In this talk we will consider the propagation of finite-amplitude acoustic waves in thermoviscous fluids that also exhibit thermal relaxation. Under the assumption that the thermal flux vector is described by the Maxwell-Cattaneo law, which is a well known generalization of Fourier's law that includes the effects of thermal inertia, we derive the weakly nonlinear equation of motion in terms of the acoustic potential. We then use singular surface theory to determine how an input signal in the form of a shock wave evolves over time, and for different values of the Mach number. Then, numerical methods are used to illustrate the analytical findings. In particular, it is shown that the shock amplitude exhibits a transcritical bifurcation; that a stable, nonzero equilibrium solution is possible; and that a Taylor shock (i.e., a diffusive soliton), in the form of a "tanh" profile, can emerge from the input shock wave. Finally, an application related to the kinematic-wave theory of traffic flow is noted. (Work supported by ONR/NRL funding.)
My title, as well as much of my talk, is paraphrased from a terrific
article by Andrew Granville (AMS Bulletin, vol. 42, 2004) that I have
been studying with Verity Crosswell, a 4th Year project student. It
refers to a recently discovered deterministic, polynomial-time test
for primality invented by Manindra Agrawal and two of his final-year
project students Neeraj Kayal and Nitin Saxena (AKS). I'll begin the
talk by discussing primality testing in general, then introduce the
main theorem behind the AKS method and discuss some delightful
algorithms that Verity and I have learned while implementing the new
test.
We propose Gauss-Seidel (GS)-type preconditioning
for solving linear systems with nonsymmetric
matrices. This preconditioning is applicable
for conventional iterative methods, e.g., BiCGStab,
GMRES(k) and IDR(s) methods and so on.
Moreover this preconditioning does not need
additional computation of incomplete-matrix-vector
multiplication as incomplete LU (ILU) factorization.
We evaluate the convergence properties of this new
preconditioning scheme through many numerical experiments.
In this presentation we outline a new, robust and efficient method for the
solution of the Boussinesq approximation of the Navier-Stokes equations. This
system of partial differential equations represents a model of thermally-buoyed
flows. A general solution strategy for a transient problem has two main
components: a predictor-corrector adaptive time stepping algorithm (with
stabilized trapezoid rule as a corrector and an explicit second-order
Adams-Bashforth method for error control), and a robust AMG-preconditioned
Krylov solver for the discrete spatial problems. The method is tested on a range
of benchmark and realistic problems in 2D and 3D (including laterally heated
cavities and the Rayleigh-Benard convection). This is a joint work with H. Elman
(University of Maryland) and D. Silvester (University of Manchester).