20111102-LANL.tex

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\title{Tightly Coupled Solvers, Loosely Coupled Software}
\subtitle{Multi-physics solvers and time integration in PETSc}

\author{Jed Brown, Emil Constantinescu, and Barry Smith}


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{
  {Argonne National Laboratory} \\
}

\date{2011-11-02}

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\subject{Talks}


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\input{slides/MultiphysicsExamples.tex}
\input{slides/MonolithicOrSplit.tex}
\input{slides/FieldSplit.tex}
%\input{slides/CoupledMultiphysics.tex}
\input{slides/SIPreconditioning.tex}

\section{Coupling software}
\input{slides/PETSc/Coupling.tex}
\input{slides/PETSc/LocalSpaces.tex}
%\input{slides/PETSc/MatGetLocalSubMatrixUsage.tex}
\input{slides/PETSc/MatGetLocalSubMatrix.tex}

\section{Applications}
%\input{slides/GroundingLine/ALEBlockForm.tex}
\subsection{Ice Flow}
\newcommand\smallterm[1]{{\color{gray} #1}}
\begin{frame}{Conservative two-phase formulation}
  Find momentum density $\rho\uu$, pressure $p$, and total energy density $E$:
  \begin{gather*}
    (\rho\uu)_t + \div (\smallterm{\rho\uu\otimes\uu} - \eta D\uu_i + p\bm 1) - \rho \bm g = 0 \\
    \rho_t + \div \rho\uu = 0 \\
    E_t + \div \big((E+p)\uu - k_T\nabla T - k_\omega\nabla\omega \big) - \eta D\uu_i\tcolon D\uu_i - \smallterm{\rho\uu\cdot\bm g} = 0
  \end{gather*}
\begin{itemize}
\item Solve for density $\rho$, ice velocity $\uu_i$, temperature $T$, and melt fraction $\omega$ using constitutive relations.
  \begin{itemize}
  \item Simplified constitutive relations can be solved explicitly.
  \item Temperature, moisture, and strain-rate dependent rheology $\eta$.
  \item High order FEM, typically $Q_3$ momentum \& energy, SUPG (yuck).
  \end{itemize}
\item DAEs solved implicitly after semidiscretizing in space.
\item Preconditioning using nested fieldsplit
\end{itemize}
\end{frame}
\input{slides/VHTSolvers.tex}
\subsection{Geodynamics}
\input{slides/DrunkenSeaman.tex}
\input{slides/DrunkenSeamanOptions.tex}

\section{Implicit-Explicit time integration}
\input{slides/PETSc/TSARKIMEX.tex}

\section{Variational Inequalities}
\input{slides/PETSc/SNESVI.tex}

\begin{frame}{Outlook}
  \begin{itemize}
  \item Unified algebraic interface for monolithic and nested formats
  \item Improves software modularity, but still manages stiff coupling
  \item Block and symmetric formats
  \item Multigrid inside or outside field splits
  \item Can use IMEX methods $g(t,x,\dot x) = f(t,x)$
  \item Variational inequalities
  \item Still to do:
    \begin{itemize}
    \item Better preallocation for off-diagonal blocks
    \item Nonlinear solvers for IMEX systems with structure
    \item General/nonsymmetric pivoting in fieldsplit
    \item Change of variables for fieldsplit \\ (e.g. low-Mach Euler in conservative variables)
    \end{itemize}
  \end{itemize}
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