STATEMENT OF RESEARCH

DAVID C. SEAL

1. Introduction

I work in numerical analysis and scientific computing. My research is interdisciplinary, and it includes aninteresting blend of elements from mathematics, computer science, physics, and chemistry. My primary focusis on fundamental algorithm development for scientific problems. I seek to define methods that makethe best use of modern computer architecture, which is often necessary to make problems tractable, especiallyif they involve dynamics that act on several scales. I am interested in multi-physics scientific models thatinvolve multi-scale phenomenon. An understanding of model derivations including first principles is essentialto create useful numerical methods. One example of a numerical challenge is that numerical methods shouldsatisfy 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 numericalsolutions with fidelity. For example, it is well known that discarding a discrete divergence criteria for themagnetohydrodynamics equations of plasma physics leads to unstable solutions. Given the interdisciplinarynature of my subject area, knowledge about current challenges and trends in computer science as well asan aptitude for software engineering is necessary to develop, implement, and demonstrate the efficiency ofthe 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 numerousphysical applications including astrophysics, gas dynamics, tsunami modeling, various geophysical applica-tions, 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 timeintegrators for PDEs that are required to make scientific problems tractable. Briefly put, I have workedon the following topics:

• Semi-Lagrangian methods for plasma physics. My thesis work involved tackling some ofthe fundamental challenges associated with kinetic plasma simulations, where I developed a semi-Lagrangian 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 singlesimulation. For example, on a high resolution grid, a classical DG solution required three days torun 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).

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2 DAVID C. SEAL

• Multiderivative time integrators. Numerical methods for PDEs are typically cast in eithera method of lines (MOL) or a Lax-Wendroff framework. Multiderivative methods unify these twoframeworks. In [2], I extended the ODE results to hyperbolic PDEs with the finite difference weightedessentially non-oscillatory (WENO) and DG methods.

• Modified flux methods. High-order multiderivative methods can be viewed as a method ofmodified fluxes. After working on them, I realized there was room for improvement over classicalfinite 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 basedon discretizing time-averaged fluxes in place of the traditional “frozen in time” approximations thatare typically used. The contribution of this work includes:

– A family of WENO methods can be constructed from the Picard integral formulation, that gobeyond the method of lines formulation. In [4], I describe Runge-Kutta and Taylor methods.

– The Picard integral formulation defines methods that reduced computational complexity, includ-ing 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 in-compressible Euler equations [5]. Our method only requires a single right hand side evaluation pertime step, and therefore is amenable to adaptive mesh refinement.

• Exponential integrators. Implicit multiderivative methods and exponential integrators yieldpromising methods for multi-scale phase-field models that I have been investigating. As a post-doc, I served as a co-mentor to Jaylan Jones (PhD, 2013), whose thesis focused on investigatingtime-stepping methods for solving variations of the Allen-Cahn and Cahn-Hilliard equations [6]. Mycontribution included working with Dr. Jones on the application of time integrators to the recentlydefined Functionalized Cahn-Hilliard equation [7], which made up a large portion of his thesis andis an ongoing topic of research.

An overarching goal with my research is to foster collaborations between mathematicians, sci-entists, and engineers by making state of the art numerical methods accessible. Beginning with mythesis work, I have become one of the primary developers of the software package DoGPack [8], which is anopen-source code for solving hyperbolic problems using the discontinuous Galerkin method on structuredand unstructured grids in multiple dimensions. During my time at Michigan state, I have developed anopen-source software package called FINite difference ESSentially non-oscillatory methods (FINESS) forthe 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 ofmatter where electrons have dissociated from their nucleons. The mathematical models describing plasmacan be broadly categorized as either kinetic or fluid descriptions. In a kinetic model, one looks to modela phase-space probability distribution function of the plasma that describes the probability of finding aparticle at a given space and velocity coordinate for all time values. Kinetic descriptions of plasma definea set of PDEs in 6+1 dimensions (3 spatial dimensions, 3 velocity dimensions, and time). Fluid modelscan be derived by taking moments in velocity space of this distribution function. These models are lowerdimensional and appropriate for plasmas near thermodynamic equilibrium. Thus far, my work has focusedon 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 · ∇xf + a · ∇vf = C(f),

where f(t,x,v) is the distribution function of some plasma species. The acceleration term a comes from theLorentz force, and the collision operator C(f) describes physical collisions between particles. For Vlasov-Poisson, the Lorentz force comes solely from the electric field: a = qs/ms E, where qs is the charge, andms is the mass of the species s under consideration, whereas for Vlasov-Maxwell, a magnetic field is alsopresent: a = qs/ms (E + v ×B). Typical numerical techniques for solving kinetic descriptions of plasmacan be classified into three categories: particle methods [9], particle-in-cell methods [10, 11, 12], and fullymesh-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 anunstructured grid. This part is solved with the classical RKDG method. Each quadra-ture 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 solvedwith the SLDG method. Conservation (and stability) is achieved through exact inte-gration of the characteristics, much like the corner-transport method.

2.2. Semi-Lagrangian discontinuous Galerkin (SLDG). A fully mesh-based description of the plasmaworks with a grid based discretization of both the plasma as well as the electromagnetic fields. The mostobvious disadvantage of such an approach is that because ‖v‖ ∈ 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 anunderlying grid representation, but single time steps are taken in a Lagrangian fashion. After evolving thesolution for a short period, the solution is then projected back onto the mesh. Semi-Lagrangian methods forVlasov equations have a long history dating back to at least 1976 [20], and have only recently been pushedto very high-order [21, 22].

