Research Statement
Terrance Pendleton
My research interests lie primarily in the field of applied nonlinear partial differential equations
(PDEs). It is dual-fold in the sense that I do work in both the fields of applied analysis and numerical
analysis. In particular, my work focuses on establishing analytical and numerical techniques on
classes of nonlinear hyperbolic evolutionary PDEs derived in the context of shallow water wave
theory.
Over the last several years, I have developed highly robust, accurate numerical schemes to prove
the existence and uniqueness of nonclassical solutions associated with these considered equations. I
have also used these same schemes to simulate the associated solutions in order to garner a deeper
understanding of the underlying dynamics associated with their evolution. By nonclassical, we
mean solutions that have some form of sharpness or non–smoothness which for us translates mathematically to a discontinuity in some derivative associated with the solution. Due to the melange of
applications that exhibit similar behavior, it is highly desirable to maximize the understanding of
these solutions. For instance, beyond the context of shallow water wave theory, in one dimension,
these equations coincide with the averaged template matching equation (ATME) for computer vision. In two dimensions, one could also use the equations to quantify growth and other changes in
shape, such as occurs in a beating heart, by providing the transformative mathematical path between the two shapes. In what follows, I provide a brief summary of my previous research results,
ongoing research and a tentative plan for future research endeavors, as they relate to potential
undergraduate research.
1. Global weak solutions to the b-family of evolutionary PDEs.
Some of my earliest research work associated with my dissertation was concerned with establishing global weak solutions for the following family of evolutionary PDEs characterized by a
bifurcation parameter, b:
mt + (um)x + (b − 1) mux = 0,
u = G ∗ m,
x ∈ R, t > 0,
(1)
where b > 1 and is subjected to a prescribed initial condition. Here, the momentum m and
velocity u are functions of a time variable t and spatial variable x, and G(x) is the Green’s
kernel. These equations arise in diverse scientific applications and enjoy several remarkable
properties both in the 1-D and multidimensional cases. To obtain our results, we applied a
particle method to the considered equations. This can be accomplished by assuming a weak
solution mN (x, t) in the form of a linear combination of N Dirac delta distributions–each having
an associated weight and location. Below, we present our main result in which we proved the
existence of a unique global weak solution in some special cases and obtained stronger regularity
properties of the solution than previously established. For a more detailed explanation, please
refer to [1] on my curriculum vitae.
Theorem 1. Suppose that (mN (x, t), uN (x, t)) is a particle solution of the system of ODEs
∗
generated by applying a particle method to (1) with the initial approximation mN (·, 0) * m0 ,
m0 ∈ M+ (R). Then there exist functions u(x, t) ∈ BV(R × R+ ) and m(x, t) ∈ M+ (R × R+ )
such that mN (x, t) and uN (x, t) converge to m(x, t) and u(x, t), respectively in the sense of
distributions as N → ∞. Furthermore, the limit (u, m) is the unique weak solution of (1), for
any b > 1 with the regularity u ∈ C(0, T ; H 1 (R)), ux ∈ BV(R × R+ ).
Using the following steps, we were able to prove 1.
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Terrance Pendleton
Research Statement
• We first derived a particle method for the b-family of fluid transport equations.
By considering a weak formulation of equation (1), we derived a system of ODEs to evolve
the particles’ locations and weights.
• We then showed that the particle solution is a weak solution to the b-family of
transport equations. Here we define a weak solution of equation (1) by using integration
by parts against a test function φ ∈ C0∞ (R × R+ ).
• Next, we established the convergence of the particle solution. Here, we showed
that uN (x, t) and uN
x (x, t), which can be computed explicitly via the particle method, are
functions of bounded variation. We then applied an associated compactness result to
show the existence of the limit.
• Finally, we showed that the limit is a unique global weak solution to the b-family
of transport equations. Here we verified that the limit obtained from the previous step
satisfies our definition of a weak solution to equation (1). Uniqueness can be proven using
an extension from previously established results.
2. Elastic collisions among peakon solutions to the Camassa-Holm equation
For a particular choice of G (the Green’s function associated with the one dimensional helmholtz
operator) and b = 2, equation (1) reduces to the Camassa-Holm (CH) equation. The CH
equation admits solutions that are nonlinear superpositions of traveling waves which have a
discontinuity in the first derivative at their peaks and therefore are called peakons. This behavior
gives rise to several diverse scientific applications, but introduces difficult numerical challenges.
