STATEMENT OF RESEARCH
DAVID C. SEAL
1. Introduction
I work in numerical analysis and scientific computing. My research is interdisciplinary, and it includes an
interesting blend of elements from mathematics, computer science, physics, and chemistry. My primary focus
is on fundamental algorithm development for scientific problems. I seek to define methods that make
the best use of modern computer architecture, which is often necessary to make problems tractable, especially
if they involve dynamics that act on several scales. I am interested in multi-physics scientific models that
involve multi-scale phenomenon. An understanding of model derivations including first principles is essential
to create useful numerical methods. One example of a numerical challenge is that numerical methods should
satisfy physically relevant information on a discrete level: conservation of mass, momentum, and energy.
Additionally, models often require a further set of constraints to be satisfied in order to obtain numerical
solutions with fidelity. For example, it is well known that discarding a discrete divergence criteria for the
magnetohydrodynamics equations of plasma physics leads to unstable solutions. Given the interdisciplinary
nature of my subject area, knowledge about current challenges and trends in computer science as well as
an aptitude for software engineering is necessary to develop, implement, and demonstrate the efficiency of
the numerical methods I work on.
My thesis work focused on developing high-order discontinuous Galerkin (DG) finite element methods
(FEMs) for hyperbolic partial differential equations (PDEs) [1]. Hyperbolic PDEs are prevalent in numerous
physical applications including astrophysics, gas dynamics, tsunami modeling, various geophysical applications, as well as in plasma physics (c.f. Figure 1). My postdoctoral work at Michigan State University
(MSU) has included branching out into new topics centered around the development of efficient time
integrators for PDEs that are required to make scientific problems tractable. Briefly put, I have worked
on the following topics:
• Semi-Lagrangian methods for plasma physics. My thesis work involved tackling some of
the fundamental challenges associated with kinetic plasma simulations, where I developed a semiLagrangian discontinuous Galerkin method (SLDG) that mitigates typically strict CFL conditions.
The benefit of doing this is that my method greatly reduced the computational runtime of a single
simulation. For example, on a high resolution grid, a classical DG solution required three days to
run on a shared workstation, whereas my method reached the same solution in under ten minutes.
Date: (last updated) October 6, 2014.
Figure 1. Solar corona and its interaction with the Earth’s magnetic field. Left: Solar eclipse
(Miloslav Druckmuller / SWNS); Right: Artist’s rendition (Steele Hill/NASA).
1
2
DAVID C. SEAL
• Multiderivative time integrators. Numerical methods for PDEs are typically cast in either
a method of lines (MOL) or a Lax-Wendroff framework. Multiderivative methods unify these two
frameworks. In [2], I extended the ODE results to hyperbolic PDEs with the finite difference weighted
essentially non-oscillatory (WENO) and DG methods.
• Modified flux methods. High-order multiderivative methods can be viewed as a method of
modified fluxes. After working on them, I realized there was room for improvement over classical
finite difference WENO methods with Lax-Wendroff time discretizations [3]. This resulted in a
(submitted) paper [4] on the Picard integral formulation (PIF) of WENO methods, which is based
on discretizing time-averaged fluxes in place of the traditional “frozen in time” approximations that
are typically used. The contribution of this work includes:
– A family of WENO methods can be constructed from the Picard integral formulation, that go
beyond the method of lines formulation. In [4], I describe Runge-Kutta and Taylor methods.
– The Picard integral formulation defines methods that reduced computational complexity, including i) smaller stencils, and ii) a reduction in the number of characteristic variable projections.
We have followed up this work with a submitted article on a positivity preserving method for incompressible Euler equations [5]. Our method only requires a single right hand side evaluation per
time step, and therefore is amenable to adaptive mesh refinement.
• Exponential integrators. Implicit multiderivative methods and exponential integrators yield
promising methods for multi-scale phase-field models that I have been investigating. As a postdoc, I served as a co-mentor to Jaylan Jones (PhD, 2013), whose thesis focused on investigating
time-stepping methods for solving variations of the Allen-Cahn and Cahn-Hilliard equations [6]. My
contribution included working with Dr. Jones on the application of time integrators to the recently
defined Functionalized Cahn-Hilliard equation [7], which made up a large portion of his thesis and
is an ongoing topic of research.
An overarching goal with my research is to foster collaborations between mathematicians, scientists, and engineers by making state of the art numerical methods accessible. Beginning with my
thesis work, I have become one of the primary developers of the software package DoGPack [8], which is an
open-source code for solving hyperbolic problems using the discontinuous Galerkin method on structured
and unstructured grids in multiple dimensions. During my time at Michigan state, I have developed an
open-source software package called FINite difference ESSentially non-oscillatory methods (FINESS) for
the purposes of rapid development, testing, and sharing of numerical methods.
