The investigator, along with his students and collaborators, will develop efficient high-order time-stepping methods that will be used in conjunction with discontinuous Galerkin spatial discretizations. In particular, adaptive space-time methods will be developed that will allow for either (1) explicit local time-stepping that allows for different time-steps in different flow regimes, or, (2) implicit time-stepping, which is sometimes required in certain applications. A key feature that will be integrated into these numerical schemes is error control in the form of adaptive mesh refinement. Several error estimators will be investigated, as well as several shock-capturing strategies. The resulting numerical schemes will be applied to a variety of model equations that arise in plasma physics, including ideal magnetohydrodynamics, two-fluid Euler-Maxwell, and kinetic Vlasov equations. Specific application problems that are of interest include the dynamics of solar coronal loops, the formation and propagation of astrophysical jets, and the simulation of collisionless magnetic reconnection.

Plasma is the fourth state of matter (after solid, liquid, and gas), which can be characterized as an ionized gas (i.e., a gas that is able to conduct electricity). Plasma appears in a wide range of applications including astrophysics and space physics, as well as in laboratory settings such as in magnetically confined fusion. Modeling and understanding the basic phenomenon in plasma have long been topics in scientific computing, yet many problems remain far too numerically intensive for modern parallel computers. The main difficulty is that plasmas span a wide range of spatial and temporal scales. The scope of this research is to develop accurate and efficient computational methods that can better solve various equations that model plasma behavior. A key aspect of this research is the development of adaptive numerical methods that are able to dynamically estimate and control the errors that are produced during the course of a computation.

Project Report

The goals of the proposed research were to develop novel computer simulation techniques for solving a variety of mathematical models that arise in the dynamics of ionized gases (commonly referred to as 'plasma'). This research was based on a class of numerical approximtion techniques that are broadly known as 'high-resolution shock-capturing' schemes. A particular emphasis of this work was on 'high-order' numerical methods, which refers to a particular approach to use high-order piecewise polynomials to approximate the sought-after true solution. The dynamics of plasma can be simulated using kinetic or fluid models. Kinetic models are valid over most of the spatial and temporal scales that are of physical relevance in many application problems; however, they are computationally expensive due to the high-dimensionality of phase space. Fluid models have a more limited range of validity, but are generally computationally more tractable than kinetic models. One critical aspect of fluid models is the question of what assumptions to make in order to close the fluid model. The research efforts of this work spanned models ranging from single-fluid magnetohydrodynamic (MHD) models to kinetic Vlasov models. This work was motivated by several application problems from the fields of astrophysical and space plasma, including fast magnetic reconnection in collisionless plasma. The research efforts resulted in several journal publications. Some key findings from these journal articles are summarized below. A novel computer simulation technique was developed for solving the single-fluid magnetohydrodynamic (MHD) equations. Consistent with our understanding of electromagnetism, the MHD equations do not allow the creation of magnetic monopoles. The key challenge is to devise numerical approaches that do not create magnetic monopoles on the discrete level. A broad framework was developed in this research to overcome this difficulty. This framework was applied to a variety of numerical discretizations, including finite difference, finite volume, and finite element schemes. Mathematical models of plasma contain variables that remain positive (e.g.,density and pressure). Numerical approximations to these models in general are not guaranteed to maintain this positivity property. This can, and indeed often does, lead to numerical instabilities. As part of this research, a new framework, which is referred to as 'outflow positivity limiting', was developed. A high-order numerical scheme was developed for the kinetic Vlasov-Poisson system. This work involved the development of specially designed positivity limiters and an efficient high-order time-stepping approach. High-order numerical schemes were developed for a novel class of fluid models of plasma. The models were derived by starting from a kinetic Vlasov model and replacing the true solution with a simple approximation that yielded a fluid model. The resulting models and numerical schemes were tested on test cases for the Vlasov-Poisson Fokker-Planck system. The research efforts involved the training of several PhD students. The work of the PI and the associated PhD students resulted in computer code that has been made publicly available. One of the purposes of this is to allow the research findings to be dessiminated more directly than through journal publications. The resulting codes are written in C++ (for the main part of the code) and Python (for visualization). These codes are freely available on the web ( through a BSD 2-Clause License from the Open Source Initiative. The actual code is stored as a Git repository through the website: These codes have been used both for research and teaching purposes and continue to be updated, improved, and expanded.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Leland M. Jameson
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University of Wisconsin Madison
United States
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