I welcome midshipmen interested in working with me for reading courses, honors projects or trident scholarships to contact me.
My work is naturally interdisciplinary, involving topics from mathematics, physics, computer science, and engineering. My research interests can be broadly stated with the following bullet list:
- Numerical analysis and scientific computing.
- Numerical methods for plasma physics:
- Kinetic plasma models (Vlasov-Poisson and Vlasov-Maxwell), and
- Fluid plasma models (single and two-fluid magnetohydrodynamics (MHD)).
- High-order numerical methods for hyperbolic conservation laws:
- Discontinuous Galerkin (DG) FEM schemes.
- Weighted essentially non-oscillatory (WENO) finite difference and finite volume methods.
- Time stepping methods (for PDEs):
- Lax-Wendroff (Taylor) methods,
- Multiderivative (strong stability preserving) methods,
- Method of lines transpose, and
- Integral/spectral deferred correction methods.
- High-order positivity preserving schemes.
- Limiters for DG and FD-WENO methods.
The primary domain I work in is the development of numerical algorithms for the purposes of solving problems of scientific merit, including plasma and gas dynamics. My research statement (current as of Fall 2014) describes a more in depth overview of the above mentioned methods.
Material contained herein is made available for the purpose of peer review and discussion and does not necessarily reflect the views of the Department of the Navy or the Department of Defense.
Supersonic shock wave. Image courtesy NASA
Solution to a two-dimensional Riemann problem for the Euler system describing gas dynamics with shocks.
Wind tunnel problem with a step.
Most of my pre-prints can be found in the arXiv.org Search Results.
- Z. Grant, S. Gottlieb, and D.C. Seal, A Strong Stability Preserving Analysis for Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions, (submitted). (Download paper from arXiv.org)
- M.F. Causley, and D.C. Seal, On the convergence of spectral deferred correction methods (under revision). (Download paper from arXiv.org)
- S. Moe, J.A. Rossmanith, and D.C. Seal, A simple and effective high-order shock-capturing limiter for discontinuous Galerkin methods (under revision). (Download paper from arXiv.org)
- Jochen Schütz, David C. Seal, and Alexander Jaust, Implicit multiderivative collocation solvers for linear partial differential equations with discontinuous Galerkin spatial discretizations, J. Sci. Com., (2017) (Download paper from arXiv.org)
- S. Moe, J.A. Rossmanith, and D.C. Seal, Positivity-preserving discontinuous Galerkin methods with Lax-Wendroff time discretizations, J. Sci. Comp., (2017) (Download paper from arXiv.org)
- Alexander Jaust, Jochen Schütz and David C. Seal, Implicit multistage two-derivative discontinuous Galerkin schemes for viscous conservation laws, J. Sci. Comp., (2016) (Download paper from arXiv.org)
- A.J. Christlieb, X. Feng, D.C. Seal, and Q. Tang, A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations J. Comp. Phys., (2016) (Download paper from arXiv.org)
- M.F. Causley, H. Cho, A.J. Christlieb, and D.C. Seal, Method of lines transpose: High order L-Stable O(N) schemes for parabolic equations using successive convolution SIAM J. Numer. Anal., (2016) (Download paper from arXiv.org)
- A.J. Christlieb, S. Gottlieb, Z. Grant, and D.C. Seal, Explicit strong stability preserving multistage two-derivative time-stepping scheme, J. Sci. Comp., (2016). (Download paper from arXiv.org)
- D.C. Seal, Q. Tang, Z. Xu, and A.J. Christlieb, An explicit high-order single-stage single-step positivity-preserving finite difference WENO method for the compressible Euler equations, J. Sci. Comp., (2015). (Download paper from arXiv.org)
- A. Jaust, J. Schütz, and D.C. Seal, Multiderivative time-integrators for the hybridized discontinuous Galerkin method, YIC GACM Conference proceedings, 2015. (Download paper)
- A.J. Christlieb, Y. Güçlü, and D.C. Seal. The Picard integral formulation of weighted essentially non-oscillatory schemes SIAM J. Numer. Anal., 53(4), 1833–1856, 2015. (Download paper from arXiv.org)
- D.C. Seal, Y. Güçlü and A.J. Christlieb. High-order multiderivative time integrators for hyperbolic conservation laws, J. Sci. Comp., Vol. 60, Issue 1, pp 101-140, 2014. (Download paper from arXiv.org)
- J.A. Rossmanith and D.C. Seal. A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations. J. Comp. Phys., 227: 9527--9553, 2011. (Download paper from arXiv.org)
Here is an incomplete list of some people that I have worked or am currently working with:
- Matt Causley, Kettering University
- Andrew Christlieb, Michigan State University
- Sigal Gottlieb, University of Massachusetts Dartmouth
- Yaman Güçlü, Max-Planck-Institut fur Plasmaphysik
- Cory Hauck, Oak Ridge National Laboratory and University of Tennesee
- Stephen Pankavich, Colorado School of Mines
- James Rossmanith, Iowa State University
- Jennifer Ryan, University of East Anglia
- Jochen Schütz, RWTH Aachen University
- Hana Cho, (PhD, MSU 2016)
- Michael Crockatt, Michigan State University
- Xiao Feng, (Expected PhD, 2016) Michigan State University
- Zachary Grant, University of Massachusetts, Dartmouth
- Alexander Jaust, RWTH Aachen University
- Jaylan Jones, (PhD, MSU 2013), Raytheon
- Scott Moe, University of Washington
- Qi Tang, (PhD, MSU 2015), First job is a postdoc at RPI.
An integral component of my work requires the development, testing and release of software for the purposes of investigating and validating new numerical methods.
I am one of the developers for the DoGPack software package. My primary contributions include the semi-Lagrangian sections of the code, as well as the bulk of the Lax-Wendroff time stepping options of the code. The name stands for `The Discontinuous Galerkin Software Package' and is written in C++. There are a number of choices for what the `o' between the `D' and `G' stands for.
Additionally, I am the lead developer for the software package FINESS, which is a FINite difference weighted ESSentially non-oscillatory (FD-WENO) software package. The basic structure was derived from DoGPack, and all of my work involving finite difference methods has been incorporated into the package.