## Section: New Results

### Hybrid time discretizations of high-order

#### High-order symmetric multistep schemes for wave equation

Participants : Juliette Chabassier, Marc Duruflé, Guillaume Marty.

We have studied high-order symmetric multistep schemes for the second-order formulation ${y}^{\text{'}\text{'}}=f(t,y)$ during the internship of Guillaume Marty. The stability condition (CFL) can be optimized for explicit schemes since they have free parameters. However, this optimization procedure is not easy since the optimum is reached for forbidden values (values for which the high-order accuracy is no longer obtained). We have proposed acceptable values of free parameters for schemes of order 4, 6 and 8. These schemes have been tested for the wave equation, they suffer from a lack of robustness with respect to rounding numerical errors. The stability of implicit schemes has also been explored. For fourth-order schemes, a family of energy-conserving schemes has been obtained. However, we have not found unconditionally stable high-order schemes, which is well-known for the first-order formulation as Dahlquist's barrier. It seems that for the second-order formulation, this barrier holds and only second-order accurate schemes are unconditionally stable. Implicit high-order schemes have a maximum CFL of $\sqrt{6}$, the same CFL as the standard $\theta $-scheme with $\theta =\frac{1}{12}$. As a result, the implicit version of these schemes does not have a practical interest.

#### High order conservative explicit and implicit schemes for wave equations.

Participants : Juliette Chabassier, Sébastien Imperiale.

In 2015 we have studied the space/time convergence of a family of high order conservative explicit and implicit schemes for wave equations. An original proof of convergence has been proposed and provides an understanding of the lack of convergence of some schemes when the time step approaches its greatest admissible value for stability (CFL condition). An article is being written and will be submitted soon.

#### Multi-level explicit local time-stepping methods for second-order wave equations

Participants : Julien Diaz, Marcus Grote.

Local mesh refinement severely impedes the efficiency of explicit time-stepping methods for numerical wave propagation. Local time-stepping (LTS) methods overcome the bottleneck due to a few small elements by allowing smaller time-steps precisely where those elements are located. Yet when the region of local mesh refinement itself contains a sub-region of even smaller elements, any local time-step again will be overly restricted. To remedy the repeated bottleneck caused by hierarchical mesh refinement, multi-level local time-stepping methods are proposed, which permit the use of the appropriate time-step at every level of mesh refinement. Based on the LTS methods from Diaz and Grote [82] , these multi-level LTS methods are explicit, yield arbitrarily high accuracy and conserve the energy.

The method was published in Computer Methods in Applied Mechanics and Engineering [24] .