tformulas.tex - pism - [fork] customized build of PISM, the parallel ice sheet model (tillflux branch)
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tformulas.tex (10927B)
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1 \newcommand{\lhsI}{
2 &\displaystyle -{{2\,w_{i+{{1}\over{2}},j}\,N_{i+{{1}\over{2}},j}\,
3 \left({{\Delta_y\left(v_{i+1,j}\right)+\Delta_y\left(v_{i,j}\right)
4 }\over{4\,\Delta y}}+{{2\,\delta_{+x}\left(u_{i,j}\right)}\over{
5 \Delta x}}\right)}\over{\Delta x}} & \\
6 &\displaystyle -{{w_{i,j+{{1}\over{2}}}\,N_{i,j+{{1}\over{2}}}\,
7 \left({{\Delta_x\left(v_{i,j+1}\right)+\Delta_x\left(v_{i,j}\right)
8 }\over{4\,\Delta x}}+{{\delta_{+y}\left(u_{i,j}\right)}\over{
9 \Delta y}}\right)}\over{\Delta y}} & \\
10 &\displaystyle {{2\,w_{i-{{1}\over{2}},j}\,N_{i-{{1}\over{2}},j}\,
11 \left({{\Delta_y\left(v_{i,j}\right)+\Delta_y\left(v_{i-1,j}\right)
12 }\over{4\,\Delta y}}+{{2\,\delta_{-x}\left(u_{i,j}\right)}\over{
13 \Delta x}}\right)}\over{\Delta x}} & \\
14 &\displaystyle {{w_{i,j-{{1}\over{2}}}\,N_{i,j-{{1}\over{2}}}\,\left(
15 {{\Delta_x\left(v_{i,j}\right)+\Delta_x\left(v_{i,j-1}\right)}\over{
16 4\,\Delta x}}+{{\delta_{-y}\left(u_{i,j}\right)}\over{\Delta y}}
17 \right)}\over{\Delta y}} & \\
18 }
19 \newcommand{\lhsII}{
20 &\displaystyle -{{w_{i+{{1}\over{2}},j}\,N_{i+{{1}\over{2}},j}\,
21 \left({{\Delta_y\left(u_{i+1,j}\right)+\Delta_y\left(u_{i,j}\right)
22 }\over{4\,\Delta y}}+{{\delta_{+x}\left(v_{i,j}\right)}\over{
23 \Delta x}}\right)}\over{\Delta x}} & \\
24 &\displaystyle -{{2\,w_{i,j+{{1}\over{2}}}\,N_{i,j+{{1}\over{2}}}\,
25 \left({{\Delta_x\left(u_{i,j+1}\right)+\Delta_x\left(u_{i,j}\right)
26 }\over{4\,\Delta x}}+{{2\,\delta_{+y}\left(v_{i,j}\right)}\over{
27 \Delta y}}\right)}\over{\Delta y}} & \\
28 &\displaystyle {{2\,w_{i,j-{{1}\over{2}}}\,N_{i,j-{{1}\over{2}}}\,
29 \left({{2\,\delta_{-y}\left(v_{i,j}\right)}\over{\Delta y}}+{{
30 \Delta_x\left(u_{i,j}\right)+\Delta_x\left(u_{i,j-1}\right)}\over{4
31 \,\Delta x}}\right)}\over{\Delta y}} & \\
32 &\displaystyle {{w_{i-{{1}\over{2}},j}\,N_{i-{{1}\over{2}},j}\,\left(
33 {{\delta_{-x}\left(v_{i,j}\right)}\over{\Delta x}}+{{\Delta_y\left(u
34 _{i,j}\right)+\Delta_y\left(u_{i-1,j}\right)}\over{4\,\Delta y}}
35 \right)}\over{\Delta x}} & \\
36 }
37 \newcommand{\lhsIII}{
38 &\displaystyle -{{2\,w_{i+{{1}\over{2}},j}\,N_{i+{{1}\over{2}},j}\,
