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Numerical Methods for Plasma Astrophysics:
From Particle Kinetics to MHD

A Joint PICSciE - CSCAMM Program Fall 2004

Fall '04 Meeting: October 25-27, 2004
PICSciE, 214 Fine Hall, Princeton University

[Spring '04 Meeting: March 22-25, 2004]

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Scientific Background Scientific Content Organizing Committee
Invited Participants Schedule

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PICSciE Workshop Page CSCAMM Spring 2004 NPA04 Page

SCIENTIFIC BACKGROUND

Most of the visible matter in the Universe is a plasma, that is a dilute gas of electrons, ions, and neutral particles. Numerical methods are the only viable way of studying the dynamics of astrophysical plasmas. Using numerical simulations, much progress has been made in recent years in understanding a variety of important problems, including the structure and evolution of accretion flows around compact objects such as neutron stars and black holes, and the decay rate and fluctuation statistics of compressible MHD turbulence. Almost without exception, such advances have used multidimensional MHD codes. However, for many astrophysical problems, the MHD approximation may not be valid. Examples include the dynamics of very dilute accretion flows, the dynamics of turbulent plasmas near the energy dissipation scale, or magnetic reconnection. In order to address fundamental problems in these areas, it will be necessary to move beyond the MHD approximation, and consider particle kinetics. However, a full time-dependent and multidimensional numerical solution to the Boltzmann equation is intractable in most circumstances, thus novel methods will be required.

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SCIENTIFIC CONTENT

The goal of this workshop is to bring together astrophysicists, plasma physicists, and applied mathematicians to discuss future developments in numerical methods for astrophysical plasmas. Topics to be covered include:

  1. reviews of the astrophysical problems that motivate future developments, including what we have learned from current techniques, and why we need new methods,
  2. reviews of modern methods for MHD, including adaptive mesh techniques for multiscale problems, and methods for non-ideal MHD, and
  3. reviews of modern methods for collisionless plasma dynamics that result from various approximations to the full collisionless Boltzmann equation. A key ingredient of the workshop is to engage plasma physicists and applied mathematicians with experience in plasma kinetics in the development of methods suitable for astrophysical plasmas.

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ORGANIZING COMMITTEE

Name

Affiliation

Email

Robert Calderbank Electrical Engineering, Princeton University
William Dorland CSCAMM and Physics, Univ. of Maryland
Jim Drake Physics/IPST, Univ. of Maryland
James Stone Astrophysics, Princeton University
Eitan Tadmor CSCAMM, Math and IPST, Univ. of Maryland
William Tang PPPL, Princeton University

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INVITED PARTICIPANTS

Jonathan Arons
Email:
Department of Astronomy
University of California, Berkeley
 
Benjamin Chandran
Email:
Department of Physics & Astronomy
Universtiy of Iowa
 
Steven Cowley
Email:
Department of Physics & Astronomy

University of California, Los Angeles
 
Robert Crockett
Email:
Department of Astronomy
University of California, Berkeley
 
James Drake
Email:
Department of Physics, Institute for Physical Science and Technology

University of Maryland
 
Thomas Gardiner
Email:
Department of Astrophysical Sciences
Princeton University
 
Gregory Hammett
Email:
Princeton Plasma Physics Laboratory

Princeton University
 
Michael Hesse
Email:
Goddard Space Flight Center

NASA
 
Gregory Howes
Email:
Department of Astronomy
University of California, Berkeley
 
Stephen Jardin
Email:
Princeton Plasma Physics Laboratory
Princeton University
 
Wei-li Lee
Email:
Princeton Plasma Physics Laboratory
Princeton University
 
David Levermore
Email:
Department of Mathematics
University of Maryland
 

Zhihong Lin
Email:
Department of Physics & Astronomy
University of California, Irvine
 

Wonchull Park
Email:
Princeton Plasma Physics Laboratory
Princeton University
 
Scott Parker
Email:
Department of Physics
University of Colorado at Boulder
 
Ian Parrish
Email:
Department of Astrophysical Sciences, Princeton Plasma Physics Laboratory
Princeton University
 
Eliot Quataert
Email:
Department of Astronomy
University of California, Berkeley
 
Ravi Samtaney
Email:
Princeton Plasma Physics Laboratory
Princeton University
 
Alexander Schekochihin
Email:
Astrophysical Fluid Dynamics Group

University of Cambridge
 
Prateek Sharma
Email:
Department of Astrophysical Sciences
Princeton University
 
Michael Shay
Email:
The Institute for Research in Electronics and Applied Physics
University of Maryland
 
