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tssa-discretization.tex (4682B)
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1 \documentclass{amsart}
2 \usepackage[margin=1in]{geometry}
3 \usepackage{underscore}
4 \date{Time-stamp: "ssa-discretization.tex, saved on Mon, Apr 18, 2011 at 12:16pm"}
5
6 % Read in formulas generated by Maxima:
7 \input{formulas.tex}
8
9 \begin{document}
10 \title{On computing coefficients in a discretization of SSA equations}
11 \author{C. Khroulev}
12 \maketitle
13
14 The SSA equations (ignoring basal drag) are
15 \begin{align}
16 -\left[ 2\bar\nu H\left( 2u_{x} + v_{y}\right)\right]_{x} - \left[\bar\nu
17 H\left(u_{y}+v_{x} \right) \right]_{y} &= - \rho gH h_{x} \label{SSA1} \\
18 -\left[ \bar\nu H\left( u_{y} + v_{x} \right)\right]_{x} - \left[2\bar\nu
19 H\left(u_{x}+2v_{y} \right) \right]_{y} &= -\rho gH h_{y} \label{SSA2}
20 \end{align}
21
22 We use centered finite differences to create a discretization of these
23 equations. For outer derivatives, these differences are centered at the current
24 $(i,j)$ point and use staggered grid locations. Hence $\bar \nu H$ is evaluated on the staggered grid.
25
26 The one-sided differences below are, in fact, centered with the center at the
27 staggered grid location.
28
29 \medskip
30 The four terms of the discretization of left hand side of the \eqref{SSA1} are
31 given below\footnote{These formulas are not as pretty as they should be because
32 they are automatically generated using Maxima.}.
33 \begin{eqnarray*}
34 \lhsI
35 \end{eqnarray*}
36
37 The discretization of the left hand side of \eqref{SSA2} is the sum of the following.
38 \begin{eqnarray*}
39 \lhsII
40 \end{eqnarray*}
41
42 Here
43 \begin{itemize}
44 \item $w_{i+\frac12,j}$ is a constant equal to $1$ if the face at $(i+\frac12,j)$ is
45 an interface between two ice-filled cells and $0$ if it is an interface
46 between ice-filled and ice-free cells.
47
48 This allows us to drop one of four terms in left hand sides of equations
49 \eqref{SSA1} and \eqref{SSA2} to implement the calving front boundary
50 condition.
51
52 \item $\delta_{+x}f_{i,j} = f_{i+1,j}-f_{i,j}$, $\delta_{-x}f_{i,j} =
53 f_{i,j}-f_{i-1,j}$, and $\Delta_{x}f_{i,j} = f_{i+1,j}-f_{i-1,j}$, and
54 similarly for $y$.
55 \item Note that the $y$-derivative at $(i+\frac12,j)$ is computed as the
56 average of two centered differences using regular grid points.
57 \item $N := \bar \nu H$.
58 \item $c_{w}$ is the value of $N$ at the western face of the current cell;
59 $c_{e}$, $c_{n}$ and $c_{s}$ are shortcuts for $N$ at the east, north and
60 south faces.
61 \end{itemize}
62
63 The next issue we need to handle is the fact that at the ice front some values
64 used in the discretization above are not available. To resolve this, we rewrite
65 $\Delta_{x}f_{i,j}$ as $\Delta_{x}f_{i,j} = \delta_{+x}f_{i,j} +
66 \delta_{-x}f_{i,j}$ and add ``weights'' $w_{\cdot,\cdot}$:
67
68 \begin{equation}
69 \label{eq:1}
70 \Delta_{x}f_{i,j} = w_{i+\frac12,j}\delta_{+x}f_{i,j} + w_{i-\frac12,j}\delta_{-x}f_{i,j}
71 \end{equation}
72 where $w_{\cdot,\cdot}$ is defined above.
73
74 If one of $w_{\cdot,j}$ in equation \eqref{eq:1} is zero and another is one,
75 this centered difference becomes a one-sided one.
76
77 Now we can write a discretization that works both in the interior and at
78 the ice margin by ``switching'' from a second-order centered difference in the
79 interior to first-order, ice-inward one-sided differences at the margin.
80
81 This yields (for the left hand side of equation \eqref{SSA1}):
82 \begin{eqnarray*}
83 \lhsIII
84 \end{eqnarray*}
85 which is almost incomprehensible.
86
87 \newpage
88 \section{Coefficients for the shelf interior (computed using Maxima)}
89 \label{sec:shelf-interior}
90
91 \begin{table}[h]
92 \centering
93 \begin{tabular}{r||c|c|c}
94 \CUfirstInterior
95 \end{tabular}
96 \caption{$u$, first equation}
97 \label{tab:u-interior-i}
98 \end{table}
99
100 \begin{table}[h]
101 \centering
102 \begin{tabular}{r||c|c|c}
103 \CUsecondInterior
104 \end{tabular}
105 \caption{$u$, second equation}
106 \label{tab:u-interior-ii}
107 \end{table}
108
109 \begin{table}[h]
110 \centering
111 \begin{tabular}{r||c|c|c}
112 \CVfirstInterior
113 \end{tabular}
114 \caption{$v$, first equation}
115 \label{tab:v-interior-i}
116 \end{table}
117
118 \begin{table}[h]
119 \centering
120 \begin{tabular}{r||c|c|c}
121 \CVsecondInterior
122 \end{tabular}
123 \caption{$v$, second equation}
124 \label{tab:v-interior-ii}
125 \end{table}
126
127 \newpage
128 \section{At the ice margin}
129 \label{sec:at-ice-margin}
130
131 \begin{table}[h]
132 \centering
133 \begin{tabular}{r||c|c|c}
134 \CUfirstMargin
135 \end{tabular}
136 \caption{$u$, first equation}
137 \label{tab:u-margin-i}
138 \end{table}
139
140 Unknown $u$ in the second equation:
141 \begin{eqnarray*}
142 \CUsecondMargin
143 \end{eqnarray*}
144
145 \newpage
146
147 Unknown $v$ in the first equation:
148 \begin{eqnarray*}
149 \CVfirstMargin
150 \end{eqnarray*}
151
152 \begin{table}[h]
153 \centering
154 \begin{tabular}{r||c|c|c}
155 \CVsecondMargin
156 \end{tabular}
157 \caption{$v$, second equation}
158 \label{tab:v-margin-ii}
159 \end{table}
160
161 \end{document}