39 \left({{w_{i+1,j-{{1}\over{2}}}\,\delta_{{\it -y}}(v_{i+1,j})+w_{i+1
40 ,j+{{1}\over{2}}}\,\delta_{{\it +y}}(v_{i+1,j})+w_{i,j-{{1}\over{2}}
41 }\,\delta_{{\it -y}}(v_{i,j})+w_{i,j+{{1}\over{2}}}\,\delta_{
42 {\it +y}}(v_{i,j})}\over{4\,\Delta y}}+{{2\,w_{i+{{1}\over{2}},j}\,
43 \delta_{{\it +x}}(u_{i,j})}\over{\Delta x}}\right)}\over{\Delta x}} & \\
44 &\displaystyle -{{w_{i,j+{{1}\over{2}}}\,N_{i,j+{{1}\over{2}}}\,
45 \left({{w_{i-{{1}\over{2}},j+1}\,\delta_{{\it -x}}(v_{i,j+1})+w_{i+
46 {{1}\over{2}},j+1}\,\delta_{{\it +x}}(v_{i,j+1})+w_{i-{{1}\over{2}},
47 j}\,\delta_{{\it -x}}(v_{i,j})+w_{i+{{1}\over{2}},j}\,\delta_{
48 {\it +x}}(v_{i,j})}\over{4\,\Delta x}}+{{w_{i,j+{{1}\over{2}}}\,
49 \delta_{{\it +y}}(u_{i,j})}\over{\Delta y}}\right)}\over{\Delta y}} & \\
50 &\displaystyle {{2\,w_{i-{{1}\over{2}},j}\,N_{i-{{1}\over{2}},j}\,
51 \left({{w_{i,j-{{1}\over{2}}}\,\delta_{{\it -y}}(v_{i,j})+w_{i,j+{{1
52 }\over{2}}}\,\delta_{{\it +y}}(v_{i,j})+w_{i-1,j-{{1}\over{2}}}\,
53 \delta_{{\it -y}}(v_{i-1,j})+w_{i-1,j+{{1}\over{2}}}\,\delta_{
54 {\it +y}}(v_{i-1,j})}\over{4\,\Delta y}}+{{2\,w_{i-{{1}\over{2}},j}
55 \,\delta_{{\it -x}}(u_{i,j})}\over{\Delta x}}\right)}\over{\Delta x
56 }} & \\
57 &\displaystyle {{w_{i,j-{{1}\over{2}}}\,N_{i,j-{{1}\over{2}}}\,\left(
58 {{w_{i-{{1}\over{2}},j}\,\delta_{{\it -x}}(v_{i,j})+w_{i+{{1}\over{2
59 }},j}\,\delta_{{\it +x}}(v_{i,j})+w_{i-{{1}\over{2}},j-1}\,\delta_{
60 {\it -x}}(v_{i,j-1})+w_{i+{{1}\over{2}},j-1}\,\delta_{{\it +x}}(v_{i
61 ,j-1})}\over{4\,\Delta x}}+{{w_{i,j-{{1}\over{2}}}\,\delta_{{\it -y}
62 }(u_{i,j})}\over{\Delta y}}\right)}\over{\Delta y}} & \\
63 }
64 \newcommand{\CUfirstInterior}{
65 &-1 & 0 & 1 \\\hline
66 \hline
67 $1$
68 &$\displaystyle 0$
69 &$\displaystyle -{{c_{n}}\over{\Delta y^2}}$
70 &$\displaystyle 0$
71 \\
72 \hline
73 $0$
74 &$\displaystyle -{{4\,c_{w}}\over{\Delta x^2}}$
75 &$\displaystyle {{c_{s}+c_{n}}\over{\Delta y^2}}+{{4\,\left(c_{w}+
76 c_{e}\right)}\over{\Delta x^2}}$
77 &$\displaystyle -{{4\,c_{e}}\over{\Delta x^2}}$
78 \\
79 \hline
80 $-1$
81 &$\displaystyle 0$
82 &$\displaystyle -{{c_{s}}\over{\Delta y^2}}$
83 &$\displaystyle 0$
84 \\
85 }
86 \newcommand{\CUsecondInterior}{
87 &-1 & 0 & 1 \\\hline
88 \hline
89 $1$
90 &$\displaystyle {{c_{w}+2\,c_{n}}\over{4\,\Delta x\,\Delta y}}$
91 &$\displaystyle {{c_{w}-c_{e}}\over{4\,\Delta x\,\Delta y}}$
92 &$\displaystyle -{{2\,c_{n}+c_{e}}\over{4\,\Delta x\,\Delta y}}$
