Propagation of 1D waves in regular discrete heterogeneous media: a Wigner measure approach. (English) Zbl 1375.81163
The article focusses on an analysis of the propagation properties of discrete waves while solving the finite difference schemes for two 1D wave propagation models, namely the first-order transport equation and second-order wave equation. The fundamental contribution of this work is the provision of a structured meaning to notions such as the principal symbol of the discrete wave operator, allowing one to construct bi-characteristic and characteristic rays propagating the energy of solutions.
Boundary controllability and identifiability properties of the solutions of the wave equation hold due to the fact that the energy of solutions is driven by characteristics that reach the boundary where controllers or observers are placed. This property is known to fail in general for numerical schemes due to high-frequency spurious numerical solutions. However this analysis has thus far only been done within the context of uniform meshes. This research extends this by considering non-uniform meshes and developing the tools and notions required for microlocal analysis.
Boundary controllability and identifiability properties of the solutions of the wave equation hold due to the fact that the energy of solutions is driven by characteristics that reach the boundary where controllers or observers are placed. This property is known to fail in general for numerical schemes due to high-frequency spurious numerical solutions. However this analysis has thus far only been done within the context of uniform meshes. This research extends this by considering non-uniform meshes and developing the tools and notions required for microlocal analysis.
Reviewer: Charis Harley (Johannesburg)
MSC:
| 81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |
| 35A21 | Singularity in context of PDEs |
| 37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 70K05 | Phase plane analysis, limit cycles for nonlinear problems in mechanics |
| 81T80 | Simulation and numerical modelling (quantum field theory) (MSC2010) |
Keywords:
variable coefficients transport/wave equations; finite difference approximations; non-uniform meshes; Wigner transform/measure; bi-characteristic rays; umklapp/internal reflectionReferences:
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