2. Numerical methods for plasma
2.1. Plasma, the “fourth” state of matter. Beyond solid, liquid, and gas, plasma represents a state of
matter where electrons have dissociated from their nucleons. The mathematical models describing plasma
can be broadly categorized as either kinetic or fluid descriptions. In a kinetic model, one looks to model
a phase-space probability distribution function of the plasma that describes the probability of finding a
particle at a given space and velocity coordinate for all time values. Kinetic descriptions of plasma define
a set of PDEs in 6+1 dimensions (3 spatial dimensions, 3 velocity dimensions, and time). Fluid models
can be derived by taking moments in velocity space of this distribution function. These models are lower
dimensional and appropriate for plasmas near thermodynamic equilibrium. Thus far, my work has focused
on pushing the boundaries of kinetic simulations of plasma in order to better understand the reduced models.
The collisional kinetic equation, sometimes referred as the the Boltzmann equation, can be written as
(1)
ft + v · ∇x f + a · ∇v f = C(f ),
where f (t, x, v) is the distribution function of some plasma species. The acceleration term a comes from the
Lorentz force, and the collision operator C(f ) describes physical collisions between particles. For VlasovPoisson, the Lorentz force comes solely from the electric field: a = qs /ms E, where qs is the charge, and
ms is the mass of the species s under consideration, whereas for Vlasov-Maxwell, a magnetic field is also
present: a = qs /ms (E + v × B). Typical numerical techniques for solving kinetic descriptions of plasma
can be classified into three categories: particle methods [9], particle-in-cell methods [10, 11, 12], and fully
mesh-based methods [13, 14, 15]. The present focus is to describe semi-Lagrangian methods [16, 17, 18],
and in particular, the extension of my discontinuous Galerkin implementation [19] to multiple dimensions.
STATEMENT OF RESEARCH
3
Figure 2. The hybrid DG method. Left: configuration space, (x, y), is represented with an
unstructured grid. This part is solved with the classical RKDG method. Each quadrature point in configuration space represents a 2-dimensional problem in velocity space,
R2 = (vx , vy ). Right: Velocity space is represented with a Cartesian grid, and solved
with the SLDG method. Conservation (and stability) is achieved through exact integration of the characteristics, much like the corner-transport method.
2.2. Semi-Lagrangian discontinuous Galerkin (SLDG). A fully mesh-based description of the plasma
works with a grid based discretization of both the plasma as well as the electromagnetic fields. The most
obvious disadvantage of such an approach is that because kvk ∈ R+ , large velocities dominate the CFL
(Courant-Frederichs-Lewy) number and severely restrict the maximum allowable time step due to numerical,
and not physical, reasons. In order to ameliorate this condition, a semi-Lagrangian method works with an
underlying grid representation, but single time steps are taken in a Lagrangian fashion. After evolving the
solution for a short period, the solution is then projected back onto the mesh. Semi-Lagrangian methods for
Vlasov equations have a long history dating back to at least 1976 [20], and have only recently been pushed
to very high-order [21, 22].
The basic idea of my proposed hybrid scheme is to apply Strang splitting to the collisionless version of
Eqn.(1), but to accomplish the spatial discretization to high-order using the discontinuous Galerkin method.
In the case of 2D-2V Vlasov-Poisson, this reduces to piecing together the following four steps:
1. Solve the ballistic operator f,t + v · ∇x f = 0 for a time step of length ∆t/2.
2. Solve Poisson’s equation −∇2 φ = ρ − ρ0 to obtain the electric field E = (E 1 , E 2 ) = −∇φ.
3. Accelerate the particles by solving f,t − E 1 f,vx − E 2 f,vy = 0 for a time step of length ∆t.
4. Repeat Step 1 to make for a full time step.
My proposed hybrid DG scheme is given by inserting the following methods into the following steps:
• Solve steps 1 and 4 using a classical Runge-Kutta DG (RKDG) method on an unstructured grid.
The purpose of using an unstructured grid is to accommodate complicated geometry.
• Solve step 2 using the high-order version of my semi-Lagrangian DG (SLDG) method.
• Solve for the potential in Step 3 using a continuous Galerkin formulation. A discontinuous Galerkin
representation for the electric field can then be constructed by differentiating the basis functions on
the interior of each triangular element.
The evolution steps and geometry for each sub-problem is presented in Figure 2.
