On the Development of a Scalable Fully-Implicit Stabilized Unstructured FE Capability for Resistive MHD with Integrated Adjoint Error-Estimate

The resistive 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. This model 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 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 3D resistive MHD with integrated adjoint error- estimation capability. The brief discussion considers the development of the stabilized FE formu- lation and the underlying fully-coupled preconditioned Newton-Krylov (NK) 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 stabilized FE formulation and robust fully-coupled NK iterative solvers enable the solution of a wide range of flow conditions that include incompressible, low Mach number approximations, Boussinesq, anelastic, and low Mach number compressible flow. In addition the fully-implicit NK formulation allows the development of adjoint-based error-estimation methods. We present some recent representative results employing the adjoint methods to simple Navier-Stokes and resistive MHD verification problems as well as a RANS turbulence model.

We then briefly consider two sets of recent simulation results with relevance to geophysical and astrophysical flows. The first is the break-up of thin Sweet-Parker current sheets into smaller plasmoids that has been the subject of attention as a possible mechanism for fast reconnection in resistive MHD. Various studies, both theoretical and numerical, have shown that the fast formation of small structures is not only possible, but in fact unavoidable for large enough Lundquist numbers. In this study, we have used state-of-the-art computational capabilities to perform simulations of the Fadeev island coalescence problem in the high Lundquist number regime to investigate if thin current sheets dynamically formed in this strongly non-uniformly driven problem are prone to break- up by fast plasmoid instabilities. Our numerical simulations confirm that plasmoid break-up of dynamically formed current sheets occur for S > 106 (with L. Chacon and D. Knoll LANL). Second we present some very recent results for high Rayleigh number thermal convection in cylindrical geometries of various aspect ratios that have relevance to aspects of the SpinLab experiments.

#This work was supported by the DOE office of Science Advanced Scientific Computing Research – Applied Math Research program at Sandia National Laboratory.

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