93 \\
94 \hline
95 $0$
96 &$\displaystyle -{{c_{s}-c_{n}}\over{2\,\Delta x\,\Delta y}}$
97 &$\displaystyle 0$
98 &$\displaystyle {{c_{s}-c_{n}}\over{2\,\Delta x\,\Delta y}}$
99 \\
100 \hline
101 $-1$
102 &$\displaystyle -{{c_{w}+2\,c_{s}}\over{4\,\Delta x\,\Delta y}}$
103 &$\displaystyle -{{c_{w}-c_{e}}\over{4\,\Delta x\,\Delta y}}$
104 &$\displaystyle {{2\,c_{s}+c_{e}}\over{4\,\Delta x\,\Delta y}}$
105 \\
106 }
107 \newcommand{\CVfirstInterior}{
108 &-1 & 0 & 1 \\\hline
109 \hline
110 $1$
111 &$\displaystyle {{2\,c_{w}+c_{n}}\over{4\,\Delta x\,\Delta y}}$
112 &$\displaystyle {{c_{w}-c_{e}}\over{2\,\Delta x\,\Delta y}}$
113 &$\displaystyle -{{c_{n}+2\,c_{e}}\over{4\,\Delta x\,\Delta y}}$
114 \\
115 \hline
116 $0$
117 &$\displaystyle -{{c_{s}-c_{n}}\over{4\,\Delta x\,\Delta y}}$
118 &$\displaystyle 0$
119 &$\displaystyle {{c_{s}-c_{n}}\over{4\,\Delta x\,\Delta y}}$
120 \\
121 \hline
122 $-1$
123 &$\displaystyle -{{2\,c_{w}+c_{s}}\over{4\,\Delta x\,\Delta y}}$
124 &$\displaystyle -{{c_{w}-c_{e}}\over{2\,\Delta x\,\Delta y}}$
125 &$\displaystyle {{c_{s}+2\,c_{e}}\over{4\,\Delta x\,\Delta y}}$
126 \\
127 }
128 \newcommand{\CVsecondInterior}{
129 &-1 & 0 & 1 \\\hline
130 \hline
131 $1$
132 &$\displaystyle 0$
133 &$\displaystyle -{{4\,c_{n}}\over{\Delta y^2}}$
134 &$\displaystyle 0$
135 \\
136 \hline
137 $0$
138 &$\displaystyle -{{c_{w}}\over{\Delta x^2}}$
139 &$\displaystyle {{4\,\left(c_{s}+c_{n}\right)}\over{\Delta y^2}}+{{
140 c_{w}+c_{e}}\over{\Delta x^2}}$
141 &$\displaystyle -{{c_{e}}\over{\Delta x^2}}$
142 \\
143 \hline
144 $-1$
145 &$\displaystyle 0$
146 &$\displaystyle -{{4\,c_{s}}\over{\Delta y^2}}$
147 &$\displaystyle 0$
148 \\
149 }
150 \newcommand{\CUfirstMargin}{
151 &-1 & 0 & 1 \\\hline
152 \hline
153 $1$
154 &$\displaystyle 0$
155 &$\displaystyle -{{c_{n}\,{\it bPP}^2}\over{\Delta y^2}}$
156 &$\displaystyle 0$
157 \\
158 \hline
159 $0$
160 &$\displaystyle -{{4\,c_{w}\,{\it aMM}^2}\over{\Delta x^2}}$
161 &$\displaystyle {{c_{n}\,{\it bPP}^2+c_{s}\,{\it bMM}^2}\over{
162 \Delta y^2}}+{{4\,\left(c_{e}\,{\it aPP}^2+c_{w}\,{\it aMM}^2\right)
163 }\over{\Delta x^2}}$
164 &$\displaystyle -{{4\,c_{e}\,{\it aPP}^2}\over{\Delta x^2}}$
165 \\
166 \hline
167 $-1$
168 &$\displaystyle 0$
169 &$\displaystyle -{{c_{s}\,{\it bMM}^2}\over{\Delta y^2}}$
170 &$\displaystyle 0$
171 \\
172 }
173 \newcommand{\CUsecondMargin}{
