Applied Seminar: Dr. Eric Cyr
Techniques for Simulating Multiple Timescales with Applications to Plasmas
A major challenge when simulating multi-physics systems is handling the diversity of time scales that impose strict stability constraints on traditional explicit time integrators. This talk will explore two classes of techniques for approaching this challenge. We conclude with a demonstration combining these methods to solve a complex multi-fluid plasma system with mixed spatial discretizations.
The first technique focuses on developing robust physics-based and block
preconditioners for use within fully implicit time integration. These techniques are derived by understanding the stiff modes of the systems. Stiff terms, like diffusion, that live on the diagonal are naturally handled by classical block preconditioners like block Jacobi and block Gauss-Seidel. Terms that live on the off diagonal are more challenging. Despite this, modes that arise as a result of strong bidirectional coupling can be essential to precondition. To address these issues, our preconditioners use approximate block factorizations and localize the coupling modes to the resulting Schur-complement. The task then is to develop an effective approximation for the Schur-complement. These ideas will be demonstrated with incompressible Navier-Stokes and magnetohydrodynamics.
The second technique tries to simplify the preconditioning problem by pursuing Implicit-Explicit Runge-Kutta (IMEX-RK) time integration. IMEX-RK methods use one implicit Butcher tableau, and one explicit Butcher tableau combined with an additive decomposition of an ODE system into stiff (implicit) and slow (explicit) modes to achieve high-order accuracy. We consider a number of advection diffusion test problems to demonstrate this approach. In addition, we develop a novel approach to solving the Euler equations in a monolithic Arbitrary Lagrangian-Eulerian (ALE) frame, where IMEX is used to allow tightly coupled multi-physics.
The final part of this talk combines the two classes of techniques to simulate multi-fluid plasmas. A central obstacle to simulating multi-fluid plasmas is the range of time scales inherent in the dynamics. In particular, very stiff time scales arising from plasma and cyclotron frequencies, and the speed of light can limit the applicability of explicit time integration. In our IMEX-RK scheme we evolve Maxwell equations implicitly to avoid the speed of light CFL restrictions.
The terms associated with the cyclotron and plasma frequencies are also handled implicitly. In this case, we use a physics-based preconditioner to construct an approximate Jacobian for use in a quasi-Newton method. The approximate Jacobian uses a carefully chosen spatial discretization to achieve computationally favorable structure. The intent is that this scheme is not only capable of stepping over the stiff modes, but also has the opportunity to be computationally efficient.
Contact Name: Jacob Schroder