An Application of 3-d Kinematical Conservation Laws: Propagation of a Three Dimensional Wavefront

3-D kinematical conservation laws (KCL) are equations of evolution of a propagating surface Ωt in three space dimensions and were first derived in 1995 by Giles, Prasad and Ravindran [15] 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.