Talk of Prof. Arieh Iserles: An overarching framework for spectral methods and dispersive equations

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:


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


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.

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