Plasmas have a huge range of applications in engineering and science. Plasmas are essential in the modern manufacturing of microchips. Plasmas are key in lighting, they are the basis of arclights that are used as street post lights as well as in overhead fluorescent tubes used in lighting most office buildings. Plasmas are finding their way into the medical industry, from sterilization through exciting breakthroughs in wound healing. Further our sun is a plasma and 99% of the visible matter in the universe is in the plasma state.
Plasmas have a huge range of dynamic scales that are coupled in both time and space. Consider time for a second. Bridging time scales includes modeling waves that propagate at the speed of light through waves that propagate at the speed of sound. It is precisely this dynamic range in scales that makes modeling plasmas difficult. This work is really focused on new techniques that help us bridge the scales offering highly accurate solvers that can advance the evolution of the models of these complex systems with greater efficiency than was previously possible. The eventual goal is to move these multiscale tools into largescale simulation frameworks that engineers and scientists use to study, model and design systems that involve plasma. This will give the engineer and scientist a better set of tools, allowing them to study much larger dynamic ranges in this class of problems than was previously possible, facilitating breakthroughs in design via a virtual laboratory.
Because of the complex nature of a plasma, there are a range of models for these systems. From the most detailed to the most basic models: plasmas can be described with fully Kinetic Models (both particle and continuum models), multi fluid models, single fluid models, zero dimensional rate equation models and many things in between. Our group works on a range of these models, seeking to develop methods which will allow us to capture long time behavior in these complex systems.
Plasmas have a huge range of dynamic scales that are coupled in both time and space. Consider time for a second. Bridging time scales includes modeling waves that propagate at the speed of light through waves that propagate at the speed of sound. It is precisely this dynamic range in scales that makes modeling plasmas difficult. This work is really focused on new techniques that help us bridge the scales offering highly accurate solvers that can advance the evolution of the models of these complex systems with greater efficiency than was previously possible. The eventual goal is to move these multiscale tools into largescale simulation frameworks that engineers and scientists use to study, model and design systems that involve plasma. This will give the engineer and scientist a better set of tools, allowing them to study much larger dynamic ranges in this class of problems than was previously possible, facilitating breakthroughs in design via a virtual laboratory.
Because of the complex nature of a plasma, there are a range of models for these systems. From the most detailed to the most basic models: plasmas can be described with fully Kinetic Models (both particle and continuum models), multi fluid models, single fluid models, zero dimensional rate equation models and many things in between. Our group works on a range of these models, seeking to develop methods which will allow us to capture long time behavior in these complex systems.

Fluid modelsWhile a kinetic model carries the most detailed information about the system, many times this is more information than we need. In this setting we resort to using either multi fluid models or single fluid models. There is a direct connection between the kinetic model and the fluid model through what is termed moments, which are specific averages over velocity. These models are three dimensional plus time. To the right is a set of images of a single fluid model describing a shock wave interacting with a dense cloud of plasma. Top to bottom shows the evolution in time.


Select Papers:
Implicit Wave Propagation and SemiImplicit Particle Methods for Plasma Simulation  Method of Lines Transpose 
In this work we developed a new class of methods for modeling wave propagation, such as acoustic or electromagnetic waves. The major advantage of this new class of methods is that it facilitates modeling problems where there is a separation of scales in time. The primary application that we had in mind when we were developing the method is in the modeling of plasma, the fourth state of matter.
Implicit Wave Propagation and SemiImplicit Particle Methods for Plasma Simulation  Method of Lines Transpose 
In this work we developed a new class of methods for modeling wave propagation, such as acoustic or electromagnetic waves. The major advantage of this new class of methods is that it facilitates modeling problems where there is a separation of scales in time. The primary application that we had in mind when we were developing the method is in the modeling of plasma, the fourth state of matter.
 M. Causley, A.J. Christlieb, E. Wolf, ''Method of Lines Transpose: An Implicit Solution to the One Dimensional Wave Equation'' , accepted Journal of Scientific Computing and on axriv
 M. Bettencourt M. Causley, A.J. Christlieb, and E. Wolf, ''A ParticelInCell Method for The Simulation of Plasmas Based on An Unconditionally Stable Field Solver'' , Journal of Computational Physics, 326, 342372, 2016
 Y. Cheng, A.J. Christlieb, W. Guo, B. Ong, ''An Asymptotic Preserving Maxwell Solver Resulting in the Darwin Limit of Electrodynamics'', Journal of Scientific Computing, 71(3), 959993, 2017
Select Papers:
Efficient Methods for Solving Kinetic Equations:
In this work we develop a series of particle methods, semiimplicit continuum methods and fully implicit continuum methods that are targeted at accurately solving kinetic models. These methods are necessarily high order in space and time so as to maintain accuracy over long time simulations of instabilities that arise in plasma.