To accurately capture these solutions, we apply the particle method to the CH equation and
demonstrate that the particle method outperforms finite volume methods in simulating these
solutions.
Two Peakons with α=1 at time=1
PM
FV
1.5
PM
FV
1.5
1
1
0.5
0.5
−10
Two Peakons with α=1 at time=3
0
10
−10
0
10
Two Peakons with α=1 at time=6
Two Peakons with α=1 at time=9
1.5
1.5
1
1
0.5
0.5
−10
0
10
−10
0
10
Figure 1: A comparsion of the propagation of peakons for both a particle method and finite volume
method
In [3], we show analytically that any collisions for the case where the initial weights are assumed
to be positive are elastic. That is, the collision is through the interaction potential in the
Hamiltonian associated with the particle solution rather than a head on collision. We then
perform several numerical simulations, which solve the CH equation under a wide range of
initial data with α = 1. In particular, we focus on peakon solutions associated with the CH
equation to showcase the merits of using a particle method. Furthermore, we provide numerical
evidence to support the claim that the shape of the peakon interaction is exchanged rather than
maintained.
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Research Statement
Terrance Pendleton
3. Tsunami Waves Simulation using a Two-Component Camassa Holm System
Adding a component for density, ρ, to the CH equation yields a two-component generalization of
the CH equation (2CH equation)–which is a physically relevant model in the context of shallow
water wave theory–and is given by:
mt + umx + 2mux = −gρρx ,
ρt + (ρu)x = 0,
(2)
which can be described in the context of shallow water waves. Here, ρ is the total depth of the
water column.
There have been many attempts to create accurate models and corresponding numerical methods
for simulating the propagation of tsunami waves. One popular model in shallow water wave
theory is the classical Saint-Venant system, which reasonably approximates the behavior of real
ocean waves and is a depth-averaged system that can be derived from the famed Navier-Stokes
equations. The Saint-Venant system is a very good simplification for lakes, rivers, and coastal
areas in which the typical time and space scales of interest are relatively short. Because the
Saint-Venant system is quite difficult to solve, it is typically simplified in a number of ways.
Tsunami waves form in deep water and travel very long distances (thousands of kilometers)
before coming to shore. Over long time, solutions of the Saint-Venant system break down,
dissipate in an unphysical manner, develop shock waves, and fail to capture the small, trailing
waves that are seen in nature and laboratory experiments. Thus, it is necessary to use a more
sophisticated model in order to preserve the wave characteristics over long time simulations.
We are interested in studying the ability for the 2CH equation to model long-time propagation
and on-shore arrival of the tsunami-type waves. We consider the situation for which a steep
ridge on the bottom of the water body breaks off and causes a submarine landslide. In this case,
the landslide creates surface waves which propagates in two directions. In one direction, this
yields a “tsunami-like” wave. We are also interested in examining the role that α plays in the
long-time propagation of water waves. More is said about this in [4] on my curriculum vitae.
We also seek to build numerically efficient and stable algorithms for simulating long term tsunami
wave propagation using the 2CH equation given by equation (2). In particular, we consider three
different numerical approaches given as follows:
• Finite Volume Method Here we use a semi-discrete central upwind scheme by considering
equation (2) as a system of conservation laws.
• Particle Method Here we solve the system via a particle method by considering a weak
solution in the form of a linear combination of Dirac delta distributions as was done for
equation (1).
• Finite Volume Particle Method Here we use the conservative form of density to solve ρ
via a finite volume method, while the momentum and velocity m, u are solved via a particle
method.
We also seek to provide a more comprehensive study on the qualitative changes in the solution
behavior given a change in the value of the length scale α.
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Terrance Pendleton
Research Statement
4. Energy Preserving Finite Difference Schemes for the CH Equation and the 2CH
Equation
Back in 1993, Camassa and Holm first observed that the CH equation is bi-Hamiltonian; that
is, the equation can be expressed in Hamiltonian form in two different ways. Recalling that
m = u − uxx (as m and u are related through the 1D helmholtz operator), the two compatible
Hamiltonian descriptions of the CH equations are given by
δH2
δH2
δH1
= −∂x 1 − α2 ∂x2
= −∂x
,
δm
δm
δu
with the following conserved quantities:
Z
Z
Z
1
1
2
2
u + ux dx and H2 =
u3 + uu2x dx.
