The Elusive Time-Space
Corner Singularity: A Multiple-Scale Phenomenon
Dr. Natasha
Flyer
Dept. of Applied
Abstract:
It
is well known that solutions of elliptic partial differential equations (PDEs)
are usually singular (not smooth) in the corners of the spatial domain. These
'corner singularities' have observable consequences in science and engineering;
in buildings, cracks radiate from the corners of walls, windows and doors;
airplanes have rounded windows to minimize cracks that can cause catastrophic
failures; small nested vortices called "Moffatt
eddies" appear in the corners of fluid-filled regions. What has gone
essentially unnoticed is that solutions of virtually all time-dependent PDEs will exhibit singularities in the corners of the
time-space domain, where the initial conditions and boundary conditions meet.
The nature of these singularities can be elusive and their impact on numerical
calculations is severe. Without special corrections, the accuracy of high-order
methods will be reduced to that of a low-order scheme. Analytical and numerical
tools to analyze and remedy these time-space singularities will be discussed in
the cases of convective-diffusive and dispersive PDEs.
Time-space singularities are a multiple-scale phenomenon to which our corner
basis functions offer a novel approach.