Carl Sovinec
Email:
Department of Engineering Physics
University of Wisconsin, Madison
 
Anatoly Spitkovsky
Email:
Kavli Institute for Particle Astrophysics and Cosmology (KIPAC)
Stanford University
 
James Stone
Email:
Department of Astrophysical Sciences

Princeton University
 
Eitan Tadmor
Email:
Center for Scientific Computation and Math Modeling

University of Maryland
 
William Tang
Email:
Princeton Plasma Physics Laboratory

Princeton University
 

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SCHEDULE

PICSciE, 214 Fine Hall, Princeton University
 
Monday, October 25
 
Kinetic Plasma Physics and Particle-in-Cell (PIC) Methods
Chair: William Tang
 
9:00 – 9:15 Coffee and Registration
9:15 – 9:30 Welcome: James Stone and William Tang
9:30 – 10:15 Jonathan Arons Relativistic Shock Waves: Astrophysical Motivation and PIC Simulations
10:15 – 11:00 Zhihong Lin Successful Particle-in-Cell Methods in Kinetic Plasma Physics Applications
11:00 – 11:30 Coffee Break
11:30 – 12:00 James Drake New developments in reconnection modeling: electron heating and the case of electron-positron reconnection
12:00 – 12:30 Michael Shay Equation free projective integration and its applicability for simulating plasma
12:30 – 2:00 Lunch Break
2:00 – 2:30 Michael Hesse On fast reconnection within large-scale magnetofluid models
2:30 – 3:00 Anatoly Spitkovsky Collisionless shocks with PIC: the final word?
3:00 – 3:30 Wei-li Lee Gyrokinetic Particle Simulation for Magnetic Fusion Plasmas
3:30 – 4:00 Benjamin Chandran Thermal conduction in strongly turbulent magnetized plasmas.


Tuesday, October 26
 
MHD with Kinetic Effects
Chair: Steve Cowley
 
9:00 – 9:30 Coffee
9:30 – 10:15 Eliot Quataert Kinetic-MHD in Astrophysical Plasmas
10:15 – 11:00 Scott Parker Successful Kinetic-MHD Methods in Plasma Physics
11:00 – 11:30 Coffee Break
11:30 – 12:00 Alexander Schekochihin MHD Turbulence in Galaxies and Clusters
12:00 – 12:30 Steven Cowley Dynamo Amplification of Field in Clusters and the Problem of Viscosity
12:30 – 2:00 Lunch Break
2:00 – 2:30 David Levermore Transition Regime Models from Kinetic Equations
2:30 – 3:00 Ian Parrish Simulation of the Magneto-Thermal Instability
3:00 – 3:30 Wonchull Park Plasma Simulation Studies using Mutilevel Physics Models
3:30 – 4:00 Gregory Hammett Properties of Landau-Fluid Models of Kinetic MHD
4:00 – 4:30 Prateek Sharma Initial Nonlinear Landau-MHD Simulations of Kinetic Effects on the MRI


Wednesday, October 27
 
MHD
Chair: James Stone
 
9:00 – 9:30 Coffee
9:30 – 10:15 James Stone Numerical Methods for Astrophysical MHD
10:15 – 11:00 Carl Sovinec Successful MHD Simulation Methods in Plasma Physics Applications
11:00 – 11:30 Coffee Break
11:30 – 12:00 Stephen Jardin High-Accuracy, Implicit Solution of the Extended-MHD Equations using High-Continuity Finite Elements
12:00 – 12:30 Ravi Samtaney The Magneto-hydrodynamic Richtmyer-Meshkov Instability
12:30 – 2:00 Lunch Break
2:00 – 2:30 Thomas Gardiner A new CT-Godunov scheme for MHD with application to the MRI
2:30 – 3:00 Gregory Howes Preliminary Results for Adaptive Particle Refinement
3:00 – 3:30 Robert Crockett An Unsplit Godunov Method for Ideal Magnetohydrodynamic Simulations of the Interstellar Medium

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ABSTRACTS

Relativistic Shock Waves: Astrophysical Motivation and PIC Simulations

Jonathan Arons, University of California, Berkeley

Abstract: I summarize the motivations for studying relativistic shocks, from observations of relativistic winds and jets emerging from compact objects. I survey the properties of such shocks as known from PIC simulations, for both the magnetized variety with shock dissipation and structure owing to cyclotron and synchrotron instabilities, and the unmagnetized variety, with structure and dissipation coming from the Weibel instability. I briefly discuss the relation of these results to the widespread belief that relativistic shocks create nonthermal particle populations through diffusive Fermi acceleration.

