homogeneous conditions.
671         \item<2-> The surface temperature has to be prescribed.
672            Alternatively, the surface heat flux can be prescribed.
673         \item<3-> Instead of MO-similarity, no-slip conditions or
674            free-slip conditions can be used
675            \begin{equation*}
676               u(z=0) = 0, \quad v(z=0)=0 \hspace{15mm}
677               \frac{\partial u}{\partial z} = 0, \quad 
678               \frac{\partial v}{\partial z} = 0 \qquad
679            \end{equation*}
680            realized by
681            \begin{flalign*}
682               u(k=0)=-u(k=1) \hspace{18mm} u(k=0)=u(k=1)\\
683               v(k=0)=-v(k=1) \hspace{18mm} v(k=0)=v(k=1)
684            \end{flalign*}
685         \item<4-> Pressure boundary condition:
686            $\dfrac{\partial p}{\partial z} = 0$ 
687            in order to guarantee $w(z=0)=0$
688         \item<5-> SGS-TKE condition $\dfrac{\partial e}{\partial z}=0$
689      \end{itemize}
690   \end{itemize}
691\end{frame}
692
693% Folie 21
694\begin{frame}
695   \frametitle{Boundary Conditions (IV)}
696
697   \begin{itemize}
698      \item<1-> Boundary conditions at the top (default)
699      \begin{itemize}
700         \item<1-> Dirichlet conditions for velocities: $u=u_{\mathrm{g}},
701            \quad v=v_{\mathrm{g}}, \quad w=0$
702         \item<2-> Neumann conditions (temporal constant gradients) for scalars:
703            $$\frac{\partial \theta}{\partial z} = 
704            \left. \frac{\partial \theta}{\partial z} \right\vert_{t=0}$$
705         \item<3-> Pressure: Dirichlet $p=0$ 
706            or Neumann $\dfrac{\partial p}{\partial z} = 0$
707         \item<4-> SGS-TKE: Neumann $\dfrac{\partial e}{\partial z} = 0$
708         \item<5-> A damping layer can be switched on in order to absorb
709            gravity waves.
710      \end{itemize}
711   \end{itemize}
712\end{frame}
713
714% Folie 22
715\begin{frame}
716   \frametitle{Initial Conditions}
717
718   All 3D-arrays are initialized with vertical profiles (horizontally homogeneous).\\
719   \quad \\
720   Two different profiles can be chosen:
721   \begin{itemize}
722      \item<2-> \textbf{constant (piecewise linear) profiles}
723         \begin{itemize}
724            \footnotesize
725            \item \textbf{e.g.} $u=0, v=0, \dfrac{\partial \theta}{\partial z}=0$ 
726               \textbf{up to} $z=\unit[1000]{m}$,
727               $\dfrac{\partial \theta}{\partial z}=+1.0$ \textbf{up to top}
728         \end{itemize}
729      \item<3-> \textbf{velocity profiles calculated by a 1D-model (which is a part of PALM)}
730         \begin{itemize}
731            \footnotesize
732            \item  \textbf{constant (piecewise linear) temperature profile is used
733               for the 1D-model}
734         \end{itemize}
735   \end{itemize}
736   \onslide<4->
737   \underline{Under horizontally homogeneous initial conditions, random}\\ 
738   \underline{fluctuations have to be added in order to generate turbulence!}
739\end{frame}
740
741\end{document}
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