# applied math seminar

### Event Description:

Title: On Scalable Solution of Implicit / IMEX FE Continuum Plasma Physics Models*

Continuum plasma physics models are used to study important phenomena in the natural world (e.g., stellar interiors, solar flares, and planetary magnetic field generation) and in technological applications such as magnetically confined fusion energy (e.g. ITER—tokamak), and pulsed fusion reactors (e.g. Z-pinch-type devices). The mathematical modeling, and computational simulation of these systems, requires the solution of the governing partial differential equations describing conservation of mass, momentum, and energy, along with various forms of approximations to Maxwell's equations. The resulting systems are characterized by strong nonlinear coupling of fluid and electromagnetic phenomena, as well as the significant range of time- and length-scales that these interactions produce. These characteristics make the scalable, robust, accurate, and efficient computational solution of these systems, over relevant dynamical time-scales of interest, extremely challenging.

This talk begins with a very brief description of our interest in continuum modeling of plasma physics applications of importance to DOE and aspects of the multiple-time-scale characteristics of these systems. In this context, we briefly discuss the mathematical models and formulations for a subset of these systems, and present representative results for plasma physics problems of current interest. The discussion then considers computational solution methods. To enable accurate and stable approximation of these complex systems a range of spatial and temporal discretization methods are employed. For finite element methods these include variational multiscale methods and structure-preserving / physics-compatible approaches. For time integration two well-structured approaches are fully-implicit and implicit-explicit type methods. The requirement to accommodate disparate spatial discretizations, and allow the flexible assignment of mechanisms as explicit or implicit operators, implies a wide variation in unknown coupling, ordering, and the conditioning of the implicit sub-system.

Our approach to help overcome these challenges has been the development of robust, scalable, and efficient fully-coupled physics-based algebraic multilevel preconditioned Newton-Krylov type iterative solvers. To discuss the structure of these algorithms, and to demonstrate the flexibility of this approach various forms of magnetohydrodynamic and multifluid electromagnetic plasma models are considered. Representative results are presented on the parallel and algorithmic scaling of the methods with weak scaling results up to 1M cores.

*This work was partially supported by the DOE Office of Science Advanced Scientific Computing Research (ASCR) - Applied Math Research program and an ASCR/Office of Fusion Energy SciDAC Partnership Project at Sandia National Laboratories.

### Event Contact

**Contact Name: **Pavel Lushnikov