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An Isaac Newton Institute Workshop

Effective Computational Methods for Highly Oscillatory Problems: The Interplay between Mathematical Theory and Applications

Splitting methods for oscillatory non-autonomous linear systems

Authors: Sergio Blanes (Universidad Politécnica de Valencia), Fernando Casas (Universitat Jaume I), Ander Murua (EHU/UPV)

Abstract

Several relevant physical phenomena are modelled (totally or partially) by linear PDEs that, once spatially discretized, give rise to systems of coupled harmonic oscillators. To diagonalise these systems is usually prohibitively expensive. Then, to use splitting methods (involving matrix-vector products and possibly FFTs) is a valid alternative. A theoretical analysis about the stability and accuracy of splitting methods on the harmonic oscillator allows us to build new methods which outperform the existing methods from the literature.

Non-autonomous problems are also of great interest but, the most efficient methods for the autonomous case are not valid in this setting. From the Magnus series expansion (as a formal solution to the non-autonomous problem) we show how to adapt these methods by treating the "time" separately from the coordinates. This technique allows us to build new methods whose performance is tested on the Schrödinger equation with time-dependent potentials.

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