Juliette Chabassier | High order schemes for linear wave equations

The publications listed here are devoted to high order time discretizations of linear wave equations. We have proposed

Fourth order $(\theta,\varphi)$-schemes, that guarantee the preservation of a discrete energy identity,
the Space-time convergence proof of polynomial-based schemes, including the usual $theta$-scheme, but also the new fourth order $(\theta,\varphi)$-schemes, and also stabilized schemes,
High order schemes for dissipative equations that guarantee the preservation of a discrete energy identity.

References

JCAM

Introduction and study of fourth order theta schemes for linear wave equations

Chabassier, Juliette, and Imperiale, Sébastien

Journal of Computational and Applied Mathematics Jan 2013

Abs URL DOI PDF

A new class of high order, implicit, three time step schemes for semi-discretized wave equations is introduced and studied. These schemes are constructed using the modified equation approach, generalizing the θ-scheme. Their stability properties are investigated via an energy analysis, which enables us to design super-convergent schemes and also optimal stable schemes in terms of consistency errors. Specific numerical algorithms for the fully discrete problem are tested and discussed, showing the efficiency of our approach compared to second order θ-schemes.
M2AN

Construction and analysis of fourth order, energy consistent, family of explicit time discretizations for dissipative linear wave equations

Chabassier, Juliette, Diaz, Julien, and Imperiale, Sébastien

ESAIM: Mathematical Modelling and Numerical Analysis May 2020

Abs URL DOI PDF

This paper deals with the construction of a family of fourth order, energy consistent, explicit time discretizations for dissipative linear wave equations. The schemes are obtained by replacing the inversion of a matrix, that comes naturally after using the technique of the Modified Equation on the second order Leap Frog scheme applied to dissipative linear wave equations, by explicit approximations of its inverse. The stability of the schemes are studied using an energy analysis and a convergence analysis is carried out. Numerical results in 1D illustrate the space/time convergence properties of the schemes and their efficiency is compared to more classical time discretizations.
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.