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{\large John Shadid \\
{\bf Performance of Implicit Fully-coupled Solution Methods for Large-scale Stabilized FE Simulations of Resistive MHD }}

Sandia National Laboratories \\ PO Box 5800 MS 0316 \\ Albuquerque \\ NM 87185 - 0316
\\
{\tt jnshadi@sandia.gov}\\
Roger Pawlowski\\
Eric Cyr\\
Ray Tuminaro\\
Paul Lin\\
Luis Chacon\end{center}

The magnetohydrodynamics (MHD) model describes the dynamics of charged
fluids in the presence of electromagnetic fields. MHD models are used to
describe important phenomena in the natural physical world and in
technological applications. The mathematical basis for the continuum
modeling of these systems is the solution of the governing partial
differential equations (PDEs) describing conservation of mass, momentum,
charge, and thermal energy augmented by Maxwell's equations for the
electric and magnetic field. This system of PDEs is non-self adjoint,
strongly coupled, highly nonlinear and characterized by multiple physical
phenomena that span a very large range of length- and time-scales. These
interacting, nonlinear multiple time-scale physical mechanisms can
balance to produce steady-state behavior, nearly balance to evolve a
solution on a dynamical time scale that is long relative to the component
time-scales, or can be dominated by just a few fast modes. These
characteristics make the scalable, robust, accurate, and efficient
computational solution of these systems over relevant dynamical time
scales of interest (or to steady-state solutions) extremely challenging.
For multiple-time-scale systems, fully-implicit methods can be an
attractive choice that can often provide unconditionally-stable time
integration techniques. The stability of these methods, however, comes at
a very significant price, as these techniques generate large and highly
nonlinear sparse systems of equations that must be solved at each time
step.

This talk describes recent progress on the development of a scalable
fully-implicit stabilized unstructured finite element (FE) capability for
low-Mach-number resistive MHD. The brief discussion considers the
development of the stabilized FE formulation and the underlying
fully-coupled preconditioned Newton-Krylov nonlinear iterative solver. To
enable robust, scalable and efficient solution of the large-scale sparse
linear systems generated by the Newton linearization, fully-coupled
multilevel preconditioners are employed. The multilevel preconditioners
are based on two differing approaches. The first algebraic multilevel
technique employs a graph-based aggressive-coarsening aggregation method
applied to the nonzero block structure of the Jacobian matrix [1,2]. The
second approach utilizes approximate block decomposition methods and
physics-based preconditioning approaches that reduce the coupled systems
into a set of simplified systems to which multilevel methods are applied
[3,4].

The performance of the multilevel preconditioners is compared to standard
variable overlap additive one-level Schwarz domain decomposition type
preconditioners. Parallel performance results are presented for a set of
challenging prototype problems that include the solution of an MHD
Faraday conduction pump, a hydromagnetic Rayleigh-Bernard linear
stability calculation, and a magnetic island coalescence problem. Initial
results that explore the scaling of the solution methods are presented on
up to 4096 processors for problems with up to 64M unknowns on a
CrayXT3/4. Additionally, a large-scale proof-of-capability calculation
for 1 billion unknowns for the MHD Faraday pump problem on 24,000 cores
is also presented.


References

[1] M. Sala, R. Tuminaro, A new Petrov-Galerkin smoothed aggregation
preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput. 31
(1) (2008) 143–166

[2] Paul T. Lin, John N. Shadid, Ray S. Tuminaro, Marzio Sala, Gary L.
Hennigan, Roger P. Pawlowski, “A Parallel Fully-Coupled Algebraic
Multilevel Preconditioner Applied to Multiphysics PDE Applications:
Drift-diffusion, Flow/transport/reaction, Resistive MHD”, Submitted to
Int. J. Numer. Meth. Fluids

[3] H. Elman, V. Howle, J. N. Shadid, R. Shuttleworth, R. Tuminaro, ”A
Taxonomy of Parallel Multilevel Block Preconditioners for the
Incompressible Navier-Stokes Equations”, JCP, v. 227, 3, pp 1790 -
1808, 2008

[4] L. Chacon, An optimal, parallel, fully implicit Newton-Krylov solver
for three-dimensional visco-resistive magnetohydrodynamics, Phys. Plasmas
15 (2008) 056103


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