The basic idea of my proposed hybrid scheme is to apply Strang splitting to the collisionless version ofEqn.(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 · ∇xf = 0 for a time step of length ∆t/2.2. Solve Poisson’s equation −∇2φ = ρ− ρ0 to obtain the electric field E = (E1, E2) = −∇φ.3. Accelerate the particles by solving f,t − E1 f,vx − E2f,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 onthe 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 thereforea 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 atquadrature points and coefficients of basis functions in a modal representation. Therefore, one can projectonto half of the quadrature points to define a single subproblem, evolve the solution, and then integrateback 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 futuretimes. Oftentimes the spatial variable is first discretized, and the time variable is left as a continuousvariable. This process is called the method of lines (MOL) formulation, and it defines a large system ofordinary differential equations (ODEs), to which an appropriate ODE time integrator can be applied. In

4 DAVID C. SEAL

TaylorRunge−Kutta

Multistage

multiderivative

Multistep multistage multiderivative

Figure 3. A simple taxonomy of multiderivative time integrators [2]. The largest class is thatas 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 typeof 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 bound-aries on new computing environments. It was reinvigorated by discovering methods that match the modernparadigm shift in computer architecture: develop parallelizable algorithms that make use of floating pointoperations in place of memory storage and reduce communication overhead. I discovered that multideriva-tive methods have a long history of development for ODEs [23, 24, 25], yet they had been largely overlookedby the PDE community. My contribution was to demonstrate how to apply them to hyperbolic PDEs, andin particular how they can be applied to the finite difference WENO method [26, 27, 28], as well as thediscontinuous Galerkin (DG) method [29, 30, 31].

Broadly defined, multiderivative methods can be thought of as a tool that unifies traditional time steppingmethods 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 methodsdeemed “multistep-multistage-multiderivative” methods [32], which accommodate all Runge-Kutta (multi-stage), Adams (multistep) and Taylor (multiderivative) methods (c.f. Fig. 3). Multiderivative methods areconspicuously missing from many popular textbooks on numerical methods for ODEs [33, 34], but not allof them [35].

The important part that extends this class of methods beyond Runge-Kutta methods is the inclusion ofextra derivatives of the right hand side function. Taylor methods already do this, but they are restrictedin the choice of coefficients for the Taylor expansion. To illustrate the idea, we consider 1D hyperbolicconservation 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

(3)qt = −f(q)x,

qtt = −f(q)tx = − (f ′(q) · qt)x = (f ′(q) · f(q)x)x .

For an ODE, the higher derivatives of q are unique, but PDEs require a definition for the spatial discretiza-tion, 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

(4)

q∗ = qn − ∆t

2f(qn)x +

(∆t/2)2

2(f ′(qn) · f(qn)x)x ,

qn+1 = qn −∆tf(qn)x +∆t2

2

(1

3(f ′(qn) · f(qn)x)x +

2

3(f ′(q∗) · f(q∗)x)x ,

),

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 centraldifferences for the higher spatial derivatives that appear. For the DG method, I modified the flux functionin 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.

STATEMENT OF RESEARCH 5

4. Ongoing and future work

My future work will be rooted to my primary research interest: develop numerical algorithms for math-ematical models for the purposes of advancing scientific knowledge. Below I list some of my ongoing workand 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 beconducted. Additionally, I would like to incorporate my semi-Lagrangian solver into an embeddedboundary framework.

• Multiderivative methods. New methods create many problems to investigate. Two topics ofinterest to me include:

– SSP properties. I am currently working with Prof. Sigal Gottlieb (University of Mas-sachusetts, Dartmouth) and her student, Mr. Zachary Grant on defining strong stability pre-serving (SSP) properties for multistage, multiderivative methods. This property has only beendefined for linear multistep and Runge-Kutta methods.

– Optimized methods. Stability analyses for multiderivative methods requires investigatingthe fully discrete system. The reason this is the case is that higher derivatives are computedusing higher spatial derivatives rather than differentiating the MOL formulation. I am currentlylooking into optimizing CFL numbers for explicit multiderivative methods for the discontinuousGalerkin 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 magnetohydro-dynamics. 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 studyingturbulence 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 throughthe application of a post-processing filter, or through leveraging low-order solves. The advantagesthese 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 thatcan be broadly construed as defect correction algorithms:

– Superconvergence extraction. Prof. Jennifer Ryan (University of East Anglia) and I areworking on the development of SIAC filters to the semi-Lagrangian method from my thesis. Theresult will reduce the memory overhead that would be required to obtain high-order accuracyfor a discontinuous Galerkin method, which can be a bottleneck for such a scheme.

– Spectral (Integral) deferred correction. Dr. Cory Hauck (Oak Ridge National Laboratoryand University of Tennessee) and I are working on the numerical methods for neutron transportproblems. The specific method we are using is based on a clever splitting of the problem intocollided and uncollided particles, which reduces the overhead for implicit solves.

• Method of lines transpose. The method of lines transpose [37] was originally developed as animplicit, embedded boundary Maxwell solver by Profs. Mathew Causley and Andrew Christlieb. Itcan be viewed as an arbitrarily high-order and L-stable extension of Rothe’s method, that is basedon similar space-time methods as what I have been working on. I am currently working with thedevelopers and Ms. Hana Cho, who is a graduate student of Prof. Christlieb, on extensions toparabolic problems. These include complicated boundary conditions for the heat equation, as wellas parabolic problems such as Allen-Cahn, Cahn-Hilliard as well as the functionalized Cahn-Hilliardequation.

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 Guclu, and Andrew J. Christlieb. High-order multiderivative time integrators for hyperbolic con-servation 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 Guclu, and Andrew J. Christlieb. The picard integral formulation of weighted essentially non-oscillatory 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

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