My research area is scientific computing. My main research areas are in
(1) partial differential equations (PDEs) on surfaces,
(2) the time evolution for PDEs and
(3) interfacial dynamics.
I also carry out some research in the area of sonoluminescence.
PDEs on Surfaces.
Partial differential equation (PDE) models for continuous processes arise throughout the applied and natural
sciences. Because analytical solutions are rarely possible, the practical importance of accurate and
efficient algorithms for approximating these equations is paramount. There has been a great
effort made to develop numerical methods for many important classes of PDEs in one, two or three spatial
dimensions. However, a remarkable variety of problems involve differential equations on surfaces,
including the physics-based modeling of computer animated objects, the mapping of
cortical change in Alzheimer's disease, brain development, and schizophrenia, the study
of pathologies in bone such as osteoporosis and bone metastasis and many other areas.
My approach to PDEs on surfaces - the Closest Point Method - is unique, in that it emphasizes methods
that (1) apply to very general PDEs, (2) apply to general surfaces, (3) are high-order accurate and efficient,
and (4) make use of existing 3D techniques and software to solve surface processes.
Ongoing work investigates methods for solving PDEs on surfaces, as well as related applications.
See Publications: PDEs on Surfaces
for some of the topics I've been recently investigating.
Time Evolution for PDEs.
Numerical methods for time-dependent PDEs are commonly derived using the method of
lines. With this technique, some spatial discretization is carried out to yield a large system
of ordinary differential equations (ODEs). The corresponding ODE system is solved by an
appropriate time-stepping method.
A method of lines discretization of a time-dependent partial differential equation may lead
to terms of different types. For example, some terms may be "stiff" (e.g., a linear diffusion
term) and require the use of an implicit time-stepping scheme (since otherwise excessively
small time steps would be needed). Other terms, however, may have mild time-stepping
restrictions when treated explicitly and may be cheaper to treat in an explicit fashion (e.g.,
a nonlinear convection term).
In other situations, we wish to maintain the positivity of certain quantities to ensure a physically reasonable result (for
example, a biological population or a chemical concentration cannot be negative), or we
may wish to avoid spurious oscillations (examples arise in many disciplines, including the
evolution of fluid flows with shock formation and the solution of option pricing models).
Strong-stability-preserving (SSP) time discretization methods have a nonlinear stability
property that makes them particularly suitable for such problems.
Alternatively, we may find that the relative merits of schemes vary according to
the spatial location. In such situations, it becomes particularly desirable to change the time-stepping
scheme according to spatial position, while still maintaining key properties such as conservation.
Ongoing work investigates the development and analysis of methods for solving PDEs in these various situations.
See Publications: Time Evolution for PDEs for further details.
Methods for Interfacial Dynamics.
There are many natural and industrial problems in which fairly sharp interfaces form and
propagate. Familiar examples include the growth of crystalline materials, the motion of
ocean waves, the processing and enhancement of images and the melting of ice in water.
Computer simulation of these processes often requires efficient numerical methods for moving
the interface with a speed that depends on the surface geometry and external physics.
Designing suitable algorithms is complicated by the fact that in many problems the interfaces
can merge or break up, form triple point junctions or more complicated interface
networks. In our work we develop and analyze methods for evolving surfaces, with particular focus on
high order motions and motions for surfaces that involve junctions.
See Publications: Interfacial Dynamics for further details.
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