We derive the conservation form of equations of evolution of a front propagating in three dimensions. We obtain a system of six conservation laws, known as 3-D kinematical conservation laws (KCL) in a ray coordinate system. The conservative variables of 3-D KCL are also constrained by a stationary vector constraint, known as geometric solenoidal constraint, which consists of three divergence-free type conditions. The 3-D KCL is an under-determined system, and therefore, additional closure relations are required to get a complete set of equations. We consider two closure relations for 3-D KCL: (1) energy transport equation of a weakly nonlinear ray theory (WNLRT) to study the propagation of a nonlinear wavefront, (2) transport equations of a shock ray theory (SRT) to study the propagation of a curved weak shock front. In both the cases we obtain a weakly hyperbolic system of balance laws. For the numerical simulation we use a high-resolution semi-discrete central scheme. The second order accuracy of the scheme is based on MUSCL type reconstructions and TVD Runge-Kutta time stepping procedures. A constrained transport technique is used to enforce the geometric solenoidal constraint and in all the test problems considered, the constraint is satisfied up to very high accuracy. We present the results of extensive numerical experiments, which confirm the efficiency and robustness of the method and also its ability to capture many physically realistic features of the fronts.

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Last updated: 09 Sep 2024