2 results
4 - Approximation of high frequency wave propagation problems
-
- By M. Motamed, O. Runborg
- Edited by Bjorn Engquist, University of Texas, Austin, Athanasios Fokas, University of Cambridge, Ernst Hairer, Université de Genève, Arieh Iserles, University of Cambridge
-
- Book:
- Highly Oscillatory Problems
- Published online:
- 07 September 2011
- Print publication:
- 02 July 2009, pp 72-97
-
- Chapter
- Export citation
-
Summary
Abstract
The numerical approximation of high frequency wave propagation is important in many applications including seismic, acoustic, optical waves and microwaves. For these problems the solution becomes highly oscillatory relative to the overall size of the domain. Direct simulations using the standard wave equations are therefore very expensive, since a large number of grid points is required to resolve the wave oscillations. There are however computationally much less costly models, that are good approximations of many wave equations at high frequencies. In this paper we review such models and related numerical methods that are used for simulations in applications. We focus on the infinite frequency approximation of geometrical optics and the finite frequency corrections given by the geometrical theory of diffraction. We also briefly discuss Gaussian beams.
Introduction
Simulation of high-frequency waves is a problem encountered in a great many engineering and science fields. Currently the interest is driven by new applications in wireless communication (cell phones, Bluetooth, WiFi) and photonics (optical fibers, filters, switches). Simulation is also used increasingly in more classical applications. Some examples in electromagnetism are antenna design, radar signature computation and base station coverage for cell phones. In acoustics simulation is used for noise reduction, underwater communication and medical ultrasonography. Finding the location of an earthquake and oil exploration are some applications of seismic wave simulation in geophysics. Nondestructive testing is another example where both electromagnetic and acoustic waves are simulated.
5 - Wavelet-based numerical homogenization
- Edited by Bjorn Engquist, University of Texas, Austin, Athanasios Fokas, University of Cambridge, Ernst Hairer, Université de Genève, Arieh Iserles, University of Cambridge
-
- Book:
- Highly Oscillatory Problems
- Published online:
- 07 September 2011
- Print publication:
- 02 July 2009, pp 98-126
-
- Chapter
- Export citation
-
Summary
Abstract
We consider multiscale differential equations in which the operator varies rapidly over fine scales. Direct numerical simulation methods need to resolve the small scales and they therefore become very expensive for such problems when the computational domain is large. Inspired by classical homogenization theory, we describe a numerical procedure for homogenization, which starts from a fine discretization of a multiscale differential equation, and computes a discrete coarse grid operator which incorporates the influence of finer scales. In this procedure the discrete operator is represented in a wavelet space, projected onto a coarser subspace and approximated by a banded or block-banded matrix. This wavelet homogenization applies to a wider class of problems than classical homogenization. The projection procedure is general and we give a presentation of a framework in Hilbert spaces, which also applies to the differential equation directly. We show numerical results when the wavelet based homogenization technique is applied to discretizations of elliptic and hyperbolic equations, using different approximation strategies for the coarse grid operator.
Introduction
In the numerical simulation of partial differential equations, the existence of subgrid scale phenomena poses considerable difficulties. With subgrid scale phenomena, we mean those processes which could influence the solution on the computational grid but which have length scales shorter than the grid size. Highly oscillatory initial data may, for example, interact with fine scales in the material properties and produce coarse scale contributions to the solution.