## Research

My research is into the development, analsyis, and implementation of numerical methods
for solving hyperbolic conservation laws. A **conservation law** is a partial differential
equation of the form:
\begin{equation}
q_{,t} + \nabla \cdot f(q) = 0,
\end{equation}
where \(q(t,x): \left(\mathbb{R}_{\ge 0}, \mathbb{R}^d \right) \mapsto \mathbb{R}^M\) is the
**vector of conserved variables** (e.g., mass, momemtum, and energy) and
\(f(q): \mathbb{R}^M \mapsto \mathbb{R}^{M \times M}\) is the **flux tensor**.
This equation is **hyperbolic** if the **flux Jacobian**:
\begin{equation}
A(q; n) = \frac{\partial}{\partial q} \left( n \cdot f(q) \right) \in \mathbb{R}^{M \times M},
\end{equation}
is **diagonalizable** with **only real eigenvalues** for any direction \( \| n \| = 1 \).

The signficance of this is that the eigenvalues of the flux Jacobian represent wave speeds in the system, and the fact that each is real, means that information in the system is propagating at a finite speed. For example, think sound waves propagating at the speed of sound, light waves propagating at the speed of light, or shallow water waves propagating at the gravity wave speed.

Specifically, in my work I am interested in solving various mathematical models from fluid dynamics, plasma physics, and astrophysics, including the following nonlinear hyperbolic systems: (1) Magnetohydrodynamics (MHD), (2) Euler-Maxwell, (3) Vlasov-Poisson, (4) Vlasov-Maxwell, and (5) the Einstein equations of general relativity. The kinds of numerical methods that I develop include the following high-order schemes: (1) Wave Propagation Schemes, (2) Residual Distribution Schemes, (3) Discontinuous Galerkin Schemes, and (4) WENO Schemes.

I welcome graduate students with a strong background or interest in applied and/or computational mathematics. I typically accept graduate students through the ISU Graduate Program in Mathematics.

- Lindsey Peterson (MS, ISU)
**Research topic:**Spectral Method Based on the Radon Transform for Models of Radiative Transfer

- Christine Wiersma (MS/PhD, ISU)
**Research topic:**DG Schemes for Quadrature-based Moment Closure Approximations of the Boltzmann Equation

- Caleb Logemann (MS/PhD, ISU)
**Research topic:**DG Schemes for Thin-film Models on Curved Manifolds

- Minwoo Shin (PhD, ISU)
**Research topic:**DG Schemes for PN Equations of Radiative Transport

- Pierson Guthrey (PhD, 2017, ISU)
**PhD Dissertation:**Regionally Implicit Discontinuous Galerkin Methods for Solving the Relativistic Vlasov-Maxwell System**First job:**Postdoc at Michigan State University (East Lansing, MI)**Current job:**Postdoc at Michigan State University (East Lansing, MI)

- Scott Moe (PhD, 2017, U. Washington)
- Co-adivsed with Prof. Randall J. LeVeque
**PhD Dissertation:**High Order Shock Capturing Methods with Compact Stencils for use with Adaptive Mesh Refinement and Mapped Grids**First job:**Senior design engineer at AMD (Bellevue, WA)**Current job:**Senior design engineer at AMD (Bellevue, WA)

- Yongtao Cheng (PhD, 2014, UW-Madison)
**PhD Dissertation:**A Mixed Fluid-Kinetic Solver for the Vlasov-Poisson Equations**First job:**Postdoc in the Department of Electrical and Electronic Engineering at University of Hong Kong (Hong Kong)**Current job:**Visiting Assistant Professor at University of Arizona (Tucson, AZ)

- Bertram Taetz (PhD, 2012, Ruhr-Universität Bochum)
- Co-advised with Prof. Christiane
Helzel

**PhD Dissertation:**Unstaggered Constrained Transport Methods for Ideal Magnetohydrodynamic Equations**First job:**Senior Researcher, Department Augmented Vision, German Research Center for Artificial Intelligence (Kaiserslautern, Germany)**Current job:**Postdoc at AG wearHEALTH, Department of Computer Science, University of Kaiserslautern (Germany)

- Co-advised with Prof. Christiane
Helzel
- David Seal (PhD, 2012,
UW-Madison)
**PhD Dissertation:**Discontinuous Galerkin Methods for Vlasov Models of Plasma**First job:**Postdoc at Michigan State University (East Lansing, MI)**Current job:**Assistant Professor at US Naval Academy (Annapolis, MD)

- E. Alec Johnson (PhD, 2011, UW-Madison)
**PhD Dissertation:**Gaussian-Moment Relaxation Closures for Verifiable Numerical Simulation of Fast Magnetic Reconnection in Plasma**First job:**Postdoc at the Centre for Mathematical Plasma-Astrophysics KU-Leuven (Leuven, Belgium)

- Erica Johnson (MS, 2017, ISU)
**MS Thesis:**A High-Order Discontinuous Galerkin Finite Element Method for a Quadrature-Based Moment-Closure Model**First job:**Data Analyst at Bexar County (San Antonio, TX)**Current job:**Data Analyst at Bexar County (San Antonio, TX)

- Milo Taylor (MS, 2016, ISU)
**MS Thesis:**An Implementation of the Relativistic Hydrodynamic Equations in Conservative Form using DoGPack**First job:**Research Associate at Johns Hopkins Applied Physics Lab (Laurel, MD)**Current job:**Research Associate at Johns Hopkins Applied Physics Lab (Laurel, MD)

- Anna Lischke (MS, 2015, ISU)
**MS Thesis:**Asymptotic-preserving Space-time Discontinuous Galerkin Methods for a Class of Relaxation Systems**First job:**PhD student in the Divison of Applied Mathematics at Brown University (Providence, RI)**Current job:**PhD student in the Divison of Applied Mathematics at Brown University (Providence, RI)

- Juan Brandi (MS, 2013, ISU)
**MS Creative Component:**Parallel Implementation of the Quasi-Steady Wave Propagation Method for the Shallow Water Equations**First job:**Cabela's Bank (Lincoln, NE)**Current job:**Scientist at Pacific Northwest National Laboratory (Richland, WA)

- Boqian Shen (BS, 2017, ISU)
**Undergraduate Research:**A Particle-Based Numerical Method for Solving Vlasov Models in Plasma Simulations**Gradudate school:**Department of Computational and Applied Mathematics, Rice University

- Scott Moe (BS, 2011, UW-Madison)
**Undergraduate Research:**Adaptive Mesh Refinement for Discontinuous Galerkin Methods**Gradudate school:**Department of Applied Mathematics, University of Washington

- Summer 2017 Mathematics REU at Iowa State University
- Camille Felton (University of Wisconsin Platteville)
- Mariana Harris (Instituto Tecnológico Autónomo de México (ITAM))
- Stefan Nelson (Minnesota State University Moorehead)
- Ian Pelakh (University of Florida)

- Summer 2015 Mathematics REU at Iowa State University
- Alan Medinger (Lewis & Clark College)
- Rachel Nevin (Augustana College)
- Emma Talis (Marist College)
- Jae-Jae Young (Iowa State University)