## Research

### Research Blurb

My research is into the development, analysis, and implementation of numerical methods for solving hyperbolic conservation laws. A conservation law is a partial differential equation of the form: $$q_{,t} + \nabla \cdot f(q) = 0,$$ 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: $$A(q; n) = \frac{\partial}{\partial q} \left( n \cdot f(q) \right) \in \mathbb{R}^{M \times M},$$ 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.

### Current Students (BS, MS, and PhD)

• 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

### Former Students (BS)

• 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)