174 C^{u,2}_{-1,-1} &=& \displaystyle -{{c_{w}\,{\it aMM}\,{\it bMw}}\over{4\,\Delta x\,\Delta y}} \\
175 &+& \displaystyle -{{c_{s}\,{\it aMs}\,{\it bMM}}\over{2\,\Delta x\,\Delta y}} \\
176 C^{u,2}_{-1,0} &=& \displaystyle {{c_{w}\,{\it aMM}\,\left({\it bMw}-{\it bPw}\right)}\over{4\,\Delta x\,\Delta y}} \\
177 &+& \displaystyle {{{\it aMM}\,\left(c_{n}\,{\it bPP}-c_{s}\,{\it bMM}\right)}\over{2\,\Delta x\,\Delta y}} \\
178 C^{u,2}_{-1,1} &=& \displaystyle {{c_{w}\,{\it aMM}\,{\it bPw}}\over{4\,\Delta x\,\Delta y}} \\
179 &+& \displaystyle {{c_{n}\,{\it aMn}\,{\it bPP}}\over{2\,\Delta x\,\Delta y}} \\
180 C^{u,2}_{0,-1} &=& \displaystyle {{c_{s}\,\left({\it aMs}-{\it aPs}\right)\,{\it bMM}}\over{2\,\Delta x\,\Delta y}} \\
181 &+& \displaystyle {{\left(c_{e}\,{\it aPP}-c_{w}\,{\it aMM}\right)\,{\it bMM}}\over{4\,\Delta x\,\Delta y}} \\
182 C^{u,2}_{0,0} &=& \displaystyle {{c_{n}\,\left({\it aPP}-{\it aMM}\right)\,{\it bPP}+c_{s}\,\left({\it aMM}-{\it aPP}\right)\,{\it bMM}}\over{2\,\Delta x\,\Delta y}} \\
183 &+& \displaystyle {{\left(c_{e}\,{\it aPP}-c_{w}\,{\it aMM}\right)\,{\it bPP}+\left(c_{w}\,{\it aMM}-c_{e}\,{\it aPP}\right)\,{\it bMM}}\over{4\,\Delta x\,\Delta y}} \\
184 C^{u,2}_{0,1} &=& \displaystyle {{c_{n}\,\left({\it aPn}-{\it aMn}\right)\,{\it bPP}}\over{2\,\Delta x\,\Delta y}} \\
185 &+& \displaystyle {{\left(c_{w}\,{\it aMM}-c_{e}\,{\it aPP}\right)\,{\it bPP}}\over{4\,\Delta x\,\Delta y}} \\
186 C^{u,2}_{1,-1} &=& \displaystyle {{c_{e}\,{\it aPP}\,{\it bMe}}\over{4\,\Delta x\,\Delta y}} \\
187 &+& \displaystyle {{c_{s}\,{\it aPs}\,{\it bMM}}\over{2\,\Delta x\,\Delta y}} \\
188 C^{u,2}_{1,0} &=& \displaystyle {{c_{e}\,{\it aPP}\,\left({\it bPe}-{\it bMe}\right)}\over{4\,\Delta x\,\Delta y}} \\
189 &+& \displaystyle {{{\it aPP}\,\left(c_{s}\,{\it bMM}-c_{n}\,{\it bPP}\right)}\over{2\,\Delta x\,\Delta y}} \\
190 C^{u,2}_{1,1} &=& \displaystyle -{{c_{e}\,{\it aPP}\,{\it bPe}}\over{4\,\Delta x\,\Delta y}} \\
191 &+& \displaystyle -{{c_{n}\,{\it aPn}\,{\it bPP}}\over{2\,\Delta x\,\Delta y}} \\
192 }
193 \newcommand{\CVfirstMargin}{
194 C^{v,1}_{-1,-1} &=& \displaystyle -{{c_{w}\,{\it aMM}\,{\it bMw}}\over{2\,\Delta x\,\Delta y}} \\
195 &+& \displaystyle -{{c_{s}\,{\it aMs}\,{\it bMM}}\over{4\,\Delta x\,\Delta y}} \\
196 C^{v,1}_{-1,0} &=& \displaystyle {{c_{w}\,{\it aMM}\,\left({\it bMw}-{\it bPw}\right)}\over{2\,\Delta x\,\Delta y}} \\
