**Date & Time:** Friday, 25th Feb, 2010 16:00-17:00.

**Venue:** Ramanujan Hall, Mathematics Department

**Title:** Shock solution for general nonlinear wave equation

**Speaker:** Dr. Francois Coulouvrat, University of Paris

**Abstract:**
Many physical phenomena are concerned with the propagation of weak
nonlinear waves that can be modeled under the form of a generalized
Burgers equation. Physical examples include nonlinearities that can be
either quadratic (nonlinear acoustical waves in fluids or longitudinal
waves in solids), cubic (nonlinear shear waves in isotropic soft
solids), or non-polynomial (Buckley-Leverett equation for diphasic
fluids, models for car traffic, Hertz contact in granular media). A
new weak shock formulation of the generalized Burgers equation using
an intermediate variable called "potential" is proposed. This
formulation is a generalization to non-quadratic nonlinearities of the
method originally proposed by Burgers himself in 1954 for his own
equation, and later applied to sonic boom applications by Hayes et al.
(1969). It is an elegant way to locate the position of a shock. Its
numerical implementation is almost exact, except for an interpolation
of Poisson's solution that can be performed at any order. It is also
numerically efficient. As it is exact, a single iteration is
sufficient to propagate the information at any distance. It
automatically manages waveform distortion, formation of shock waves,
and shock wave evolution and merging. The theoretical formulation and
the principle of the algorithm are detailed and illustrated by various
examples of applications.