An overarching framework for spectral methods and dispersive equations

27 Set 2023
10:00
Aula Magna da Facultade de Matemáticas, USC
CITMAga

​​​​​​"An overarching framework for spectral methods and dispersive equations", organizado polo CITMAga. Será impartido por Arieh Iserles (University of Cambridge).

Data: mércores 27 de setembro

Hora: 10:00 h.

Duración: 1 hora

Lugar: Aula Magna da Facultade de Matemáticas, ou ben en liña a través do enlace Teams meeting. Conferenciante por Teams.

Abstract:

Every spectral method commences from an orthonormal system Φ = {𝜑𝜑𝑛𝑛}𝑛𝑛∈ℤ+ over an interval (𝑎𝑎, 𝑏𝑏) and its major feature is the differentiation matrix 𝒟𝒟 such that



Once 𝒟 is skew-Hermitian, exp(𝑡𝒟) is unitary and the method is stable and preserves the L2 norm.


The nature of the interval (𝑎𝑎, 𝑏𝑏) and the boundary conditions is critical: periodic boundary conditions in a compact interval are a no-brainer: we use a Fourier basis. This leaves out three important scenarios: the real line (−∞, ∞), the half-line (0, ∞) and the interval (−1,1), the latter two with zero Dirichlet boundary conditions. In the case of the real line we show that the additional requirement that 𝒟 is tridiagonal leads to a complete characterisation of all orthonormal and complete sets Φ by means of a Fourier transform, the Favard theorem and
the Plancherel theorem. The half-line and a compact interval require a completely different approach. We show that in each case there exists an orthonormal set which leads to a skew-symmetric matrix 𝒟 which, although dense, lends itself to very fast algebraic computations, and a magical value of the parameter that results in much improved accuracy.