Remark 1. The basis functions in the DG representation of the solution contain cross-terms, and therefore
a projection step onto the so-called ‘sub-problems’ in steps 1,3 and 4 needs to be defined.
My strategy is to first recognize that there is a one-to-one correspondence between function values at
quadrature points and coefficients of basis functions in a modal representation. Therefore, one can project
onto half of the quadrature points to define a single subproblem, evolve the solution, and then integrate
back up to full solution.
3. High-order time integrators for PDEs
The term ‘time-integrator’ refers to the manner in which an initial value problem is advanced to future
times. Oftentimes the spatial variable is first discretized, and the time variable is left as a continuous
variable. This process is called the method of lines (MOL) formulation, and it defines a large system of
ordinary differential equations (ODEs), to which an appropriate ODE time integrator can be applied. In
4
DAVID C. SEAL
Multistep multistage multiderivative
Multistage
multiderivative
Runge−Kutta
Taylor
Figure 3. A simple taxonomy of multiderivative time integrators [2]. The largest class is that
as defined by Hairer and Wanner [32], and includes all General Linear Methods [36],
Taylor and Runge-Kutta methods.
many solvers, the time integrator and spatial discretization are developed separately. The space-time type
of methods that I have been looking at [2, 4] open up new possibilities beyond the MOL formulation.
3.1. Multiderivative methods. My original interest in multiderivative methods was to push the boundaries on new computing environments. It was reinvigorated by discovering methods that match the modern
paradigm shift in computer architecture: develop parallelizable algorithms that make use of floating point
operations in place of memory storage and reduce communication overhead. I discovered that multiderivative methods have a long history of development for ODEs [23, 24, 25], yet they had been largely overlooked
by the PDE community. My contribution was to demonstrate how to apply them to hyperbolic PDEs, and
in particular how they can be applied to the finite difference WENO method [26, 27, 28], as well as the
discontinuous Galerkin (DG) method [29, 30, 31].
Broadly defined, multiderivative methods can be thought of as a tool that unifies traditional time stepping
methods that are commonly taught in undergraduate numerical analysis courses: Runge-Kutta methods,
linear multistep, and Taylor method. An important paper on the topic defines a large class of methods
deemed “multistep-multistage-multiderivative” methods [32], which accommodate all Runge-Kutta (multistage), Adams (multistep) and Taylor (multiderivative) methods (c.f. Fig. 3). Multiderivative methods are
conspicuously missing from many popular textbooks on numerical methods for ODEs [33, 34], but not all
of them [35].
The important part that extends this class of methods beyond Runge-Kutta methods is the inclusion of
extra derivatives of the right hand side function. Taylor methods already do this, but they are restricted
in the choice of coefficients for the Taylor expansion. To illustrate the idea, we consider 1D hyperbolic
conservation law
(2)
qt + f (q)x = 0,
q(0, x) = q0 (x),
x ∈ [a, b].
Higher derivatives can be computed via the Cauchy-Kowalewski procedure
qt = −f (q)x ,
(3)
qtt = −f (q)tx = − (f 0 (q) · qt )x = (f 0 (q) · f (q)x )x .
For an ODE, the higher derivatives of q are unique, but PDEs require a definition for the spatial discretization, which has many options. A numerical scheme in a pure MOL formulation uses linear combinations of
−f (q)x at different stage values. A Lax-Wendroff (Taylor) method takes linear combinations of qt , qtt , . . .
with fixed coefficients. An example of a fourth-order multistage, multiderivative method is
∆t
(∆t/2)2 0 n
f (q n )x +
(f (q ) · f (q n )x )x ,
2
2
(4)
∆t2 1 0 n
2 0 ∗
n+1
n
n
n
∗
q
= q − ∆tf (q )x +
(f (q ) · f (q )x )x + (f (q ) · f (q )x )x , ,
2
3
3
which can be viewed as a method of modified fluxes. The spatial discretization happens after applying Eqn.
(3) to each q (i) term. The method I devised for the finite difference WENO scheme was based on central
differences for the higher spatial derivatives that appear. For the DG method, I modified the flux function
in order to accommodate the extra terms. Higher derivatives came from differentiating the basis functions,
and as a result, we ended up with an effectively higher-order Riemann solution at each interface.
q∗ = qn −
STATEMENT OF RESEARCH
5
4. Ongoing and future work
My future work will be rooted to my primary research interest: develop numerical algorithms for mathematical models for the purposes of advancing scientific knowledge. Below I list some of my ongoing work
and future research agendas I find interesting.
