Computation of high frequency fields near caustics

It is well known that although the usual harmonic ansatz of geometrical opt ics fails near a caustic, uniform expansions can be found which remain vali d in the neighborhood of the caustic, and reduce asymptotically to the usua l geometric field far enough from it. Such expansions can be constructed by several methods which make essentially use of the symplectic structure of the phase space. In this paper we efficiently apply the Kravtsov-Ludwig met hod of relevant functions, in conjunction with Hamiltonian ray tracing to d efine the topology of the caustics and compute high-frequency scalar wave f ields near smooth and cusp caustics. We use an adaptive Runge-Kutta method to successfully retrieve the complete ray field in the case of piecewise sm ooth refraction indices. We efficiently match the geometric and modified am plitudes of the multi-valued field to obtain numerically the correct asympt otic behavior of the solution. Comparisons of the numerical results with an alytical calculations in model problems show excellent accuracy in calculat ing the modified amplitudes using the Kravtsov-Ludwig formulas.