Narrow Results

A computational procedure is developed for adaptive modeling of transient wave propagation in an unbounded acoustic layer. The adaptivity herein is concerned with controlling the error caused by imperfect modeling of unbounded domains, and the procedure aims at determining the optimal order of the absorbing boundary condition (ABC) under consideration to control the error in the solution within a predetermined tolerance. The perfectly matched layer (PML) is used as an ABC, and the size of the PML domain controls the error associated with modeling unbounded domains. As a key technique in the adaptive method, a local solution procedure is designed based on the method of bicharacteristics to extract, from the computed solution, the information flowing into the computational domain. Numerical experiments show that, under wide circumstances, the maximum pressure associated with the inflow evaluated near the truncated boundary of the PML domain can be related with the maximum error in the pressure and thus can be used as an error estimator. It is demonstrated that a simple adaptive algorithm based on this error estimator can automatically determine the PML domain size to produce solutions with controlled errors.

Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics and in Physics. I will review the theory of characteristics and bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e. those aspects which are invariant under general changes of coordinates. After a basically analytic introduction, I will pass to a modern, geometric point of view, presenting characteristics within the jet space approach to PDEs. In particular, I will discuss the relationship between characteristics and singularities of solutions and observe that: "wave-fronts are characteristic surfaces and propagate along bicharacteristics". This remark may be understood as a mathematical formulation of the wave/particle duality in optics and/or quantum mechanics. The content of the paper reflects the three-hour mini-course that I gave at the XXII International Fall Workshop on Geometry and Physics, September 2–5, 2013, Évora, Portugal.

The scalar wave equation in Kasner spacetime is solved, first for a particular choice of Kasner parameters, by relating the integrand in the wave packet to the Bessel functions. An alternative integral representation is also displayed, which relies upon the method of integration in the complex domain for the solution of hyperbolic equations with variable coefficients. In order to study the propagation of wave fronts, we integrate the equations of bicharacteristics which are null geodesics, and we are able to express them, for the first time in the literature, with the help of elliptic integrals for another choice of Kasner parameters. For generic values of the three Kasner parameters, the solution of the Cauchy problem is built through a pair of integral operators, where the amplitude and phase functions in the integrand solve a coupled system of partial differential equations. The first is the so-called transport equation, whereas the second is a nonlinear equation that reduces to the Eikonal equation if the amplitude is a slowly varying function. Remarkably, the analysis of such a coupled system is proved to be equivalent to building first an auxiliary covariant vector having vanishing divergence, while all nonlinearities are mapped into solving a covariant generalization of the Ermakov–Pinney equation for the amplitude function. Last, from a linear set of equations for the gradient of the phase one recovers the phase itself. This is the parametrix construction that relies upon Fourier–Maslov integral operators, but with a novel perspective on the nonlinearities in the dispersion relation. Furthermore, the Adomian method for nonlinear partial differential equations is applied to generate a recursive scheme for the evaluation of the amplitude function in the parametrix. The resulting formulas can be used to build self-dual solutions to the field equations of noncommutative gravity, as has been shown in the recent literature.