Unconditionally stable space-time isogeometric discretization for the wave equation in Hamiltonian formulation

TL;DR

This paper introduces an unconditionally stable space-time isogeometric discretization for the wave equation in Hamiltonian form, avoiding CFL conditions and ensuring stability through spectral analysis, validated by numerical tests.

Contribution

It presents a novel unconditionally stable space-time discretization for the wave equation using maximal regularity splines, extending stability analysis without CFL constraints.

Findings

Method is unconditionally stable without CFL restrictions

Spectral analysis confirms stability properties

Numerical tests demonstrate effective performance

Abstract

We consider a family of conforming space-time discretizations for the wave equation based on a first-order-in-time formulation employing maximal regularity splines. In contrast with second-order-in-time formulations, which require a CFL condition to guarantee stability, the methods we consider here are unconditionally stable without the need for stabilization terms. Along the lines of the work by M. Ferrari and S. Fraschini (2024), we address the stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. Numerical tests validate the performance of the method.