Simulations of Fusion Energy and Corecollapse
Supernova
(Algorithmic Design and Software for Robust & Efficient
Multiphysics 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 largescale,
multirate 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 corecollapse 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,
with the lowfrequency resistive Maxwell equations that model the
evolution of the surrounding electromagnetic fields,
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 SweetParker Magnetic Reconnection
from fully implicit simulations of a viscoresistive MHD
plasma.

A distinct feature of these type of multiphysics models is that
each variable or group of variables will often evolve on
drastically different time and space scales. As a result,
standardpractice explicit and operatorsplit time integration
techniques fail to efficiently and accurately track some of the
more slowlyevolving 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 largescale
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
highaccuracy 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 slowlyevolving 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 viscoresistive MHD processes for stability and
refueling studies [1, 5]. Moreover, in a collaboration with
astrophysicists from the Stony
Brook University in the context of radiationhydrodynamics
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,
advectiondominated, multiphysics 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 corecollapse supernova.
Figure 3:
Pellet ablation and mass deposition for
pelletinjection 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 constrainthandling properties
of the associated models. As exemplified in the MHD system
(2)(3), many multiphysics 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 largescale nonlinear problems." Journal of
Physics: Conference Series, 46:433442, 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. "Operatorbased
preconditioning of stiff hyperbolic systems." SIAM
J. Sci. Comput. 32:150170, 2010. 
[4] 
D.R. Reynolds, R. Samtaney, and C.S. Woodward. "Physicsbased
preconditioning of resistive MHD systems." (in
preparation). 
[5] 
D.R. Reynolds, R. Samtaney, and C.S. Woodward. "A fully implicit
numerical method for singlefluid resistive magnetohydrodynamics."
Journal of Computational Physics, 219:144162,
2006. 
[6] 
D.R. Reynolds, F.D. Swesty, and C.S. Woodward. "A
NewtonKrylov 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 ViscoResistive
Magnetohydrodynamics." Lecture Notes in Computational
Science and Engineering, vol. 87, 2012. 
[8] 
D.R. Reynolds, R. Samtaney and H.C. Tiedeman. "A fully
implicit NewtonKrylovSchwarz 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, 27:483,
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), 20112016.
DOE FASTMath SciDAC Grant (coI; with
L. Diachin et al.), 20112016.
LBL Subcontract 6925354 (PI), 20102011.
DOE TOPS SciDAC Grant ER25785 (coPI; with
D.E. Keyes et al.), 20062011.
LLNL Subcontract B555750 (coPI; with
M. Holst), 2005.
