Juliette Chabassier | Space time convergence of conservative schemes for the linear wave equation

In this page are gathered several integration strategies guaranteeing energy identity at the discrete level, in order to solve weakly nonlinear partial differential equations involved in piano modeling. Two contributions are adapted to wave propagation in the piano string under large deformations, leading to a model with nonlinear potential energy. Moreover a contribution is tailored to wave propagation in the rotating hammer neck, leading to a model with additional non-linear kinetic energy.

References

preprint

Stability and space/time convergence of Störmer-Verlet time integration of the mixed formulation of linear wave equations

Chabassier, Juliette

Aug 2023

Abs URL PDF

This work focuses on the mixed formulation of linear wave equations. It provides a proof of stability and convergence of time discretisation of a semi discrete linear wave equation in mixed form with Störmer-Verlet time integration, that is uniform as the time step reaches its largest allowed value for stability (Courant-Friedrich-Levy condition), contrary to the proofs recalled here from the literature.
RR

Space time convergence of implicit discretization strategies for the mixed formulation of linear wave equations

Chabassier, Juliette

Research Report Nov 2023

URL PDF
RR

Two implicit parametrized discretization strategies for the mixed formulation of linear wave equations

Chabassier, Juliette

Research Report Oct 2023

Abs URL PDF

This report presents two parametric implicit strategies for discretizing the mixed formulation of linear wave equations, based for the first on the Störmer-Verlet scheme and for the second on the Crank-Nicolson scheme. If the initial conditions are well chosen, these two strategies are equivalent, and equivalence formulas allow the variables of one scheme to be reconstructed from the other. These two schemes are also equivalent to the 𝜃 θ -scheme discretization of the wave equation formulated at order 2. However, the first scheme has a lower complexity than the second one. Numerical results illustrate these equivalences and these efficiency comparisons.
CRAS

Space/Time convergence analysis of a class of conservative schemes for linear wave equations

Chabassier, Juliette, and Imperiale, Sebastien

Comptes Rendus. Mathématique Mar 2017

Abs URL DOI PDF

This paper concerns the space/time convergence analysis of conservative two-step time discretizations for linear wave equations. Explicit and implicit, second- and fourth-order schemes are considered, while the space discretization is given and satisfies minimal hypotheses. Convergence analysis is done using energy techniques and holds if the time step is upper-bounded by a quantity depending on space discretization parameters. In addition to showing the convergence for recently introduced fourth-order schemes, the novelty of this work consists in the independency of the convergence estimates with respect to the difference between the time step and its greatest admissible value.
WaMot

Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string.

Chabassier, Juliette, and Imperiale, Sébastien

Wave Motion Nov 2012

Abs URL DOI PDF

We study the implicit time discretization of piano strings governing equations within the Timoshenko prestressed beam model. Such model features two different waves, namely the flexural and shear waves, that propagate with very different velocities. We present a novel implicit time discretization that reduces the numerical dispersion while allowing the use of a large time step in the numerical computations. After analyzing the continuous system and the two branches of eigenfrequencies associated with the propagating modes, the classical -scheme is studied. We present complete new proofs of stability using energy-based approaches that provide uniform results with respect to the featured time step. A dispersion analysis confirms that reduces the numerical dispersion, but yields a severely constrained stability condition for the application considered. Therefore we propose a new -like scheme, which allows to reduce the numerical dispersion while relaxing this stability condition. Stability proofs are also provided for this new scheme. Theoretical results are illustrated with numerical experiments corresponding to the simulation of a realistic piano string.
M2AN

Construction and convergence analysis of conservative second order local time discretisation for linear wave equations

Chabassier, Juliette, and Imperiale, Sébastien

ESAIM: Mathematical Modelling and Numerical Analysis Jul 2021

Abs URL DOI PDF

In this work we present and analyse a time discretisation strategy for linear wave equations t hat aims at using locally in space the most adapted time discretisation among a family of implicit or explicit centered second order schemes. The proposed family of schemes is adapted to domain decomposition methods such as the mortar element method. They correspond in that case to local implicit schemes and to local time stepping. We show that, if some regularity properties of the solution are satisfied and if the time step verifies a stability condition, then the family of proposed time discretisations provides, in a strong norm, second order space-time convergence. Finally, we provide 1D and 2D numerical illustrations that confirm the obtained theoretical results and we compare our approach on 1D test cases to other existing local time stepping strategies for wave equations.