# Preprint 2008-025

# An application of 3-D kinematical conservation laws: Propagation of a three dimensional wavefront

## K.R. Arun, M. Lukacova-Medvidova, S.V. Raghurama and Phoolan Prasad

**Abstract:**
3-D kinematical conservation laws (KCL) are equations of evolution
of a **propagating** surface
Ω_{t}
in three space dimensions and were first derived in 1995
by Giles, Prasad and Ravindran
assuming the motion of the surface to be isotropic.
We start with a brief introduction to 3-D KCL and mention
some properties relevant to this paper.
The 3-D KCL, a system of 6 conservation laws,
is an under-determined system to which we add
an energy transport equation for a small amplitude disturbance
to study the propagation of a three dimensional nonlinear wavefront
in a polytropic gas in a uniform state and at rest.
We call the enlarged system (3-D KCL and the energy transport equation)
equations of weakly nonlinear ray theory – WNLRT.
We highlight some interesting properties of the eigenvalues
of the equations of the WNLRT but main aim of this paper
is to test the numerical efficacy of this system
of 7 conservation laws.
We take initial shape of the front to be cylindrically symmetric
with a suitable amplitude distribution on it
and let it evolve according to the 3-D WNLRT.
The 3-D WNLRT is a weakly hyperbolic 7×7 system
that is highly nonlinear.
Due to a possibility of appearance of δ waves and shocks
it is a challenging task to develop an appropriate numerical method.
Here we use the Lax–Friedrichs scheme
and Nessyahu–Tadmor central scheme
and have obtained some very interesting shapes of the wavefronts
for two cases – in one case kink lines
and another case a point singularity appear in the physical space
though the results remain single-valued in the ray coordinates.
Thus we find the 3-D KCL to be suitable to solve many
complex problems for which there seems to be no other method
which at present can give these physically realistic features.