SOUTHERN METHODIST UNIVERSITY DEPARTMENT OF MATHEMATICS
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SMU

Simulations of Fusion Energy and Core-collapse Supernova

(Algorithmic Design and Software for Robust & Efficient Multi-physics Problems)

Through collaborations with DOE scientists at Princeton Plasma Physics Laboratory, Lawrence Livermore National Laboratory, Stony Brook University, Columbia University and others, we have been investigating the use of fully implicit computational approaches for time evolution of large-scale, multi-rate PDE systems. These efforts have been in the context of resistive magnetohydrodynamics (MHD), arising in studies of fusion plasma stability and refueling, and radiation hydrodynamics (RHD), used to model core-collapse supernova explosions. These applications involve the solution of coupled PDE systems for modeling multiple interacting physical processes. For example, the resistive MHD model couples the compressible viscous Euler equations for modeling plasma hydrodynamics,

   
(2)
   
with the low-frequency resistive Maxwell equations that model the evolution of the surrounding electromagnetic fields,

   
(3)
   
   
Here is the density, is the velocity, is the magnetic induction, is the electric current, is the electric field, is the total energy, is the pressure, is the plasma temperature, is the plasma viscosity, and and correspond to the coefficients of viscosity and resistivity. Similarly, in the model for radiation hydrodynamics, additional PDE models for nonlinear radiation diffusion are coupled with either the hydrodynamical system (2), or even the full MHD system (2)-(3) for increased modeling accuracy.

Figure 1:  Snapshots of Sweet-Parker Magnetic Reconnection from fully implicit simulations of a visco-resistive MHD plasma.

A distinct feature of these type of multi-physics models is that each variable or group of variables will often evolve on drastically different time and space scales. As a result, standard-practice explicit and operator-split time integration techniques fail to efficiently and accurately track some of the more slowly-evolving processes of interest, such as the macroscopic stability of a fusion plasma or the energy partitioning within a collapsing star. Moreover, for any consistent mathematical system to successfully model problems with large disparities in scale, it requires spatial resolutions that may only be tractable on the world's largest supercomputers, along with novel computational algorithms that may efficiently operate on such large-scale machines.

Figure 2:  Fully implicit simulations of a twisting magnetic field due to the Kelvin Helmholtz instability.

   

For such problems, we have been investigating the use of high-accuracy fully implicit approaches for solving the MHD and RHD systems of equations. Due to the disparity of time scales involved in such problems they are typically very stiff, requiring advances in nonlinear solvers and preconditioning approaches that allow the methods to efficiently "step over" their stiff transient components while still accurately tracking the more slowly-evolving quantities of interest. Through our collaboration with computational physicists from Princeton Plasma Physics Laboratory and computational mathematicians from Lawrence Livermore National Laboratory, we have successfully developed implicit simulations of visco-resistive MHD processes for stability and refueling studies [1, 5]. Moreover, in a collaboration with astrophysicists from the Stony Brook University in the context of radiation-hydrodynamics astrophysics simulations, we have honed techniques for fully implicit solution of hydrodynamics systems in the presence of strong shocks [6].

We have also worked to develop new preconditioning approaches for fully implicit solution of stiff PDE systems. These preconditioners are designed to tackle stiff, advection-dominated, multi-physics problems, and are demonstrating promising results in their efficacy and parallel scalability [2, 3, 4]. Moreover, in our collaborations with SUNY astrophysicists, we are working toward completion of the first fully implicit solver approaches for RHD simulations of core-collapse supernova.

Figure 3:  Pellet ablation and mass deposition for pellet-injection tokamak refueling [image courtesy of collaborator R. Samtaney].

These collaborations are funded through a DOE SciDAC grant, which aims to move scientific fields ahead in both computational efficiency (through scalable algorithms), and in the incorporation of additional physical realism and constraint-handling properties of the associated models. As exemplified in the MHD system (2)-(3), many multi-physics PDE models involve both evolution and constraint equations that must be simultaneously satisfied for their accurate solution. Unfortunately, many numerical models violate these constraints at the discrete level, though certain algorithms may retain such quantities given a satisfactory initial state. In our work to develop scalable solvers for such systems, we also examine these issues in detail, specifically in regards to the development of preconditioning techniques that retain desirable algorithmic properties, while also allowing efficient solution at large computational scale.


References

[1] D.E. Keyes, D.R. Reynolds, and C.S. Woodward. "Implicit solvers for large-scale nonlinear problems." Journal of Physics: Conference Series, 46:433--442, 2006.
[2] D.R. Reynolds. "On the improvement of splitting methods for fully implicit systems of equations." (in preparation).
[3] D.R. Reynolds, R. Samtaney, and C.S. Woodward. "Operator-based preconditioning of stiff hyperbolic systems." SIAM J. Sci. Comput. 32:150-170, 2010.
[4] D.R. Reynolds, R. Samtaney, and C.S. Woodward. "Physics-based preconditioning of resistive MHD systems." (in preparation).
[5] D.R. Reynolds, R. Samtaney, and C.S. Woodward. "A fully implicit numerical method for single-fluid resistive magnetohydrodynamics." Journal of Computational Physics, 219:144--162, 2006.
[6] D.R. Reynolds, F.D. Swesty, and C.S. Woodward. "A Newton-Krylov solver for implicit solution of hydrodynamics in core collapse supernovae." Journal of Physics: Conference Series, 125, 2008.
[7] D.R. Reynolds and R. Samtaney. "Sparse Jacobian Construction for Mapped Grid Visco-Resistive Magnetohydrodynamics." Lecture Notes in Computational Science and Engineering, vol. 87, 2012.
[8] D.R. Reynolds, R. Samtaney and H.C. Tiedeman. "A fully implicit Newton-Krylov-Schwarz method for tokamak MHD: Jacobian construction and preconditioner formulation." Computational Science & Discovery, 5:014003, 2012.
[9] D.E. Keyes et al. "Multiphysics Simulations: Challenges and Opportunities." International Journal of High Performance Computing Applications, 27:4-83, 2013.
[9] D.J. Gardner, C.S. Woodward, D.R. Reynolds, G. Hommes, S. Aubry and A.T. Arsenlis. "Implicit integration methods for dislocation dynamics." accepted to Modelling and Simulation in Materials Science and Engineering, 2014.

Funding Support

    LLNL Subcontract B603971 (PI), 2013.

    LLNL Subcontract B598130 (PI), 2011-2016.

    DOE FASTMath SciDAC Grant (co-I; with L. Diachin et al.), 2011-2016.

    LBL Subcontract 6925354 (PI), 2010-2011.

    DOE TOPS SciDAC Grant ER25785 (co-PI; with D.E. Keyes et al.), 2006-2011.

    LLNL Subcontract B555750 (co-PI; with M. Holst), 2005.





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