Piecewise discretization of monodromy operators of delay equations on adapted meshes

TL;DR

This paper introduces a piecewise discretization method for monodromy operators of delay equations on adapted meshes, improving stability analysis accuracy in computational methods.

Contribution

It develops a piecewise pseudospectral technique that incorporates adapted meshes, enhancing the computation of Floquet multipliers for delay equations.

Findings

Piecewise discretization improves stability analysis accuracy.

Including adapted meshes is essential for strong mesh adaptation.

The method is experimentally validated for delay equations.

Abstract

Periodic solutions of delay equations are usually approximated as continuous piecewise polynomials on meshes adapted to the solutions' profile. In practical computations this affects the regularity of the (coefficients of the) linearized system and, in turn, the effectiveness of assessing local stability by approximating the Floquet multipliers. To overcome this problem when computing multipliers by collocation, the discretization grid should include the piecewise adapted mesh of the computed periodic solution. By introducing a piecewise version of existing pseudospectral techniques, we explain why and show experimentally that this choice is essential in presence of either strong mesh adaptation or nontrivial multipliers whose eigenfunctions' profile is unrelated to that of the periodic solution.