• Semi-Lagrangian methods for plasma simulations. First, collisional effects should be added,
and an investigation into boundary conditions and simulations of Langmuir probes should be
conducted. Additionally, I would like to incorporate my semi-Lagrangian solver into an embedded
boundary framework.
• Multiderivative methods. New methods create many problems to investigate. Two topics of
interest to me include:
– SSP properties. I am currently working with Prof. Sigal Gottlieb (University of Massachusetts, Dartmouth) and her student, Mr. Zachary Grant on defining strong stability preserving (SSP) properties for multistage, multiderivative methods. This property has only been
defined for linear multistep and Runge-Kutta methods.
– Optimized methods. Stability analyses for multiderivative methods requires investigating
the fully discrete system. The reason this is the case is that higher derivatives are computed
using higher spatial derivatives rather than differentiating the MOL formulation. I am currently
looking into optimizing CFL numbers for explicit multiderivative methods for the discontinuous
Galerkin and finite difference WENO spatial discretizations.
• Modified flux methods. There are many extensions of the Picard integral formulation [4].
– MHD equations. I am currently involved with co-advising Mr. Xiao Feng, who is a PhD
student of Prof. Christlieb on developing a single-stage, single-step method for magnetohydrodynamics. Our method will be high-order and provably positivity-preserving.
– Adaptive Mesh Refinement (AMR). The Taylor discretization of the PIF is single-stage,
single-step and high-order. This means it is amenable to AMR, which is important for studying
turbulence problems in gas and fluid dynamics as well as astronomical simulations.
• Defect correction algorithms. In some cases, it is possible to extract high-order methods through
the application of a post-processing filter, or through leveraging low-order solves. The advantages
these methods provide is the ability to take large time steps of the size a low-order method provides,
as well as ease of implementation with existing codes. I am actively involved in two projects that
can be broadly construed as defect correction algorithms:
– Superconvergence extraction. Prof. Jennifer Ryan (University of East Anglia) and I are
working on the development of SIAC filters to the semi-Lagrangian method from my thesis. The
result will reduce the memory overhead that would be required to obtain high-order accuracy
for a discontinuous Galerkin method, which can be a bottleneck for such a scheme.
– Spectral (Integral) deferred correction. Dr. Cory Hauck (Oak Ridge National Laboratory
and University of Tennessee) and I are working on the numerical methods for neutron transport
problems. The specific method we are using is based on a clever splitting of the problem into
collided and uncollided particles, which reduces the overhead for implicit solves.
• Method of lines transpose. The method of lines transpose [37] was originally developed as an
implicit, embedded boundary Maxwell solver by Profs. Mathew Causley and Andrew Christlieb. It
can be viewed as an arbitrarily high-order and L-stable extension of Rothe’s method, that is based
on similar space-time methods as what I have been working on. I am currently working with the
developers and Ms. Hana Cho, who is a graduate student of Prof. Christlieb, on extensions to
parabolic problems. These include complicated boundary conditions for the heat equation, as well
as parabolic problems such as Allen-Cahn, Cahn-Hilliard as well as the functionalized Cahn-Hilliard
equation.
References
[1] David C. Seal. Discontinuous Galerkin methods for Vlasov models of plasma. PhD thesis, University of Wisconsin,
Madison, WI, 2012.
[2] David C. Seal, Yaman Güçlü, and Andrew J. Christlieb. High-order multiderivative time integrators for hyperbolic conservation laws. J. Sci. Comput., pages 1–40, 2013.
6
DAVID C. SEAL
[3] Jianxian Qiu and Chi-Wang Shu. Finite difference WENO schemes with Lax-Wendroff-type time discretizations. SIAM
J. Sci. Comput., 24(6):2185–2198, 2003.
[4] David C. Seal, Yaman Güçlü, and Andrew J. Christlieb. The picard integral formulation of weighted essentially nonoscillatory schemes. http://arxiv.org/abs/1403.1282v2, 2014.
[5] David C. Seal, Qi Tang, and Andrew J. Christlieb. A high-order single step positivity preserving finite difference WENO
method for the incompressible Euler equations. (submitted), 2014.
[6] Jaylan A. Jones. Development of a fast and accurate time stepping scheme for the functionalized Cahn-Hilliard equation
and application to a graphics processing unit. PhD thesis, Michigan State University, East Lansing, MI, 2013.
[7] K. Promislow and B. Wetton. Pem fuel cells: a mathematical overview. SIAM J. Appl. Math, 70(2):369–409, 2009.
[8] J.A. Rossmanith. DoGPack software, 2013. Available from http://www.dogpack-code.org.
[9] A.J. Christlieb, R. Krasny, J.P. Verboncoeur, J.W. Emhoff, and Iain D. Boyd. Grid-free plasma simulation techniques.
