Section: New Results

Cost reduction for numerical methods

In [22], S. Lemaire et al. design and analyze (in the periodic setting) nonconforming multiscale methods for highly oscillatory elliptic problems, which are applicable on coarse grids that may feature general polytopal cells. Two types of methods are introduced: a Finite Element-type method, that generalizes classical nonconforming multiscale Finite Element methods to general meshes and to arbitrary-order polynomial cell boundary conditions, and a Virtual Element-type method, that allows, up to the computation of an adequate projection, to compute less oscillatory basis functions for equivalent precision. The Virtual Element-type method is based on the Hybrid High-Order framework [78]. As standard with such multiscale approaches, the general workflow of the method splits into an offline, massively parallelizable stage, where all fine-scale computations are performed, and the online, fully-coarse-scale stage.

In [25], T. Rey et al. extend the Fast Kinetic Scheme (FKS) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique, supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamics.

In [43], T. Rey et al. present high-order, fully explicit time integrators for nonlinear collisional kinetic equations, including the full Boltzmann equation. The methods, called projective integration, first take a few small steps with a simple, explicit method (forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in a Runge–Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. We illustrate the method with numerical results in one and two spatial dimensions.