The course will study the numerical approximation for partial differential equations in the high frequency regime, when solutions have the form
 

where  is the large high frequency parameter. Examples include the wave equation, the Schr ̈odinger equation, or Maxwell equations. High frequency equations pose difficult multiscale problems, since the wavelength  is short compared to the overall size of the computational domain, and direct simulation techniques are very expensive if not unfeasible for such problems.

1–2 Geometrical optics approximation of high frequency waves.

 

The geometrical optics approximation describes the infinite frequency limit of wave equations. We will derive mathematical models which capture different facets of this limit, such as the eikonal and transport partial differential equations, the ray tracing ordinary differential equations and the phase space Liouville equation. We will also discuss limitations of geometrical optics, mainly in connection with caustics and boundaries. Furthermore, we will discuss different classes of computational techniques for the mathematical models. The numerical challenges include computing solutions with crossing waves (i.e. multivalued phases), controlling the accuracy of solutions with rapidly diverging rays, the nonlinearity of the partial differential equations, and the high dimensionality of the phase space.

 

Reference article:

 

3 Gaussian beams.

 

Gaussian beams form another high frequency asymptotic model for waves. It is related to geometrical optics but unlike geometrical optics, gaussian beams do not suffer from breakdown at caustics. In this lecture we will discuss their mathematical formulation, how they can be used for numerical computations, and their accuracy in terms of the frequency.

 

Reference articles: