### Research Overview

Broadly, I am interested in applied mathematics and partial differential equations. Within these fields, I have a variety of active research projects, summarized below:

### Accelerated Dynamics

Since joining the Math/Mechanics/Materials PIRE, I have become interested in accelerated dynamics, methods for accelerating the computation of many body systems, such as those found in atomistically modeled materials.

While techniques like the quasicontiuum method have helped to reduce the degrees of freedom required to accurately model a material, the time scales required to see infrequent events'', may be intractable. Studying the algorithms developed by A. Voter and his collaborators (Parallel Replica Dynamics, Hyperdynamics, Temperature Accelerated Dynamics) is one of my active projects.

Publications:
• G. Simpson, M. Luskin. Numerical Analysis of Parallel Replica Dynamics. Submitted [arXiv]

### Well-Posedness of PDEs

Studying the well-posedness of equations tells us whether they have solutions, in a mathematical sense, and how those solutions behave. Do they always exist? Do they develop singularities? Answering these questions is of interest not just in the pure sense, but also in applications modeled by the PDE. For example, does the appearance of a singularity in an equation reflect a true singularity of the system, or a failure of the assumptions used to model the system?

Some of the problems I have worked on in this context include a nonlinear wave equation arising in Earth science, the Zakharov equations from plasma physics, and the nonlinear Schrodinger equation.

Publications:
• G. Simpson, I. Zwiers. Vortex Collapse for the L2-Critical Nonlinear Schrodinger Equation. Journal of Mathematical Physics, 52(8):083503, 2011. [pdf][arXiv][codes]
• G. Simpson, C. Sulem, and P.L. Sulem. Arrest of Langmuir wave collapse by quantum effects. PRE , 80:5, 056405, 2009. [pdf]
• G. Simpson, M. Spiegelman, and M.I. Weinstein. Degenerate dispersive equations arising in the study of magma dynamics. Nonlinearity, 20:21--49, 2007. [pdf]

### Nonlinear Waves

Much of my rigorous work on PDEs has been inspired by solitary waves, localized, persistent solutions to nonlinear wave equations. These distinguished solutions are of interest both as mathematical objects and in applications, where they may be used in the transmission of energy, information, and mass.

But before a solitary wave could be used for such a purpose, we must consider its stability. If we perturb a solitary wave, will it stay close to the initial state, or will it break up into smaller waves? The stability of solitary waves is a topic of great interest to me, and I look to a variety of techniques in studying it, including variational analysis and spectral theory.

Publications:
• X. Liu, G. Simpson, C. Sulem. Stability of Solitary Waves for a Generalized Derivative Nonlinear Schrodinger Equation. Submitted [arXiv]
• R. Asad, G. Simpson. Embedded Eigenvalues and the Nonlinear Schrodinger Equation. Submitted [pdf][arXiv][codes]
• J.L. Marzuola, G. Simpson. Spectral Analysis for Matrix Hamiltonian Operators. Nonlinearity, 24:389-429, 2011. [pdf] [arXiv][codes]
• J. Marzuola, S. Raynor, and G. Simpson. A System of ODEs for a Perturbation of a Minimal Mass Soliton. Journal of Nonlinear Science , 20:425--461, 2010. [pdf]
• G. Simpson and M.I. Weinstein. Asymptotic stability of ascending solitary magma waves. SIAM J. Math. Anal., 40:1337--1391, 2008. [pdf]
• G. Simpson, M.I. Weinstein and P. Rosenau. On a Hamiltonian PDE arising in magma dynamics. DCDS-B, 10:903--924, 2008. [pdf]

### Scientific Computing

Some of the problems that I study are treated by direct numerical simulation. This includes studying stability of structures, assessing singularity formation, and examining the limits of rigorous predictions. For these problems, great care must be taken to distinguish between "real" and "numerical" effects. This was the case in some recent work on supercritical wave equations, and for forthcoming work on singularity formation.

I am also interested in algorithms for problems on unbounded domains. This is motivated by my research in solitary waves, which usually live on R^d. If one wishes to investigate their stability numerically, a high quality approximation is required.

Publications:
• G. Simpson, M. Luskin. Numerical Analysis of Parallel Replica Dynamics. Submitted [arXiv]
• G. Simpson, M.I. Weinstein. Coherent Structures and Carrier Shocks in the Nonlinear Periodic Maxwell Equations. Submitted [pdf][arXiv]
• G. Simpson, M. Spiegelman. Solitary Wave Benchmarks in Magma Dynamics. Journal of Scientific Computing. [pdf][SpringerLink]
• J. Colliander, G. Simpson, and C. Sulem. Numerical simulations of the energy- supercritical Nonlinear Schrodinger equation. JHDE, 7:279--296, 2010. [pdf]
See the supercritical simulations from the Colliander, Simpson, and Sulem paper and the nonlinear maxwell simulations from the Simpson and Weinstein paper in the gallery.

### Applications

As a graduate student, I was funded by an NSF IGERT combining Earth science and applied mathematics. This has led me to become quite interested in problems in the solid Earth, particularly the rheological properties of Earth materials. I am particularly interested in magma migration, how molten rock flows in the Earth's interior, as this may be fundamental to fully understanding the interaction between plate tectonics and mantle convection.

The closure problem inherent to systems with multiple temporal and spatial scales is particularly challenging. Macroscopic variations should impact the fine scale, but can the fine scale dynamics influence the large scale? If so how does one model the fine scale effect on the macroscopic scale, without having to resolve the fine scale? In addition to the magma problem, I am also interested in modeling challenges in fluid mechanics, nonlinear optics, and materials science.

Publications:
• G. Simpson, M. Spiegelman, and M.I. Weinstein. A Multiscale Model of Partial Melts 2: Numerical Results. Journal of Geophysical Research -- Solid Earth, 115, B04411. [pdf]
• G. Simpson, M. Spiegelman, and M.I. Weinstein. A Multiscale Model of Partial Melts 1: Effective Equations. Journal of Geophysical Research -- Solid Earth, 115, B04410. [pdf]
See the cell problems in the gallery for some example computations from this work.