Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach
In this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce a...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2014 |
| País: | España |
| Institución: | Basque Center for Applied Mathematics (BCAM) |
| Repositorio: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/237 |
| Acceso en línea: | http://hdl.handle.net/20.500.11824/237 |
| Access Level: | acceso abierto |
| Palabra clave: | Hamiltonians Inverse problems Mathematical transformations Analysis and simulation Diffeomorphic transformation Finite difference approximations Finite difference scheme Non-uniform mesh Propagation properties Variable coefficients Wigner transforms Finite difference method |
| Sumario: | In this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce and give a rigorous meaning to notions like the principal symbol of the discrete wave operator and the corresponding bi-characteristic rays. The main mathematical tool we employ is the discrete Wigner transform, which, in the limit as the mesh size parameter tends to zero, yields the so-called Wigner (semiclassical) measure. This measure provides the dynamics of the bi-characteristic rays, i.e., the solutions of the Hamiltonian system describing the propagation, in both physical and Fourier spaces, of the energy of the solution to the wave equation. We show that, due to dispersion phenomena, the high-frequency numerical dynamics does not coincide with the continuous one. Our analysis holds for the class (Formula presented.) of globally Lipschitz coefficients and non-uniform grids obtained by means of (Formula presented.)-diffeomorphic transformations of a uniform one. We also present several numerical simulations that confirm the predicted paths of the space–time projections of the bi-characteristic rays. Based on the theoretical analysis and simulations, we describe some of the pathological phenomena that these rays might exhibit as, for example, their reflection before touching the boundary of the space domain. This leads, in particular, to the failure of the classical properties of boundary observability of continuous waves, arising in control and inverse problems theory. |
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