Efficient Methods for Solving Kinetic Equations:
In this work we develop a series of particle methods, semiimplicit continuum methods and fully implicit continuum methods that are targeted at accurately solving kinetic models. These methods are necessarily high order in space and time so as to maintain accuracy over long time simulations of instabilities that arise in plasma.
 A.J. Christlieb, R. Krasny, J.P. Verboncoeur, J. Emhoff and I.D. Boyd , "GridFree Plasma Simulation Techniques'', IEEE Trans. on Plasma Science, 34 (2): 149165 Part 1 APR 2006
 S. Olson and A.J. Christlieb, "GridFree Direct Simulation Monte Carlo'', Journal of Computational Physics, 227{17}, 80358064, 2008
 A.J. Christlieb, W.N.G. Hitchon, J.E. Lawler and G.G. Lister, "Integral and Lagrangian Simulations of Particle and Radiation Transport in Plasma.'', Journal of Physics D: Applied Physics, 42 (2009) 194007
 J. Qiu and A.J. Christlieb, "A Conservative high order semiLagrangian method for the Vlasov Equation”, Journal of Computational Physics, 229(4), 1130–1149, 2010.
 A.J. Christlieb, W. Guo, M. Mortion and J. Qiu, "A High Order Time Splitting Method Based on Integral Deferred Correction for SemiLagrangian Vlasov Simulations”, Journal of Computational Physics, 267, 7–27, 2014
 Y. Guclu, A.J. Christlieb, W.N.G. Hitchon, "Arbitrarily high order Convected Scheme so lution of the VlasovPoisson systemr”, Journal of Computational Physics, 270, 711–752, 2014
 Y. Cheng, A.J. Christlieb, X. Zhong, "Energyconserving Discontinuous Galerkin Methods for the VlasovMaxwell System”, Journal of Computational Physics, 279, 145–173, 2014
 Y. Cheng, A.J. Christlieb, X. Zhong, "Numerical study of the twospecies VlasovAmp‘ere system: energyconserving schemes and the currentdriven ionacoustic instability,”, Journal of Computational Physics, 288, 66–85, 2015
 A.J. Christlieb, Y. Jang, W. Gou, "A WENObased Method of Line Transpose Approach for Vlasov Simulations” Journal of Computational Physics, 327, 337367, 2016
Select Papers:
High Order Constrained Transport for Magnetohydrodynamics:
In this work we have developed a series of high order methods for single fluid models of plasmas. This class of fluid models is known as Magnetohydrodynamics. It is essential in these models that one maintains the divergence free property of the magnetic field, else the numerical simulation will go unstable. The particular type of method we have developed is known as unstaggered constrained transport. In all forms of methods for Magnetohydrodynamics, some form of auxiliary equation is solved to ensure that the magnetic field is divergence free. The form we have developed in our work is very efficient because we cast the auxiliary equation in the form of a conservation law that facilitates the use of a fast solver to update the solution at each time step.
High Order Constrained Transport for Magnetohydrodynamics:
In this work we have developed a series of high order methods for single fluid models of plasmas. This class of fluid models is known as Magnetohydrodynamics. It is essential in these models that one maintains the divergence free property of the magnetic field, else the numerical simulation will go unstable. The particular type of method we have developed is known as unstaggered constrained transport. In all forms of methods for Magnetohydrodynamics, some form of auxiliary equation is solved to ensure that the magnetic field is divergence free. The form we have developed in our work is very efficient because we cast the auxiliary equation in the form of a conservation law that facilitates the use of a fast solver to update the solution at each time step.
 A.J. Christlieb, J.A. Rossmanith, Q. Tang, “Finite Difference Weighted Essentially NonOscillatory Schemes with Constrained Transport for Ideal Magnetohydrodynamics”, Journal of Computational Physics, 268, 302–325, 2014
 A.J. Christlieb, Y. Liu, Q. Tang, Z. Xu, “PositivityPreserving Finite Difference WENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations”, SIAM Journal on Scientific Computing 37(4), A1825–A1845, 2015
 A.J.Christlieb, X. Feng, D.C.Seal, Q. Tang, "A highorder positivitypreserving singlestage singlestep method for the ideal magnetohydrodynamic equations'', Journal of Computational Physics, 316, 218242, 2016
 A.J. Christlieb, X. Feng, Y. Jiang, Q. Tang, "A highorder finite difference WENO scheme for ideal magnetohydrodynamics on curvilinear meshes", submitted/ axriv