An application of 3-D kinematical conservation laws: propagation of a 3-D wavefront

Arun, K. R. ; Lukáčová-Medviďová, M. ; Prasad, Phoolan ; Raghurama Rao, S. V. (2010) An application of 3-D kinematical conservation laws: propagation of a 3-D wavefront SIAM Journal on Applied Mathematics, 70 (7). pp. 2604-2626. ISSN 0036-1399

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Three-dimensional (3-D) kinematical conservation laws (KCL) are equations of evolution of a propagating surface Omega(t) in three space dimensions. We start with a brief review of the 3-D KCL system and mention some of its properties relevant to this paper. The 3-D KCL, a system of six conservation laws, is an underdetermined system to which we add an energy transport equation for a small amplitude 3-D nonlinear wavefront propagating in a polytropic gas in a uniform state and at rest. We call the enlarged system of 3-D KCL with the energy transport equation equations of weakly nonlinear ray theory (WNLRT). We highlight some interesting properties of the eigenstructure of the equations of WNLRT, but the main aim of this paper is to test the numerical efficacy of this system of seven conservation laws. We take several initial shapes for a nonlinear wavefront 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. Here we use the staggered Lax–Friedrichs and Nessyahu–Tadmor central schemes and have obtained some very interesting shapes of the wavefronts. We find the 3-D KCL to be suitable for solving many complex problems for which there presently seems to be no other method capable of giving such physically realistic features.

Item Type:Article
Source:Copyright of this article belongs to Society for Industrial and Applied Mathematics.
Keywords:Kinematical Conservation Laws; Ray Theory; Nonlinear Waves; Kinks; Weakly Hyperbolic system; Finite Difference Scheme
ID Code:37677
Deposited On:25 Apr 2011 10:20
Last Modified:17 May 2016 20:34

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