Plasma Science, IEEE Transactions on, 34(2):149–165, 2006.
[10] R.W. Hockney and J.W. Eastwood. Computer simulation using particles. Institute of Physics Publishing, 1988.
[11] C.K. Birdsall and A.B. Langdon. Plasma physics via computer simulation. Taylor & Francis Group, 1985.
[12] G.B. Jacobs and J.S. Hesthaven. High-order nodal discontinuous Galerkin particle-in-cell method on unstructured grids.
J. Comp. Phys., 96–121(214), 2006.
[13] F. Filbet and E. Sonnendrücker. Comparison of Eulerian Vlasov solvers. Comp. Phys. Comm., 150:247–266, 2003.
[14] R. E. Heath, I. M. Gamba, P. J. Morrison, and C. Michler. A discontinuous Galerkin method for the Vlasov-Poisson
system. J. Comput. Phys., 231(4):1140–1174, 2012.
[15] J.W. Banks and J.A.F. Hittinger. A new class of nonlinear finite-volume methods for Vlasov simulation. IEEE Transactions
on Plasma Science, 38:2198–2207, 2010.
[16] G.J. Parker and W.N.G. Hitchon. Convected scheme simulations of the electron distribution function in a positive column
plasma. Jpn. J. Appl. Phys., 36:4799–4807, 1997.
[17] N. Besse, J. Segre, and E. Sonnendrücker. Semi-Lagrangian schemes for the two-dimensional Vlasov-Poisson system on
unstructured meshes. Transp. Theory and Stat. Phys., 34:311–332, 2005.
[18] N. Crouseilles, T. Respaud, and E. Sonnendrücker. A forward semi-Lagrangian method for the numerical solution of the
Vlasov equation. Comp. Phys. Comm., 180:1730–1745, 2009.
[19] James A. Rossmanith and David C. Seal. A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin
scheme for the Vlasov-Poisson equations. J. Comput. Phys., 230(16):6203–6232, 2011.
[20] C. Cheng and G. Knorr. The integration of the Vlasov equation in configuration space. J. Comp. Phys., 22:330–351, 1976.
[21] J. Qiu and A.J. Christlieb. A conservative high order semi-Lagrangian method for the Vlasov equation. J. Comp. Phys.,
2010. Article in press.
[22] Y. Güçlü and W. N. G. Hitchon. A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows. J. Comput. Phys., 231(8):3289–3316, 2012.
[23] N. Obreschkoff. Neue Quadraturformeln. Abh. Preuss. Akad. Wiss. Math.-Nat. Kl., 1940(4):20, 1940.
[24] Erwin Fehlberg. Neue genauere Runge-Kutta-Formeln für Differentialgleichungen n-ter Ordnung. Z. Angew. Math. Mech.,
40:449–455, 1960.
[25] Robert P. K. Chan and Angela Y. J. Tsai. On explicit two-derivative Runge-Kutta methods. Numer. Algorithms, 53(23):171–194, 2010.
[26] Xu-Dong Liu, Stanley Osher, and Tony Chan. Weighted essentially non-oscillatory schemes. J. Comput. Phys., 115(1):200–
212, 1994.
[27] Guang-Shan Jiang and Chi-Wang Shu. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126(1):202–
228, 1996.
[28] Chi-Wang Shu. High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev.,
51(1):82–126, 2009.
[29] W.H. Reed and T.R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479,
Los Alamos Scientific Laboratory, 1973.
[30] Bernardo Cockburn and Chi-Wang Shu. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems.
J. Sci. Comput., 16(3):173–261, 2001.
[31] J.S. Hesthaven and T. Warburton. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications.
Springer, 2007.
[32] E. Hairer and G. Wanner. Multistep-multistage-multiderivative methods of ordinary differential equations. Computing
(Arch. Elektron. Rechnen), 11(3):287–303, 1973.
[33] J. D. Lambert. Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley, 1991.
[34] David F. Griffiths and Desmond J. Higham. Numerical Methods for Ordinary Differential Equations. Springer-Verlag,
2010.
[35] Ernst Hairer, Syvert P. Nørsett, and Gerhard Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems
(Springer Series in Computational Mathematics) (v. 1). Springer, 3rd edition, December 2009.
[36] J. C. Butcher. General linear methods. Acta Numer., 15:157–256, 2006.
[37] Matthew F. Causley and Andrew J. Christlieb. Higher order A-stable schemes for the wave equation using a successive
convolution approach. SIAM J. Numer. Anal., 52(1):220–235, 2014.