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.
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Acknowledgements
The work of D. Reynolds was supported by the U.S. Department of Energy, in grants DOE-ER25785 and LBL-6925354. R. Samtaney acknowledges funding from KAUST for the portion of this work performed at KAUST.
We would also like to thank Steve Jardin for insightful discussions on preconditioning techniques for toroidal plasmas, and Mike Fagan for help in building and using OpenAD for this work.
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Reynolds, D.R., Samtaney, R. (2012). Sparse Jacobian Construction for Mapped Grid Visco-Resistive Magnetohydrodynamics. In: Forth, S., Hovland, P., Phipps, E., Utke, J., Walther, A. (eds) Recent Advances in Algorithmic Differentiation. Lecture Notes in Computational Science and Engineering, vol 87. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30023-3_2
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DOI: https://doi.org/10.1007/978-3-642-30023-3_2
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