# Diff of /trunk/doc/user/linearPDE.tex

revision 109 by jgs, Thu Jan 27 06:21:48 2005 UTC revision 110 by jgs, Mon Feb 14 04:14:42 2005 UTC
# Line 381  returns the residual when insering \var{ Line 381  returns the residual when insering \var{
381  In cases of PDEs dominated by the advection terms $B$ and $C$ against the diffusion term $A$  In cases of PDEs dominated by the advection terms $B$ and $C$ against the diffusion term $A$
382  up-winding has been used.  up-winding has been used.
383  The \AdvectivePDE class applies upwinding to the advective terms, see \Ref{SUPG}.  The \AdvectivePDE class applies upwinding to the advective terms, see \Ref{SUPG}.
384
385    In the following we set
386    \begin{eqnarray}\label{LINEARPDE.UPWIND.Z}
387    Z\hackscore{j}=C\hackscore{j}-B\hackscore{j}
388    \mbox{ or } \\
389    Z\hackscore{ikl}=C\hackscore{ikl}-B\hackscore{ilk}
390    \end{eqnarray}
391  To measure the dominance of the advective terms over the diffusive term $A$ the  To measure the dominance of the advective terms over the diffusive term $A$ the
392  Pelclet number is used \index{Pelclet number}. The are defined as  Pelclet number is used \index{Pelclet number}. The are defined as
393  \begin{eqnarray}\label{LINEARPDE.Peclet.single}  \begin{eqnarray}\label{LINEARPDE.Peclet.single}
394  P^{B}=\frac{h\|B\hackscore{:}\|}{2\|A\hackscore{::}\|}  P=\frac{h\|Z\hackscore{:}\|}{2\|A\hackscore{::}\|}
\mbox{ and }
P^{C}=\frac{h\|C\hackscore{:}\|}{2\|A\hackscore{::}\|}
395  \end{eqnarray}  \end{eqnarray}
396  \begin{eqnarray}\label{LINEARPDE.Peclet.system}  \begin{eqnarray}\label{LINEARPDE.Peclet.system}
397  P^{B}_{ik}=\frac{h\|B\hackscore{i:k}\|}{2\|A\hackscore{i:k:}\|}  P\hackscore{ik}=\frac{h\|Z\hackscore{i:k}\|}{2\|A\hackscore{i:k:}\|}
\mbox{ and }
P^{C}_{ik}=\frac{h\|C\hackscore{ik:}\|}{2\|A\hackscore{i:k:}\|}
398  \end{eqnarray}  \end{eqnarray}
399  where $h$ is the local cell size and  where $h$ is the local cell size and
401  \|C\hackscore{:}\|^2=C\hackscore{j}C\hackscore{j} \\  \|Z\hackscore{:}\|^2=Z\hackscore{j}Z\hackscore{j} \\
402  \|A\hackscore{::}\|^2=A\hackscore{jl}A\hackscore{jl} \\  \|A\hackscore{::}\|^2=A\hackscore{jl}A\hackscore{jl} \\
403  \|C\hackscore{i:k}\|^2=C\hackscore{ijk}C\hackscore{ijk} \\  \|Z\hackscore{i:k}\|^2=\delta\hackscore{in} \delta\hackscore{km} Z\hackscore{njm}Z\hackscore{njm} \\
404  \|A\hackscore{i:k:}\|^2=A\hackscore{ijkl}A\hackscore{ijkl} \; .  \|A\hackscore{i:k:}\|^2=\delta\hackscore{in} \delta\hackscore{km} A\hackscore{njml}A\hackscore{njml} \; .
405  \end{eqnarray}  \end{eqnarray}
406  From the Pelclet number the stabilization parameters $\Xi^{B}$ and $\Xi^{C}$ are calculated:  In the case that it is $\|A\hackscore{::}\|$ we set $P=0$ if $\|Z\hackscore{:}\|=0$ and
407    $P=\infinity$ if $\|Z\hackscore{:}\|=0$. Analogously for $P$ in the case of a systems of PDEs.
