RESEARCH STATEMENT
My
interests lie in the study of nonlinear dynamical systems which exhibit wave
phenomena. G.B. Whitham once stated, "Almost any field of science or
engineering involves some questions of wave motion," [19] and studies of the
equations that model these phenomena are essential for answering many of these
questions. Due to the complex nature of these systems, it is often necessary to
employ techniques from various disciplines, such as numerical analysis,
approximation theory, operator theory and dynamical systems theory, in order to
understand their behavior. Using these techniques, I have focused my efforts on
problems in two particular fields, namely multi-symplectic integration and
traveling wave solutions.
Multi-Symplectic Integration. From a dynamical systems point of view, one is able
to understand the behavior of numerical methods and develop schemes, called
geometric integrators, which preserve the structure of the original equations
[7]. In particular, it is often advantageous to preserve the symplectic
structure of a Hamiltonian system when solving numerically. Such methods have
been shown to yield accurate and efficient long term simulation of the
dynamical system. These methods are also well understood through backward error
analysis, a tool which uses asymptotic expansions to derive a modified
differential equation, whose exact solution represents the numerical solution,
to study the behavior of the method. Due to the success of symplectic methods
for ODEs, there is a recent and growing interest in multi-symplectic
integrators, methods that preserve the symplectic structure in both space and
time, for Hamiltonian PDEs, and there are many open problems in this field.
In
joint work with S. Reich, we studied finite difference methods [14, 15]
that preserve a multi-symplectic conservation law for a general class of
nonlinear Hamiltonian systems known as multi-symplectic PDEs [2]. We have shown
that these schemes can be derived from a discrete variational principle, making
our analysis comparable to previous research in this field [11]. In addition,
we showed that these methods exactly preserve semi-discrete, local, energy and
momentum conservation laws that are inherited from the original PDE, noting
that local conservation is a stronger result than the global results of
previous studies [12].
Since
energy and momentum conservation laws are generally not exactly preserved by
these schemes, we have also developed a modified equations analysis, in
order to better understand the effect of discretization on these conservation
laws [14, 15]. This is done by deriving a modified PDE that is solved by the
numerical solution to higher order accuracy, and we have shown that the
modified equations for these methods are always multi-symplectic, meaning the
method solves a nearby multi-symplectic PDE to high order accuracy and thereby
reproduces some of the qualitative solution behavior of the original PDE. Since
the modified equations are multi-symplectic they also have local energy and
momentum conservation laws, showing that the numerical schemes nearly preserve
these local conservation laws and yielding explanation about the excellent long
term performance of the schemes. We have also demonstrated these results
numerically by showing that the modified conservation law is preserved to
higher order accuracy by the numerical solution.
Furthermore,
we have considered traveling wave solutions for multi-symplectic PDEs in
order to demonstrate error propagation for these schemes [15]. Initializing a
scheme with the exact traveling wave solution of the PDE yields an error of the
same order as that of the scheme. However, we have shown in the specific case
of the sine-Gordon equation that initializing the scheme with a traveling wave
solution of the modified equation yields a numerical solution of higher order
accuracy, providing confirmation that the modified equations do, in fact,
represent the numerical method.
Working
with J. Frank andS. Reich, we have
performed an in-depth study of dispersion relations for numerical
methods that satisfy a multi-symplectic conservation law, giving a complete
description of the numerical solution behavior [6]. The focus of this work is
to perform an in-depth and thorough comparison of these methods and show that
not all methods of this type should be labeled multi-symplectic. Using
numerical dispersion relations we are able to show that some schemes yield
spurious modes, and prove that for certain parameter values one mode is
unstable while the other is diffusive, contrary to what should be expected of a
multi-symplectic integrator. For these methods, the analog for a spatial
discretization of spurious modes that arise from a time discretization is
having dispersion relations that are non-monotonic.Hence, this aspect of the numerical
dispersion relations must also be considered, and we have shown that certain
methods in this class have non-monotonic dispersion relations.In effect, we have presented a class of
methods, known as Runge-Kutta collocation methods [17], which are truly
multi-symplectic because they satisfy a multi-symplectic conservation law, they
have no spurious modes, and the dispersion relations are monotonic.
My five year plan for future research in this field
includes development of multi-symplectic splitting methods [7] for Hamiltonian
PDEs with added dissipation, an idea first introduced in my thesis [13].By solving the conservative and dissipative
parts of the equations separately with structure-preserving methods, the flow
maps are composed to obtain a numerical solution, which preserves some of the
qualitative solution behavior.In
addition, I plan to combine the ideas of multi-symplectic integration with the
structure offered by traveling wave equations (also introduced in [13]).One intent of this endeavor is to further
develop a backward error analysis, and to derive results concerning long-time
behavior of the numerical scheme, leading to a better understanding of the
numerical methods as well as the dynamics of conservative differential-difference
equations.
Traveling Wave Solutions. I am also currently working on various problems
concerning differential-difference equations. The focus of a current project
with A.R. Humphries and E.S. Van Vleck is to find traveling wave solutions for
a spatially discrete Nagumo equation with inhomogeneous diffusion [4, 16], an idealized system
used to understand traveling fronts through myelinated nerves with a
deteriorated region, characteristic of diseases that affect the nervous system.
