tformulas.tex - pism - [fork] customized build of PISM, the parallel ice sheet model (tillflux branch)
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       tformulas.tex (3217B)
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            1 \newcommand{\R}{\frac{\Delta t\,K^{n}_{k}}{\Delta z^2\,\rho_{i}}}
            2 \newcommand{\mufactor}{\frac{\Delta t}{2\,\Delta z}}
            3 \newcommand{\discretization}{\rho_{i}\,\left(\frac{w^{n}_{k}\,\Delta\left(E^{n+1}_{k}\right)\,\lambda}{2\,\Delta z}+\frac{w^{n}_{k}\,\operatorname{Up}\left(E^{n+1}_{k} , w^{n}_{k}\right)\,\left(1-\lambda\right)}{\Delta z}+\frac{E^{n+1}_{k}-E^{n}_{k}}{\Delta t}\right)-\left(\frac{K^{n}_{k+\frac{1}{2}}\,\mathbf{\delta}_{+}\left(E^{n+1}_{k}\right)}{\Delta z}-\frac{K^{n}_{k-\frac{1}{2}}\,\mathbf{\delta}_{-}\left(E^{n+1}_{k}\right)}{\Delta z}\right)\,\frac{1}{\Delta z}=\Phi^{n}_{k}}
            4 % Values of E on the grid, for the generic equation.
            5 \newcommand{\E}{E^{n+1}}
            6 \newcommand{\El}{E^{n+1}_{k-1}}
            7 \newcommand{\Eu}{E^{n+1}_{k+1}}
            8 % Matrix entries, w > 0.
            9 \newcommand{\Lp}{w^{n}_{k}\,\mu\,\left(\lambda-2\right)-R^{n}_{k-\frac{1}{2}}}
           10 \newcommand{\Dp}{-2\,w^{n}_{k}\,\mu\,\left(\lambda-1\right)+R^{n}_{k+\frac{1}{2}}+R^{n}_{k-\frac{1}{2}}+1}
           11 \newcommand{\Up}{w^{n}_{k}\,\mu\,\lambda-R^{n}_{k+\frac{1}{2}}}
           12 \newcommand{\Bp}{\frac{\Delta t\,\Phi^{n}_{k}}{\rho_{i}}+E^{n}_{k}}
           13 % Matrix entries, w < 0.
           14 \newcommand{\Lm}{-w^{n}_{k}\,\mu\,\lambda-R^{n}_{k-\frac{1}{2}}}
           15 \newcommand{\Dm}{2\,w^{n}_{k}\,\mu\,\left(\lambda-1\right)+R^{n}_{k+\frac{1}{2}}+R^{n}_{k-\frac{1}{2}}+1}
           16 \newcommand{\Um}{-w^{n}_{k}\,\mu\,\left(\lambda-2\right)-R^{n}_{k+\frac{1}{2}}}
           17 \newcommand{\Bm}{\frac{\Delta t\,\Phi^{n}_{k}}{\rho_{i}}+E^{n}_{k}}
           18 % Neumann B. C. at the base.
           19 \newcommand{\neumannb}{\frac{E^{n+1}_{1}-E^{n+1}_{-1}}{2\,\Delta z}=G_{0}}
           20 \newcommand{\Dpb}{-2\,w^{n}_{0}\,\mu\,\left(\lambda-1\right)+R^{n}_{\frac{1}{2}}+R^{n}_{-\frac{1}{2}}+1}
           21 \newcommand{\Upb}{2\,w^{n}_{0}\,\mu\,\left(\lambda-1\right)-R^{n}_{\frac{1}{2}}-R^{n}_{-\frac{1}{2}}}
           22 \newcommand{\Bpb}{2\,G_{0}\,\Delta z\,\left(w^{n}_{0}\,\mu\,\lambda-2\,w^{n}_{0}\,\mu-R^{n}_{-\frac{1}{2}}\right)+\frac{\Phi^{n}_{0}\,\Delta t}{\rho_{i}}+E^{n}_{0}}
           23 \newcommand{\Dmb}{2\,w^{n}_{0}\,\mu\,\left(\lambda-1\right)+R^{n}_{\frac{1}{2}}+R^{n}_{-\frac{1}{2}}+1}
           24 \newcommand{\Umb}{-2\,w^{n}_{0}\,\mu\,\left(\lambda-1\right)-R^{n}_{\frac{1}{2}}-R^{n}_{-\frac{1}{2}}}
           25 \newcommand{\Bmb}{-2\,G_{0}\,\Delta z\,\left(w^{n}_{0}\,\mu\,\lambda+R^{n}_{-\frac{1}{2}}\right)+\frac{\Phi^{n}_{0}\,\Delta t}{\rho_{i}}+E^{n}_{0}}
           26 % Neumann B. C. at the surface.
           27 \newcommand{\neumanns}{\frac{E^{n+1}_{k_{s}+1}-E^{n+1}_{k_{s}-1}}{2\,\Delta z}=G_{k_{s}}}
           28 \newcommand{\Lps}{2\,w^{n}_{k_{s}}\,\mu\,\left(\lambda-1\right)-R^{n}_{k_{s}+\frac{1}{2}}-R^{n}_{k_{s}-\frac{1}{2}}}
           29 \newcommand{\Dps}{-2\,w^{n}_{k_{s}}\,\mu\,\left(\lambda-1\right)+R^{n}_{k_{s}+\frac{1}{2}}+R^{n}_{k_{s}-\frac{1}{2}}+1}
           30 \newcommand{\Bps}{-2\,\Delta z\,G_{k_{s}}\,\left(w^{n}_{k_{s}}\,\mu\,\lambda-R^{n}_{k_{s}+\frac{1}{2}}\right)+\frac{\Delta t\,\Phi^{n}_{k_{s}}}{\rho_{i}}+E^{n}_{k_{s}}}
           31 \newcommand{\Lms}{-2\,w^{n}_{k_{s}}\,\mu\,\left(\lambda-1\right)-R^{n}_{k_{s}+\frac{1}{2}}-R^{n}_{k_{s}-\frac{1}{2}}}
           32 \newcommand{\Dms}{2\,w^{n}_{k_{s}}\,\mu\,\left(\lambda-1\right)+R^{n}_{k_{s}+\frac{1}{2}}+R^{n}_{k_{s}-\frac{1}{2}}+1}
           33 \newcommand{\Bms}{2\,\Delta z\,G_{k_{s}}\,\left(w^{n}_{k_{s}}\,\mu\,\lambda-2\,w^{n}_{k_{s}}\,\mu+R^{n}_{k_{s}+\frac{1}{2}}\right)+\frac{\Delta t\,\Phi^{n}_{k_{s}}}{\rho_{i}}+E^{n}_{k_{s}}}