An application of 3-D Kinematical Conservation Laws: propagation of a three dimensional wavefront

Arun, KR and Lukacova, Medvidova M and Rao, Raghurama SV (2008) An application of 3-D Kinematical Conservation Laws: propagation of a three dimensional wavefront. [Preprint]

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Abstract

We use the newly formulated 3-D kinematical conservation laws (KCL) to study the propagation of a three dimensional nonlinear wavefront in a polytropic gas in a uniform state and at rest in order to test the numerical efficacy of the 3-D KCL theory. We take initial shape of the front to be cylindrically symmetric with a suitable amplitude distribution and let it evolve according to the 3-D weakly nonlinear ray theory (WNLRT), which is obtained by adding to 3-D KCL a transport equation (in conservation form) for small amplitude. The 3-D WNLRT is a weakly hyperbolic $7\times7$ system that is highly nonlinear. Due to a possibility of appearance of $\delta$ 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 some cases - in one case a kink line 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 at present which can give these physically realistic features.

Item Type: Preprint http://math.iisc.ernet.in/~eprints/prepr... This paper has been submitted to SIAM Journal on Applied Mathematics and is under review. kinematical conservation laws;ray theory;nonlinear waves;kinks;weakly hyperbolic system;finite difference scheme Division of Physical & Mathematical Sciences > Mathematics 06 Jun 2008 19 Sep 2010 04:45 http://eprints.iisc.ernet.in/id/eprint/14216