We consider the problem on an infinite spatial domain, and using a piecewise
linear approximation of the nonlinearity, we are able to obtain an analytic
solution by way of Fourier transform. This has proved to be a particularly
difficult problem because these waves are not translational invariant and the
unknown wave speeds depend on the lattice and on time. Yet, once we have an
analytic expression for the solution, we are able to determine wave speeds for
which this expression satisfies a necessary set of conditions needed to ensure
that it is, in fact, a solution of the differential equation. Using Jacobi
operator theory, we are also able to determine necessary and sufficient
conditions, for which traveling waves exhibit propagation failure, based on the
amount of deterioration [10].
For
the spatially discrete Nagumo equation with homogeneous diffusion there are
analytic results concerning propagation failure of traveling waves for
the problem with a piecewise linear approximation to the nonlinearity [3], and
there are numerical results that suggest propagation failure for the nonlinear
problem [9], depending on known parameters. However, there are currently no
analytic results in this respect for the fully nonlinear problem, and this is
the focus of another current project with A.R. Humphries [8]. For small
parameter values, the system with zero wave speed becomes a symplectic
discretization of a Hamiltonian ODE, for which the modified equations that
represent the discretization are easily derived. Showing that the modified equations
are an exact representation of the discretization and proving there are
heteroclinic connections for this modified system would provide sufficient
explanation of propagation failure for the original problem.
My five year plan for future research in this
field is threefold.First, in collaboration with C.E.
Elmer, I intend to consider traveling waves of a spatially discrete Nagumo
equation coupled to a system of harmonic oscillators as a more realistic model
of crystal growth.In collaboration with
A.R. Humphries, the second and third projects concern the use of new numerical
solvers [1] for a nonlinear spatially discrete FitzHugh-Nagumo equation, and
the electromagnetic two-body problem.
Our main intent is to gain insight into the dynamics in order to aid in
future analysis of these problems.With
regard to the former, the oscillatory nature in the tails of traveling
waves [5] makes it difficult to find the best boundary conditions for numerical
simulation, but we have begun to develop a remedy.Concerning the later, I have begun to develop
a functional differential equation model for this problem, which was first
discussed by Wheeler and Feynman (see [18] and references therein).
[1] K.A. Abell, C.E. Elmer, A.R.
Humphries, and E.S. VanVleck, Computation of mixed type functional differential
boundary value problems, SIAM Journal on Applied Dynamical Systems, 4:745-771,
2005.
[2]
[3] J.W. Cahn, J. Mallet-Paret, and E.S. Van Vleck. Traveling
wave solutions for systems of ODEs on a two-dimensional spatial lattice.
[4] C.E. Elmer and E.S. Van Vleck. Traveling wave solutions for
bistable differential-difference equations with periodic diffusion.
[5] C.E. Elmer and E.S. Van Vleck. Spatially discrete
FitzHugh-Nagumo equations.
[6] J. Frank, B.E. Moore
and S. Reich, Linear PDEs and numerical methods that preserve a
multi-symplectic conservation law, SIAM Journal on Scientific Computing
28:260-277, 2006.
[7] E. Hairer, Ch. Lubich, and G. Wanner. Geometric Numerical
Integration: Structure Preserving Algorithms for Ordinary Differential
Equations.
[8] A.R. Humphries and B.E. Moore, Modified equations and
propagation failure of traveling waves, preprint.
[9] J.P. Keener. Propagation and its failure in coupled systems
of discrete excitable cells.
[10] T.J. Lewis and J.P. Keener. Wave-block in excitable media due
to regions of depressed excitability.
[11] J.E. Marsden, G.P. Patrick, and S. Shkoller. Multi-symplectic
geometry, variational integrators, and nonlinear PDEs. Communications in
Mathematical Physics, 199:351-395, 1999.
[12] R.I. McLachlan. Symplectic integration of Hamiltonian wave
equations. Numerische Mathematik, 66:465-492, 1994.
[13] B.E. Moore, A Modified
Equations Approach for Multi-Symplectic Integration Methods, Ph.D. Thesis,
Department of Mathematics and Statistics, University of Surrey, 2003.
[14] B.E. Moore and S. Reich. Backward error analysis for
multi-symplectic integration methods. Numerische Mathematik, 95:625-652,
2003.
[15] B.E. Moore and S. Reich. Multi-symplectic integration methods
for Hamiltonian PDEs. Future Generation Computer Systems, 19:395-402,
2003.
[16] B.E. Moore, E.S. VanVleck and A.R. Humphries, Waves for
bistable differential-difference equations with inhomogeneous diffusion,
preprint.
[17] S. Reich. Multi-symplectic Runge-Kutta collocation methods
for Hamiltonian wave equations. Journal of Computational Physics,
157:473-499, 2000.
[18] A. Schild. Electromagnetic two-body problem. Physical
Review, 131:2762-2766, 1963.
[19] G.B. Whitham. Linear and Nonlinear Waves.