H0 =
m(x, t) dx, H1 =
2 R
2 R
R
mt = − (m∂x + ∂x m)
(3)
(4)
Because of this unique property, the CH equation possesses an infinite number of conservation
laws. In this area, we proposed a new energy conserving finite difference scheme (ECFD) for
simulating solutions associated with the CH equation that conserves H0 and H1 and showed the
advantages of using such a scheme for the long time propagation of peakon solutions. In particular, we showed that the proposed numerical schemes numerically produced wave solutions with a
smaller phase error over a long time period than those generated by other conventional methods.
Equipped with such a scheme for the CH equation, we were able to extend our numerical scheme
to simulate solutions to the 2CH equation which preserves the following invariants:
H0 =
Z
m dx,
R
1
H1 =
2
Z
2
u +
u2x
2
+ gρ
dx
R
and H2 =
Z
ρ dx.
(5)
R
The results in [4] suggest that the preservation of these invariants may be essential to producing
numerical solutions which exhibit properties conducive to satisfying long time behavior, thus
revealing the desired dispersal pattern.
5. Future Goals And Undergraduate Research Plans
• Euler-Poincare Equation Up until now, my research has focused on the one-dimensional
b-family of evolutionary PDEs. However for b = 2, there is a two dimensional analogue
called the Euler-Poincare (EPDiff) equation associated with the diffeomorphism group. In
the d-dimensional case on Rd , this equation is given by:
∂m
+ u · ∇m + ∇uT · m + m(div u) = 0.
(6)
∂t
Here the momentum m and velocity u are vector functions of the time variable t and the
d spatial variables x = (x1 , . . . xd ) and are related by a second-order Helmhotz operator.
Some of my immediate future research goals involve establishing similar convergence results
for a particle method applied to (6) and studying relevant applications of the equations.
• Modified Camassa-Holm Equation The CH equation has a modified version which
incorporates a cubic nonlinearity. In particular, the modified Camassa-Holm equation
(MCH) equation is given as:
mt + ((u2 − u2x )m)x = 0,
(7)
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Research Statement
Terrance Pendleton
where the momentum m is related to the velocity u by m = u−uxx . The MCH equation has
recently been proposed as a new integrable system by applying the general method of triHamiltonian duality to the bi-Hamiltonian representation of the modified kDV (KortewegdeVries) equation. It was also shown that the MCH equation may be obtained from the 2-D
Euler equations. One of my immediate future research goals involves developing a highly
robust, efficient, and accurate numerical scheme to simulate solutions to the MCH equation
with the eventual goal of establishing the existence of a global weak solution for (7).
Beyond my immediate future goals, I aim to continue to develop highly robust, efficient, and accurate numerical scheme for a diverse class of PDEs with wide ranging applications in areas such
as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, and quantum mechanics.
While I was initially trained in particle methods and finite volume/ finite difference methods,
I have since extended my knowledge to include finite element methods such as discontinuous
Galerkin methods. This will allow me to widen the class of PDEs that I am able to study.
Several components of numerical methods for PDEs are accessible to undergraduate research
projects. Projects in this area will combine elements from multi-variable Calculus, and an
introductory course in numerical analysis. Additional work in Analysis and Partial Differential
Equations is ideal but unnecessary to begin work in this area. Many of the topics that my work
focuses on is open-ended and thus highly adaptable. For instance, a possible suitable project
for an undergraduate student includes the analysis of solitons–a special class of solutions to
time-dependent nonlinear PDEs. Over time, even under natural disturbances and noise, these
solutions preserve their shape and speed of travel. Students working in this area can study
various solutions to soliton–generating PDEs such as the celebrated Korteweg-deVries equation.
In particular, students can focus on such dynamics as a multiple-soliton interaction, or collisions
between solitons and other objects. While my work focuses on the shallow water wave regime,
one may use similar techniques in PDE modeling in the realm of population dynamics or traffic
flow.
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