Section: New Results

Hybrid time discretizations of high-order

Space-Time Discretization of Elasto-Acoustic Wave Equation in Polynomial Trefftz-DG Bases

Participants : Elvira Shishenina, Hélène Barucq, Julien Diaz.

In the context of the strategic action "Depth Imaging Partnership" between Inria and Total we have investigated to the development of an explicit Trefftz-DG formulation for elasto-acoustic problem, solving the global sparse matrix by constructing an approximate inverse obtained from the decomposition of the global matrix into a block-diagonal one. The inversion is then justified under a CFL-type condition. This idea allows for reducing the computational costs but its accuracy is limited to small computational domains. According to the limitations of the method, we have investigated the potential of Tent Pitcher algorithms following the recent works of Gopalakrishnan et al. It consists in constructing a space-time mesh made of patches that can be solved independently under a causality constraint. We have obtained very promising numerical results illustrating the potential of Tent Pitcher in particular when coupled with a Trefftz-DG method involving only surface terms. In this way, the space-time mesh is composed of elements which are 3D objects at most. It is also worth noting that this framework naturally allows for local time-stepping which is a plus to increase the accuracy while decreasing the computational burden. The results of this work have been published in the Applicable Analysis Journal [6], in the book of proceedings for European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2017) (due date April 27, 2019), and in the PhD thesis [2], as well as presented during the International Conference on Spectral and High-Order Methods (ICOSAHOM 2018, London - UK), the 13^th World Congress on Computational Mechanics (WCCM 2018, New-York - USA), and during the annual seminar on Computational Science Engineering and Data Science organized by TOTAL (MATHIAS 2018, Serris - France).

Performance analysis of local time-stepping schemes for wave propagation

Participants : Julien Diaz, Rose-Cloé Meyer.

The efficiency of numerical simulation of wave propagation is highly dependent of the quality of the mesh. For complex simulations, the size of the cells in the mesh can strongly vary, either because of the geometry or because of the different propagation celerity of the waves. To ensure stability, explicit numerical schemes must match with the CFL conditions of every cells of the mesh. When significant disparities appear in the domain, the time step used on big cells is not optimal, which can cause heavy calculation cost and result in a loss of efficiency. To improve the performance of the programs, local time-stepping methods based on a spatial Discontinuous Galerkin discretization have been implemented. In this work, we compared three local time-stepping methods: a conservative method, a recursive method, and an asynchron method. The two first methods use local time steps that are fractions of the global time step, while the third method can use independent time steps on each cell of the mesh. The accuracy of the solution, the computation cost and the speedup of local-time stepping are presented on cases in two and three dimensions on configurations as fine slot or domains with geometric singularities. The results are presented in Rose-Cloé Meyer Master thesis [41]. This work has been achieved in collaboration with Guillaume Dufour and Xavier Ferrières (Onera)

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

Participants : Juliette Chabassier, Julien Diaz.

We submitted a paper to M2AN [37]. This paper deals with the construction of a fourth order, energy preserving, explicit time discretization for dissipative linear wave equations. This family of schemes 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 series can be truncated at different orders, which leads to several schemes. The stability of the schemes is studied. 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).

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

Participant : Juliette Chabassier.

In this work we present and analyse a time discretisation strategy for linear wave propagation that aims at using locally in space the most adapted time discretisation among a family of implicit or explicit centered second order schemes. The domain of interest being decomposed into several regions, different time discretisations can be chosen depending on the local properties of the spatial discretisations (mesh size and quality) or the physical parameters (high wave speed, low density). We show that, under some conditions on the time step, the family of time discretisations obtained combined with standard finite elements methods in space ensures a second order space-time convergence. This work has been done in collaboration with Sébastien Imperiale (Inria Project-Team M3DISIM). It has been submitted to Numerische Mathematik.

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 [8]) 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 is a chapter of the thesis defended by Mamadou N'diaye under the joint supervision of Hélène Barucq and Marc Duruflé. In 2018, the implemented method has been parallelized in Montjoie and 3-D numerical results have been obtained. An article is in preparation.