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Successful Particle-in-Cell Methods in Kinetic Plasma Physics Applications

Zhihong Lin, University of California, Irvine

Abstract: Particle-in-cell (PIC) simulation have proven to be a powerful computational approach for studying nonlinear physics in both fusion and space plasma applications. These application areas typically involve nonlinear kinetic effects, disparate spatial-temporal scales, multiple physical processes, non-local interactions, and complex geometry. In particular, impressive progress in the massively parallel gyro-kinetic particle simulation capability during the past two decades have led to significant advances in the fundamental understanding of turbulence and transport in fusion plasmas. Examples include new paradigms of turbulence self-regulation by zonal flows and of spectral cascade via non-local interactions in the wave-vector space. In PIC simulations, all nonlinearities are treated on the same footing and systematically delineated. For example, while the wave-wave interactions determine the fluctuation characteristics, the turbulent transport is driven by wave-particle interactions. In addition to reviewing the physics models, numerical algorithms, and parallel computation for gyro-kinetic turbulence simulation of fusion plasmas, recent progress and future directions for space plasma applications of gyro-kinetic PIC simulations will be presented.

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New developments in reconnection modeling: electron heating and the case of electron-positron reconnection

James Drake, University of Maryland

Abstract: (abstract not available for print.)

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Equation free projective integration and its applicability for simulating plasma

Michael Shay, University of Maryland

Abstract: (abstract not available for print.)

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On fast reconnection within large-scale magnetofluid models

Michael Hesse, NASA

Abstract: During the quest to understand the microphysics of magnetic reconnection, it was discovered that the Hall-term, the manifestation of scale separation between ions and electrons, also provided a mechanism for reconnection to proceed at rates much higher than in conventional MHD models. This feature was explained by the dispersion of the Whistler mode, which permits the stationary, quadrupolar, magnetic field structure near the reconnection region to always develop the proper gradient scales for reconnection to proceed at Petschek-like rates. MHD simulations with constant or weakly localized resistivity, however, exhibited a much slower, Sweet Parker-like behavior. Similar results were found for simulations with magnetic guide fields. Recently, however, we investigated the idea that a valuable electric field representation for MHD models might be obtained directly from the ion dynamics. The equality between the electric fields felt by ions on one and electrons on the other hand implies that the ion equation of motion might yield a larger-scale description of a nonideal electric field, which should provide larger reconnection rates in MHD models also. In this presentation, we describe this idea, and present a number of particle-in-cell simulations for equal ion and electron masses to test it further. In particular, we will show that fast reconnection indeed results in all cases considered, and we will discuss how the system circumvents the Sweet Parker limit of the reconnection rate. We emphasize that these results do not question the validity of the Hall-dynamics in collisionless reconnection – it is essential to provide the proper cross-scale coupling. The results do provide, however, a mechanism by which it may be possible to obtain fast reconnection in MHD models also.

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Collisionless shocks with PIC: the final word?

Anatoly Spitkovsky, Stanford University

Abstract: I discuss the 3D simulations of relativistic collisionless shocks in electron-positron pair plasmas with the particle-in-cell (PIC) method. The shock structure is mainly controlled by the shock's magnetization ("sigma" parameter). I will demonstrate how the structure of the shock varies as a function of sigma for both perpendicular and oblique shocks. At low magnetizations the shock is mediated mainly by the Weibel instability which generates transient magnetic fields which can exceed the initial field. At larger magnetizations the shock is dominated by magnetic reflections. I demonstrate where the transition occurs and argue that it is impossible to have very low magnetization collisionless shocks in nature (in more than 1 spatial dimension). I further discuss the acceleration properties of these shocks, and show that higher magnetization perpendicular shocks do not efficiently accelerate particles in 3D. Among other astrophysical applications, this poses a restriction on the structure and composition of pulsar wind outflows.

