Stability of space-time isogeometric methods for wave propagation problems

TL;DR

This thesis explores the development of an unconditionally stable space-time isogeometric method for the linear acoustic wave equation, demonstrating its stability and effectiveness through numerical tests and proposing extensions for broader application.

Contribution

It introduces a stabilized space-time isogeometric method based on splines of maximal regularity, addressing stability issues in wave propagation simulations.

Findings

The method achieves unconditional stability in wave propagation problems.

Numerical tests confirm the effectiveness of the stabilization approach.

Proposed extension of stabilization to full space-time formulations.

Abstract

This thesis aims at investigating the first steps toward an unconditionally stable space-time isogeometric method, based on splines of maximal regularity, for the linear acoustic wave equation. The unconditional stability of space-time discretizations for wave propagation problems is a topic of significant interest, by virtue of the advantages of space-time methods compared with more standard time-stepping techniques. In the case of continuous finite element methods, several stabilizations have been proposed. Inspired by one of these works, we address the stability issue by studying the isogeometric method for an ordinary differential equation closely related to the wave equation. As a result, we provide a stabilized isogeometric method whose effectiveness is supported by numerical tests. Motivated by these results, we conclude by suggesting an extension of this stabilization tool to the space-time isogeometric formulation of the acoustic wave equation.