A Numerical Scheme for Three-Dimensional Front Propagation and Control of Jordan Mode

We study the propagation of a three-dimensional weakly nonlinear wavefront into a polytropic gas in a uniform state and at rest. The successive positions and geometry of the wavefront are obtained by solving the conservation form of equations of a weakly nonlinear ray theory. The proposed set of equations forms a weakly hyperbolic system of seven conservation laws with an additional vector constraint analogous to the scalar solenoidal condition in the equations of ideal magnetohydrodynamics. The new divergence-free type constraint is termed as geometric solenoidal constraint. The analysis of a Cauchy problem for the linearised system shows that when this constraint is satisfied initially, the solution does not exhibit any Jordan mode. For the numerical simulation of the conservation laws we employ a high resolution central scheme. The second order accuracy of the scheme is achieved by using MUSCL type reconstructions and Runge-Kutta time discretisations. A constrained transport technique is used to enforce the geometric solenoidal constraint. The results of several numerical experiments are presented which confirm the efficiency and robustness of the proposed numerical method and the control of Jordan mode.