Publication: Sparse Jacobian Construction for Mapped Grid Visco-Resistive Magnetohydrodynamics
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Sparse Jacobian Construction for Mapped Grid Visco-Resistive Magnetohydrodynamics

- incollection -
 

Author(s)
Daniel R. Reynolds , Ravi Samtaney

Published in
Recent Advances in Algorithmic Differentiation

Editor(s)
Shaun Forth, Paul Hovland, Eric Phipps, Jean Utke, Andrea Walther

Year
2012

Publisher
Springer

Abstract
We apply the automatic differentiation tool OpenAD toward constructing a preconditioner for fully implicit simulations of mapped grid visco-resistive magnetohydrodynamics (MHD), used in modeling tokamak fusion devices. Our simulation framework employs a fully implicit formulation in time, and a mapped finite volume spatial discretization. We solve this model using inexact Newton-Krylov methods. Of critical importance in these iterative solvers is the development of an effective preconditioner, which typically requires knowledge of the Jacobian of the nonlinear residual function. However, due to significant nonlinearity within our PDE system, our mapped spatial discretization, and stencil adaptivity at physical boundaries, analytical derivation of these Jacobian entries is highly nontrivial. This paper therefore focuses on Jacobian construction using automatic differentiation. In particular, we discuss applying OpenAD to the case of a spatially-adaptive stencil patch that automatically handles differences between the domain interior and boundary, and configuring ad for reduced stencil approximations to the Jacobian. We investigate both scalar and vector tangent mode differentiation, along with simple finite difference approaches, to compare the resulting accuracy and efficiency of Jacobian construction in this application.

Cross-References
Forth2012RAi

AD Tools
OpenAD

AD Theory and Techniques
Sparsity

BibTeX
@INCOLLECTION{
         Reynolds2012SJC,
       title = "Sparse {J}acobian Construction for Mapped Grid Visco-Resistive Magnetohydrodynamics",
       doi = "10.1007/978-3-642-30023-3_2",
       author = "Daniel R. Reynolds and Ravi Samtaney",
       abstract = "We apply the automatic differentiation tool OpenAD toward constructing a
         preconditioner for fully implicit simulations of mapped grid visco-resistive magnetohydrodynamics
         (MHD), used in modeling tokamak fusion devices. Our simulation framework employs a fully implicit
         formulation in time, and a mapped finite volume spatial discretization. We solve this model using
         inexact Newton-Krylov methods. Of critical importance in these iterative solvers is the development
         of an effective preconditioner, which typically requires knowledge of the Jacobian of the nonlinear
         residual function. However, due to significant nonlinearity within our PDE system, our mapped
         spatial discretization, and stencil adaptivity at physical boundaries, analytical derivation of
         these Jacobian entries is highly nontrivial. This paper therefore focuses on Jacobian construction
         using automatic differentiation. In particular, we discuss applying OpenAD to the case of a
         spatially-adaptive stencil patch that automatically handles differences between the domain interior
         and boundary, and configuring AD for reduced stencil approximations to the Jacobian. We investigate
         both scalar and vector tangent mode differentiation, along with simple finite difference approaches,
         to compare the resulting accuracy and efficiency of Jacobian construction in this application.",
       pages = "11--21",
       crossref = "Forth2012RAi",
       booktitle = "Recent Advances in Algorithmic Differentiation",
       series = "Lecture Notes in Computational Science and Engineering",
       publisher = "Springer",
       address = "Berlin",
       volume = "87",
       editor = "Shaun Forth and Paul Hovland and Eric Phipps and Jean Utke and Andrea Walther",
       isbn = "978-3-540-68935-5",
       issn = "1439-7358",
       year = "2012",
       ad_tools = "OpenAD",
       ad_theotech = "Sparsity"
}


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