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Gyrokinetic Particle Simulation for Magnetic Fusion Plasmas

Wei-li Lee, Princeton University

Abstract: Princeton Plasma Physics Laboratory (PPPL) has a long history in using Particle-in-Cell (PIC) methods in studying magnetic fusion plasmas and is currently leading a national laboratory/university collaboration for the investigation of turbulent transport and kinetic-MHD physics relevant to the International Tokamak Experimental Reactor (ITER). The effort consists of solving the gyrophased-averaged (gyrokinetic) Vlasov-Maxwell equations on massively parallel computers with the aid of modern-day parallelization, data management and visualization techniques. The present talk will briefly review the project as well as the methodology in solving these equations and the evolution of perturbative particle simulation schemes in reducing the inherent noise problem in PIC codes. The use of our Global Gyrokinetic Toroidal Code (GTC) for transport physics and the extension of the code for kinetic-MHD physics will also be discussed.

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Thermal conduction in strongly turbulent magnetized plasmas.

Benjamin Chandran, Universtiy of Iowa

Abstract: Turbulence can affect the diffusion of particles and heat in a number of ways. This talk will focus on how "tangled" magnetic field lines in a turbulent plasma affect the trajectories of diffusing particles. I will describe a physical paradigm for fast-particle diffusion and thermal conduction based on the theory of Rechester and Rosenbluth, as well as analytic and numerical results on thermal conduction. I will also briefly describe the implications of this work for the study of galaxy clusters.

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Kinetic-MHD in Astrophysical Plasmas

Eliot Quataert, University of California, Berkeley

Abstract: I describe several astrophysics problems in which plasma is macroscopically collisionless (e.g., the solar wind, accretion onto black holes, etc.). In these cases kinetic MHD (rather than 'regular' MHD) governs the large-scale dynamics of the system. I describe the stability properties of collisionless plasmas governed by kinetic-MHD, highlighting the differences relative to the short mean free path limit. I then discuss several of the waves (e.g., cyclotron) and instabilities (e.g., mirror and firehose) that limit the effective mean free path of particles in collisionless plasmas, and try to assess to what extent this enforces "MHD-like" dynamics on macroscopically collisionless plasmas.

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Successful Kinetic-MHD Methods in Plasma Physics

Scott Parker, University of Colorado at Boulder

Abstract: Recent advances in the direct kinetic simulation of fusion plasma turbulence now lay the groundwork and provide an enormous impetus for kinetic closure (using kinetic simulation) of MHD computational models. The topic is lively and is still very much an open research area. This is because efficient and practical nonlinear MHD and kinetic methods require subtle underlying orderings, equations and numerical methods (gyrokinetics, gyro-Landau fluid, semi- implicit, finite-element, particle-in-cell, drift-ordering, etc.) all of which must properly meld together into one grand simulation. Even on a particular and well-defined MHD problem, (e.g. internal kink instability, edge-localized modes, tearing modes) knowing whether it is better to use kinetic closure of MHD or solve the problem directly using kinetics is very much unanswered at this point. In this talk we will discuss the possible ways to close MHD equations using kinetics, as well as more direct MHD-like kinetic models. This talk will highlight some recent successes in kinetic-MHD, including modeling of energetic particle effects in fusion plasmas. We will also discuss recent kinetic and kinetic-MHD models of tearing mode behavior.

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MHD Turbulence in Galaxies and Clusters

Alexander Schekochihin, University of Cambridge

Abstract: I will review the main points of our recent work on small-scale dynamo and MHD turbulence in media with large magnetic Prandtl number --- with application to interstellar and intracluster media in mind. The properties of the magnetic fields generated by small-scale dynamo, the saturation mechanism and the nature of the fully developed state of isotropic MHD turbulence will be discussed. The key idea is that interactions in isotropic MHD turbulence may be nonlocal in wavenumber space, so Kolmogorov-style dimensional theories based on the idea of energy cascade of Alfven-wave packets may not be applicable. I will also discuss briefly the modern state of observational evidence relevant to this subject. A detailed account of this work is contained in Schekochihin et al. 2004, ApJ 612, 276.

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Dynamo Amplification of Field in Clusters and the Problem of Viscosity

Steven Cowley, University of California, Los Angeles

Abstract: (abstract not available for print.)

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Transition Regime Models from Kinetic Equations

David Levermore, University of Maryland

Abstract: (abstract not available for print.)

