## Section: New Results

### Hybrid time discretizations of high-order

#### Construction and analysis of a fourth order, energy preserving, explicit time discretization for dissipative linear wave equations.

Participants : Juliette Chabassier, Julien Diaz, Anh-Tuan Ha.

We submitted a paper to M2AN. This paper deals with the construction of a fourth order, energy preserving, explicit time discretization for dissipative linear wave equations. This scheme is 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 an explicit approximation of its inverse. The stability of the scheme is studied first using an energy analysis, then an eigenvalue analysis. Numerical results in 1D illustrate the good behavior regarding space/time convergence and the efficiency of the newly derived scheme compared to more classical time discretizations. A loss of accuracy is observed for non smooth profiles of dissipation, and we propose an extension of the method that fixes this issue. Finally, we assess the good performance of the scheme for a realistic dissipation phenomenon in Lorentz's materials. This work has been done in collaboration with Sébastien Imperiale (Inria Project-Team M3DISIM) and Alain Anh-Tuan Ha (Internship at Magique 3D in 2016).

#### Higher-order optimized explicit Runge-Kutta schemes for linear ODEs

Participants : Hélène Barucq, Marc Duruflé, Mamadou N'Diaye.

In this work, we have constructed optimized explicit Runge-Kutta schemes for linear ODEs that we called Linear-ERK. Theses schemes can be applied to the following ODE

where ${M}_{h}$ is the mass matrix, ${K}_{h}$ the stiffness matrix and $F\left(t\right)$ a source term. Linear-ERK schemes are constructed using polynomial stability functions which are obtained by maximizing the CFL number. We have considered a polynomial stability function based on the Taylor series expansion of an exponential function. Then, we have added extra terms beyond the terms of the Taylor expansion without changing the order of accuracy. The coefficients of those extra terms have been computed by optimizing the CFL number such that the stability region of the developed scheme includes a typical spectrum. This spectrum has been obtained by computing eigenvalues of the matrix ${M}_{h}^{-1}{K}_{h}$ for the wave equation solved on a square with Hybrid Discontinuous Formulation (HDG). The optimization is performed by using the algorithm developed by D. Ketcheson and coworkers. By proceeding this way, we have obtained optimized explicit schemes up to order 8. We have also determined the CFL number and the efficiency on the typical spectrum for each explicit scheme. We have provided algorithms to implement these schemes and numerical results to compare them.

This work is a chapter of the thesis defended by Mamadou N'diaye on December 8, 2017, under the joint supervision of Hélène Barucq and Marc Duruflé.

#### High-order locally implicit time schemes for linear ODEs

Participants : Hélène Barucq, Marc Duruflé, Mamadou N'Diaye.

In this work we have proposed a method that combines optimized explicit schemes and implicit schemes to form locally implicit schemes for linear ODEs, including in particular ODEs coming from the space discretization of wave propagation phenomena. This method can be applied to the following ODE

Like in the local time-stepping developed by Grote and co-workers, the computational domain is split into a fine region and a coarse region. The matrix ${A}_{h}$ is given as

where $P$ is the projector on the fine region of the computational domain. Then the proposed locally implicit method is obtained from the combination of the A-stable implicit schemes we have developed in 2016 (Padé schemes or Linear-SDIRK schemes detailed in [17]) on the fine region and explicit schemes with optimal CFL number in the coarse region. The developed method has been used to solve the acoustic wave equation and we have checked the convergence in time of these schemes for order 4, 6 and 8.

This work has been presented at the Mathias annual Total seminar and is a chapter of the thesis defended by Mamadou N'diaye on December 8, 2017, under the joint supervision of Hélène Barucq and Marc Duruflé.