408
409    From the Pelclet number the stabilization parameters $\Xi$ and $\Xi$ are calculated:
410  \begin{eqnarray}\label{LINEARPDE.Peclet.2}  \begin{eqnarray}\label{LINEARPDE.Peclet.2}
411  \Xi^{B}=\frac{\xi(P^{B}) h}{\|B\hackscore{:}\|}  \Xi=\xi(P) \frac{h}{\|Z\hackscore{:}\|}  \\
412  \mbox{ and }  \mbox{ or } \\
413  \Xi^{C}=\frac{\xi(P^{C}) h}{\|C\hackscore{:}\|} \\  \Xi\hackscore{ik}=\xi(P\hackscore{ik}) \frac{h}{\|Z\hackscore{i:k}\|}
\mbox{ or }
\Xi^{B}\hackscore{ik}=\frac{\xi(P^{B}\hackscore{ik}) h}{\|B\hackscore{i:k}\|}
\mbox{ and }
\Xi^{C}\hackscore{ik}=\frac{\xi(P^{C}\hackscore{ik}) h}{\|C\hackscore{ik:}\|}
414  \end{eqnarray}  \end{eqnarray}
415  where $\xi$ is a suitable function of the Peclet number.  where $\xi$ is a suitable function of the Peclet number.
416  In the case of a single PDE the coefficient are up-dated in the following way:  In the case of a single PDE the coefficient are up-dated in the following way:
418  A\hackscore{jl} \leftarrow A\hackscore{jl} + \Xi^{B} B\hackscore{j} B\hackscore{l} + \Xi^{C} C\hackscore{j} C\hackscore{l} \\  A\hackscore{jl} \leftarrow A\hackscore{jl} + \Xi Z\hackscore{j} Z\hackscore{l} \\
419  B\hackscore{j} \leftarrow B\hackscore{j} + \Xi^{C} C\hackscore{j} D \\  B\hackscore{j} \leftarrow B\hackscore{j} + \Xi C\hackscore{j} D \\
420  C\hackscore{j} \leftarrow C\hackscore{j} + \Xi^{B} B\hackscore{j} D \\  C\hackscore{j} \leftarrow C\hackscore{j} + \Xi B\hackscore{j} D \\
421  X\hackscore{j} \leftarrow X\hackscore{j} + (Xi^{B} B\hackscore{j} + \Xi^{C} C\hackscore{j}) Y \\  X\hackscore{j} \leftarrow X\hackscore{j} + \Xi Z\hackscore{j} Y \\
422  \end{eqnarray}  \end{eqnarray}
423  Similar for the case of a systems of PDEs:  Similar for the case of a systems of PDEs:
425  A\hackscore{ijkl} \leftarrow A\hackscore{ijl} + \Xi^{B}\hackscore{ik} B\hackscore{ijk} B\hackscore{ilk} +  A\hackscore{ijkl} \leftarrow A\hackscore{ijl} + \delta\hackscore{pm} \Xi\hackscore{mi} Z\hackscore{pij} Z\hackscore{mkl} \\
426                                                  \Xi^{C}\hackscore{ik} C\hackscore{ikj} C\hackscore{ikl} \\  B\hackscore{ijk} \leftarrow B\hackscore{ijk} +  \delta\hackscore{pm} \Xi\hackscore{mi} D\hackscore{pk} C\hackscore{mij} \\
427  B\hackscore{ijk} \leftarrow B\hackscore{ijk} + \Xi^{C}\hackscore{ij} C\hackscore{ikj} D\hackscore{ik} \\  C\hackscore{ikl} \leftarrow C\hackscore{ikl} +  \delta\hackscore{pm} \Xi\hackscore{mi} D\hackscore{pk} B\hackscore{mli} \\
428  C\hackscore{ikl} \leftarrow C\hackscore{ikl} + \Xi^{B}\hackscore{ik} B\hackscore{ilk} D\hackscore{ik} \\  X\hackscore{ij} \leftarrow X\hackscore{ij} + \delta\hackscore{pm} \Xi\hackscore{mi}  Y\hackscore{p} Z\hackscore{mij}\\
X\hackscore{ij} \leftarrow X\hackscore{ij} + (Xi^{B}\hackscore{ik} B\hackscore{ij} + \Xi^{C}\hackscore{ik} C\hackscore{ij}) Y\hackscore{i} \\
429  \end{eqnarray}  \end{eqnarray}
430  Using upwinding in this form, introduces an additonal error which is proprtional to the cell size $h$  Using upwinding in this form, introduces an additonal error which is proprtional to the cell size $h$
431  but with the intension to stabilize the solution.  but with the intension to stabilize the solution.
# Line 435  and \var{numSolutions} gives the number Line 436  and \var{numSolutions} gives the number
436  If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations  If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations
437  and the number solutions, respectively, stay undefined until a coefficient is  and the number solutions, respectively, stay undefined until a coefficient is
438  defined. \var{xi} defines a function which returns for any given  Preclet number $P\ge 0$ the  defined. \var{xi} defines a function which returns for any given  Preclet number $P\ge 0$ the
439  $\xi$-value used to define the stabilization parameters $\Xi^{B}$ and $\Xi^{C}$.  $\xi$-value used to define the stabilization parameters $\Xi$ and $\Xi$.