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Simulation of the Magneto-Thermal Instability

Ian Parrish, Princeton University

Abstract: Recent work by Balbus has shown that there exists an instability in convectively stable atmospheres with weak magnetic fields in the presence of anisotropic thermal conductivity.1,2 An analogy can be made between this instability and the well-studied magneto- rotational instability.3 Specifically, in a Keplerian disk with a stable angular momentum gradient (conserved quantity) the MRI is able to extract energy from the angular velocity gradient (source of free energy). Analogously, in a convectively stable atmosphere with a monotonically decreasing entropy gradient (conserved quantity), the MTI is able to extract energy from the temperature gradient (source of free energy). We simulate this instability by using the Athena magnetohydrodynamics code with the addition of anisotropic heat conduction.4 We have verified the analytical expression for the linear behavior of this instability with the computational model. The computational results are extended to the non-linear regime to examine the astrophysical importance of this instability.

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Plasma Simulation Studies using Mutilevel Physics Models

Wonchull Park, Princeton University

Abstract: The M3D (Multilevel 3D) Project carries out plasma simulation studies using a code package which solves a hierarchy of physics models with increasing realism. The available physics levels are fluid models: MHD and two-fluids, and hybrid models: Gyrokinetic- energetic-particle/MHD, and Gyrokinetic-particle-ion/Fluid-electron. The M3D code uses finite element unstructured meshes on poloidal planes and finite difference in toroidal direction. It runs with good parallel scaling to more than 1000 processors. The rationale of the project and recent results of tokamak and stellarator studies will be presented.

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Properties of Landau-Fluid Models of Kinetic MHD

Gregory Hammett, Princeton University

Abstract: (abstract not available for print.)

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Initial Nonlinear Landau-MHD Simulations of Kinetic Effects on the MRI

Prateek Sharma, Princeton University

Abstract: Preliminary simulations of the collisionless magnetorotational instability (MRI) will be presented. We have modified the widely used ZEUS code to include anisotropic pressure. A Landau fluid prescription is used for heat conduction parallel to the magnetic field. The equations and their numerical implementation will be discussed. Conservation of adiabatic invariants will lead to pressure anisotropy in a collisionless plasma as the magnetic field amplifies. Implications of anisotropy on the nonlinear evolution of the MRI will be discussed.

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Numerical Methods for Astrophysical MHD

James Stone, Princeton University

Abstract: Studying the multidimensional, time-dependent and/or highly nonlinear dynamics of astrophysical plasmas usually requires numerical methods, however developing accurate and robust methods for compressible MHD is still an active area of research. I will describe some problems in astrophysics which motivate the development and application of numerical methods for MHD. Next, I will describe both standard numerical methods used in astrophysics that have proven to be reliable and robust, as well as recent advances in algorithm development that promise new and more accurate methods. Finally, I will describe some of what we have learned from application of the methods.

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Successful MHD Simulation Methods in Plasma Physics Applications

Carl Sovinec, University of Wisconsin, Madison

Abstract: Magnetohydrodynamics is frequently used for assessing the global force-balance and ideal stability properties of laboratory plasmas. It also provides valuable information on the nonlinear evolution of macroscopic instabilities—if not quantitatively, then either qualitatively or as a starting point for multi-fluid and kinetic treatments. The need for numerical computation arises from strong nonlinear effects and from behavior that is sensitive to geometry. While the magnetic Mach number is small in magnetically confined plasmas, numerical algorithms must deal with extreme stiffness and anisotropy in nearly dissipation-free conditions (large Lundquist number). Presently, several techniques for solving the MHD system with a large separation of time- scales are being used or are undergoing continued development. The merits of methods described as 'partially implicit,' 'semi-implicit,' and 'fully implicit' will be discussed and compared. Modeling extreme anisotropy benefits greatly from high-order spatial representations, as illustrated through examples from a finite-element application.

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High-Accuracy, Implicit Solution of the Extended-MHD Equations using High-Continuity Finite Elements

Stephen Jardin, Princeton University

Abstract: It has been recognized for some time that it is necessary to go beyond the simple "resistive MHD" description of the plasma in order to get the correct quantitative results for the growth and saturation of global dissipative modes in a fusion device. The inclusion of a more complete "generalized Ohms law" and the off-diagonal terms in the ion pressure tensor introduce Whistler waves, Kinetic Alfven waves, and gyro-viscous waves, all of which are dispersive and require special numerical treatment. We have developed a new numerical approach to solving these Extended-MHD equations using a compact representation that is specifically designed to yield efficient high-order-of-accuracy, implicit solutions of a general formulation of the compressible Extended-MHD equations. The representation is based on a triangular finite element with fifth order accuracy that is constructed to have continuous derivatives across element boundaries, allowing its' use with systems of equations containing complex spatial derivative operators of up to 4th order. The final set of equations are solved using the parallel sparse direct solver, SuperLU, which makes linear solutions exceptionally efficient, since only a one-time LU decomposition is required. The magnetic and velocity fields are decomposed without loss of generality in in a potential, stream function form. Subsets of the full set of 6 equations describing unreduced compressible extended MHD yield (1) the two variable reduced MHD equations, and (2) the 4-field Fitzpatrick-Porcelli equations. Applications are presented in straight and toroidal geometry showing the accuracy and efficiency of the method in computing highly anisotropic heat conduction, toroidal equilibrium, and the effect of "two-fluid" effects on resistive instabilities.

