El pròxim, dimarts 25 d’abril, a les 12:30 en el I.U. de Matemàtica Multidisciplinaria, el Prof. Arieh Iserles, Professor Emèrit d’Anàlisi Numèric d’Equacions Diferencials a la Universitat de Cambridge farà una xarrada sobre mètodes espectrals i equacions dispersives.
Autor:
Arieh Iserles
Emeritus Professor in Numerical Analysis of Differential Equations
Department of Applied Mathematics and Theoretical Physics
Centre for Mathematical Sciences
University of Cambridge
Academic Portfolio
Abstract:
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.