197 &+& \displaystyle {{{\it aMM}\,\left(c_{n}\,{\it bPP}-c_{s}\,{\it bMM}\right)}\over{4\,\Delta x\,\Delta y}} \\
198 C^{v,1}_{-1,1} &=& \displaystyle {{c_{w}\,{\it aMM}\,{\it bPw}}\over{2\,\Delta x\,\Delta y}} \\
199 &+& \displaystyle {{c_{n}\,{\it aMn}\,{\it bPP}}\over{4\,\Delta x\,\Delta y}} \\
200 C^{v,1}_{0,-1} &=& \displaystyle {{c_{s}\,\left({\it aMs}-{\it aPs}\right)\,{\it bMM}}\over{4\,\Delta x\,\Delta y}} \\
201 &+& \displaystyle {{\left(c_{e}\,{\it aPP}-c_{w}\,{\it aMM}\right)\,{\it bMM}}\over{2\,\Delta x\,\Delta y}} \\
202 C^{v,1}_{0,0} &=& \displaystyle {{c_{n}\,\left({\it aPP}-{\it aMM}\right)\,{\it bPP}+c_{s}\,\left({\it aMM}-{\it aPP}\right)\,{\it bMM}}\over{4\,\Delta x\,\Delta y}} \\
203 &+& \displaystyle {{\left(c_{e}\,{\it aPP}-c_{w}\,{\it aMM}\right)\,{\it bPP}+\left(c_{w}\,{\it aMM}-c_{e}\,{\it aPP}\right)\,{\it bMM}}\over{2\,\Delta x\,\Delta y}} \\
204 C^{v,1}_{0,1} &=& \displaystyle {{c_{n}\,\left({\it aPn}-{\it aMn}\right)\,{\it bPP}}\over{4\,\Delta x\,\Delta y}} \\
205 &+& \displaystyle {{\left(c_{w}\,{\it aMM}-c_{e}\,{\it aPP}\right)\,{\it bPP}}\over{2\,\Delta x\,\Delta y}} \\
206 C^{v,1}_{1,-1} &=& \displaystyle {{c_{e}\,{\it aPP}\,{\it bMe}}\over{2\,\Delta x\,\Delta y}} \\
207 &+& \displaystyle {{c_{s}\,{\it aPs}\,{\it bMM}}\over{4\,\Delta x\,\Delta y}} \\
208 C^{v,1}_{1,0} &=& \displaystyle {{c_{e}\,{\it aPP}\,\left({\it bPe}-{\it bMe}\right)}\over{2\,\Delta x\,\Delta y}} \\
209 &+& \displaystyle {{{\it aPP}\,\left(c_{s}\,{\it bMM}-c_{n}\,{\it bPP}\right)}\over{4\,\Delta x\,\Delta y}} \\
210 C^{v,1}_{1,1} &=& \displaystyle -{{c_{e}\,{\it aPP}\,{\it bPe}}\over{2\,\Delta x\,\Delta y}} \\
211 &+& \displaystyle -{{c_{n}\,{\it aPn}\,{\it bPP}}\over{4\,\Delta x\,\Delta y}} \\
212 }
213 \newcommand{\CVsecondMargin}{
214 &-1 & 0 & 1 \\\hline
215 \hline
216 $1$
217 &$\displaystyle 0$
218 &$\displaystyle -{{4\,c_{n}\,{\it bPP}^2}\over{\Delta y^2}}$
219 &$\displaystyle 0$
220 \\
221 \hline
222 $0$
223 &$\displaystyle -{{c_{w}\,{\it aMM}^2}\over{\Delta x^2}}$
224 &$\displaystyle {{4\,\left(c_{n}\,{\it bPP}^2+c_{s}\,{\it bMM}^2
225 \right)}\over{\Delta y^2}}+{{c_{e}\,{\it aPP}^2+c_{w}\,{\it aMM}^2
226 }\over{\Delta x^2}}$
227 &$\displaystyle -{{c_{e}\,{\it aPP}^2}\over{\Delta x^2}}$
228 \\
229 \hline
230 $-1$
231 &$\displaystyle 0$
232 &$\displaystyle -{{4\,c_{s}\,{\it bMM}^2}\over{\Delta y^2}}$
233 &$\displaystyle 0$
234 \\
235 }