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The Magneto-hydrodynamic Richtmyer-Meshkov Instability

Ravi Samtaney, Princeton University

Abstract: In the past two decades the Richtmyer-Meshkov (RM) instability has become the subject of extensive experimental, theoretical and computational research due to its importance in technological applications such as inertial confinement fusion, as well as astrophysical phenomena such as shock interactions with intersteller clouds. In this talk we will present recent results from nonlinear simulations of the Richtmyer-Meshkov instability in the presence of a magnetic field. The seminar will be divided into three segments. In the first segment, we will present a brief primer on discontinuities in MHD. In the second segment we will present numerical evidence that the growth of the Richtmyer-Meshkov instability is suppressed in the presence of a magnetic field. This is due to a bifurcation which occurs during the refraction of the incident shock on the density interface. The result is that baroclinically generated vorticity is transported away from the interface to a pair of slow or intermediate magnetosonic shocks. Consequently, the density interface is devoid of vorticity and its growth and associated mixing is completely suppressed. We will present analytical results on the structure of the singular solution in the limit of vanishing magnetic field to the hydrodynamics case. The third segment on the talk will focus on the numerical method to obtain the aforementioned results. We will discuss the implementation of an unsplit upwinding method to solve the ideal MHD equations with adaptive mesh refinement (AMR) using the Chombo framework. The solenoidal property of the magnetic field is enforced using a projection method which is solved using a multigrid technique. This work was supported by USDOE Contract no. DE-AC020-76-CH03073. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

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A new CT-Godunov scheme for MHD with application to the MRI

Thomas Gardiner, Princeton University

Abstract: In recent years there has been an increased emphasis on applying high order Godunov-type algorithms to the system of ideal MHD. This is motivated by their strong shock capturing and their conservation properties which make them ideally suited for use in combination with adaptive mesh refinement. Such efforts, however, have traditionally met with difficulty owing to the divergence free constraint on the magnetic field. We describe a new, unsplit MHD Godunov-type integration algorithm which uses the Constrained Transport approach to ensure the divergence free character of the magnetic field. The algorithm includes two novel features, 1) the incorporation of MHD source terms in the PPM-type reconstruction procedure and 2) an upwind CT-algorithm for combining the Godunov fluxes to calculate the electric fields needed for CT. We present test calculations comparing this algorithm against previously published results. Finally, we highlight recent progress in applying this algorithm to problems of astrophysical interest.

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Preliminary Results for Adaptive Particle Refinement

Gregory Howes, University of California, Berkeley

Abstract: Adaptive Particle Refinement (APR) is a new scheme providing a generally adaptive framework for Lagrangian particle methods for numerical simulation of astrophysical phenomena. The strategy for error estimation and refinement in APR is described. Initial results of this generally adaptive approach are presented.

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An Unsplit Godunov Method for Ideal Magnetohydrodynamic Simulations of the Interstellar Medium

Robert Crockett, University of California, Berkeley

Abstract: The Interstellar Medium (ISM) can, to good approximation, be treated as a non-resistive magnetized fluid. In order to accurately simulate this highly turbulent, compressible fluid requires numerical schemes that faithfully reproduce shocks and other nonlinear structures. These are handled very well in general by finite-volume Godunov methods. However, certain types of nonlinearities can cause problems wherein the divergence-free constraint, div.B=0, is not maintained. This can have deleterious effects, causing incorrect dynamics and field topologies, and numerical instabilities. I outline several related Godunov schemes with for ensuring that the effects of not maintaining the divergence-free condition are minimized, even for highly nonlinear structures. All these schemes are based on an unsplit, second-order corner-transport upwind method. Our chosen scheme retains the cell-centering of variables, making extension to adaptive meshes easier. Preliminary results using our adaptive mesh code, both on relatively simple test problems and on simulations of magnetized turbulence, are presented.

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