D.A. Knoll

On balanced approximations for time integration of multiple time scale systems

The effect of various numerical approximations used to solve linear and nonlinear problems with multiple time scales is studied in the framework of modified equation analysis (MEA). First, MEA is used to study the effect of linearization and splitting in a simple nonlinear ordinary differential equation (ODE), and in a linear partial differential equation (PDE). Several time discretizations of the ODE and PDE are considered, and the resulting truncation terms are compared analytically and numerically. It is demonstrated quantitatively that both linearization and splitting can result in accuracy degradation when a computational time step larger than any of the competing (fast) time scales is employed. Many of the issues uncovered on the simple problems are shown to persist in more realistic applications. Specifically, several differencing schemes using linearization and/or time splitting are applied to problems in nonequilibrium radiation-diffusion, magnetohydrodynamics, and shallow water flow, and their solutions are compared to those using balanced time integration methods.

A 2D high-ß Hall MHD implicit nonlinear solver

A nonlinear, fully implicit solver for a 2D high-β (incompressible) Hall magnetohydrodynamics (HMHD) model is proposed. The task in non-trivial because HMHD supports the whistler wave. This wave is dispersive (ω ˜ k2) and therefore results in diffusion-like numerical stability limits for explicit time integration methods. For HMHD, implicit approaches using time steps above the explicit numerical stability limits result in diagonally submissive Jacobian systems. Such systems are difficult to invert with iterative techniques. In this study, Jacobian-free Newton-Krylov iterative methods are employed for a fully implicit, nonlinear integration, and a semi-implicit (SI) preconditioner strategy, developed on the basis of a Schur complement analysis, is proposed. The SI preconditioner transforms the coupled hyperbolic whistler system into a fourth-order, parabolic, diagonally dominant PDE, amenable to iterative techniques. Efficiency and accuracy results are presented demonstrating that an efficient fully implicit implementation (i.e., faster than explicit methods) is indeed possible without sacrificing numerical accuracy.

Hall MHD effects on the 2D Kelvin–Helmholtz/tearing instability

Abstract The Kelvin–Helmholtz (KHI)/tearing (TMI) instability is studied with a 2D incompressible Hall MHD model. In the equilibrium configuration of interest, the magnetic and ion velocity fields are parallel and identically sheared. While in resistive MHD simultaneous growth of a TMI and a KHI is precluded, Hall physics, by decoupling electrons and ions, destabilizes both modes, leading to a more complex interaction. Nonlinearly, saturation occurs with the formation of a magnetic island and an ion flow vortex in both sub- and super-Alfvenic regimes. For moderately large c/ωpi, the electron flow shows good alignment with the magnetic field, while demagnetized ions still show KH activity.

New physics-based preconditioning of implicit methods for non-equilibrium radiation diffusion

This note presents an extension of previous work on physics-based preconditioning of the non-equilibrium radiation diffusion equations. The new physics-based preconditioner presented in this manuscript is a minor modification to the operator-split preconditioner presented previously. Results show that the new preconditioner is more effective on test problems that are more nonlinear.

Coalescence of magnetic islands, sloshing, and the pressure problem

The coalescence of magnetic islands in the high Lundquist number regime is studied. Within the resistive magnetohydrodynamics model, the coalescence rate is known to stall (sloshing) in the limit of high Lundquist number. Previously, this stalling has been associated with the presence of a secondary tearing mode. Here it is shown that this stalling results from high magnetic pressure gradients, which result from thin current sheets at high Lundquist number. This phenomenon is frequently referred to as the “pressure problem” in flux pile-up reconnection studies. Sensitivity of the dynamic solution is presented, over a wide range of resistivities.

The Kelvin–Helmholtz instability, differential rotation, and three-dimensional, localized, magnetic reconnection

Results are presented from a study of three-dimensional magnetic reconnection caused by a Kelvin–Helmholtz instability and differential rotation. Specifically, subsonic and sub-Alfvenic flow is considered, which is Kelvin–Helmholtz stable in the direction of the magnetic field, but unstable perpendicular to the magnetic field. The flow is modeled by the resistive magnetohydrodynamics equations in three dimensions with constant resistivity. As a result of differential rotation (a gradient in vorticity parallel to the initial field), localized transient reconnection is observed on the Kelvin–Helmholtz time scale. Current amplification is observed along with the generation of parallel current. Results indicate that the observed transient reconnection rate is insensitive to resistivity (even with a constant resistivity model), but is sensitive to the initial flow shear.

Energy gain calculations in Penning fusion systems using a bounce-averaged Fokker-Planck model

In spherical Penning fusion devices, a spherical cloud of electrons, confined in a Penning-like trap, creates the ion-confining electrostatic well. Fusion energy gains for these systems have been calculated in optimistic conditions (i.e., spherically uniform electrostatic well, no collisional ion-electron interactions, single ion species) using a bounce-averaged Fokker–Planck (BAFP) model. Results show that steady-state distributions in which the Maxwellian ion population is dominant correspond to lowest ion recirculation powers (and hence highest fusion energy gains). It is also shown that realistic parabolic-like wells result in better energy gains than square wells, particularly at large well depths (>100u200akV). Operating regimes with fusion power to ion input power ratios (Q-value) >100 have been identified. The effect of electron losses on the Q-value has been addressed heuristically using a semianalytic model, indicating that large Q-values are still possible provided that electron particle losses are...

Nonlinear study of the curvature-driven parallel velocity shear–tearing instability

The nonlinear regime of the parallel velocity shear–tearing instability is studied numerically using a two-dimensional reduced, resistive magnetohydrodynamics model. In this instability, a sheared parallel velocity profile interacts with the perpendicular dynamics via the magnetic field curvature. Linearly, it has been shown [J. M. Finn, Phys. Plasmas 2, 4400 (1995)] that, in the inviscid limit, such interaction alters the classical behavior of the tearing instability, resulting in increased growth rates for classically tearing-unstable regimes (Δ′>0), and destabilizing classically tearing-stable regimes, leading to an electrostatic mode as Δ′→−∞. These trends are seen to hold with finite viscosity as long as the perpendicular plasma viscosity is of the order or smaller than the plasma resistivity. Nonlinearly, it is found that a self-consistent perpendicular shear flow and a reversed (stabilizing) density gradient develop. For favorable curvature, the latter implies an anomalous pinch effect. The shear f...

Two-body similarity and its violation in tokamak edge plasmas

Scaling laws found under the assumption that two‐body collisions dominate can be effectively used to benchmark complex multi‐dimensional codes dedicated to investigating tokamak edge plasmas. The applicability of such scaling laws to the interpretation of experimental data, however, is found to be restricted to the relatively low plasma densities (<1019 m−3) at which multistep processes, which break the two‐body collision approximation, are unimportant.

A Multigrid Newton-Krylov Solver for Non-linear Systems

We present a technique which solves systems of nonlinear equations. The technique couples two solution methods together, multigrid and Newton-Krylov, producing in a method which efficiently uses the strengths of each technique. A form of distributed relaxation multigrid is used to solve systems of scalar linear equations which are then combined to provide efficient preconditioners for the Newton-Krylov method. This new method can be viewed as an alternative to other nonlinear multigrid solvers. Results will be presented for a steady state fluid flow and transient heat conduction.