Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems

On discreteness of the Hopf equation

“…We make no attempt to comment on these methods, but refer to [21] for a seminar survey on computational high-frequency wave propagation. More recently, a geometric view point has been adopted in place of the kinetic one in phase space; here we shall outline the corresponding level set methods developed in [8,[35][36][37]51,52]. Traditionally the level set method has been a highly successful computational technique for capturing the evolution of curves and surfaces [67,68] with applications in diverse areas such as multi-phase fluids, computer vision, imaging processing, optimal shape design, etc.…”

“…Based on the level set framework in the phase space, the amplitude is evaluated by ρ(t, x) = f (t, x, p)δ(φ)dp, (4.13) where the quantity f also solves the same Liouville equation (4.12) but with f (0, x, p) = ρ 0 (x) as initial data. The multi-valued higher moments can be also resolved by integrating f along the bi-characteristic manifold in the phase directions (see [35,36]). We refer the reader to the review article [52] for further details.…”

Computing high frequency solutions of symmetric hyperbolic systems with polarized waves

“…The computational methods are based on solving a coupled system of inhomogeneous Liouville equations which is the high frequency limit of the underlying hyperbolic systems by using the Wigner transform [15]. We first extend the level set methods developed in [6] for the homogeneous Liouville equation to the coupled inhomogeneous system, and find an efficient simplification in one space dimension for the Eulerian formulation which reduces the computational cost of two-dimesnional phase space Liouville equations into that of two one-dimensional equations. For the Lagrangian formulation, we introduce a geometric method which allows a significant simplification in the numerical evaluation of the energy density and flux.…”

Computation of transmissions and reflections in geometrical optics via the reduced Liouville equation

“…(1.4) Recently several phase space based level set methods are based on the equation (1.2) or (1.1), see [13,16,23,33,36]. It was used to compute the multivalued phase or velocity beyond caustics.…”

“…It was used to compute the multivalued phase or velocity beyond caustics. The computations of multivalued solution in geometrical optics, or more generally for nonlinear PDEs, have been an active area of research in recent years, see [2,3,5,4,8,15,10,11,13,12,18,19,16,24,37,41,6,36,23]. However, all these works were developed without the interface.…”