Next Tuesday, April 25th, at 12:30 at the I.U. de Matemática Multidisciplinar, Prof. Arieh Iserles, Emeritus Professor in Numerical Analysis of Differential Equations of University of Cambridge will present a talk about spectral methods and dispersive equations.
You can follow the talk online at: https://teams.microsoft.com/l/meetup-join/19%3aac3291a8974744de9ff32dbfa56c36c4%40thread.tacv2/1681991092801?context=%7b%22Tid%22%3a%22be4655df-ac73-401f-a7ae-198c3b72d0c6%22%2c%22Oid%22%3a%22bfd5fc16-28c0-4ab1-b7c1-451c80a4affb%22%7d
Emeritus Professor in Numerical Analysis of Differential Equations
Department of Applied Mathematics and Theoretical Physics
Centre for Mathematical Sciences
University of Cambridge
Every spectral method commences from an orthonormal system Φ=φₙ n ∈ ℤ₊ over an interval (a,b) and its major feature is the differentiation matrix 𝒟 such that
Once 𝒟 is skew-Hermitian, exp(t·𝒟) is unitary and the method is stable and preserves the L₂ norm.
The nature of the interval (a,b) 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.