## Section: New Results

### Palindromic BGK methods

Since two years we work on implicit relaxation methods to solve hyperbolic PDE without CFL and without matrices to invert. The Palindromic BGK method allows to approximate a hyperbolic system by a larger set of transport equations coupled by a nonlinear source term which relaxes the variables on an equilibrium. Using a splitting scheme, we can solve these transport equations in parallel and solve the local relaxation in a second step. The high-order extension is obtained by a symmetric modified Strang splitting and composition methods.

#### Boundary condition for Palindromic BGK method

**Participants**: Florence Drui, Emmanuel Franck, Philippe Helluy, Laurent Navoret.

One of the drawbacks to the Palindromic BGK model is the treatment of the boundary conditions. Indeed the BGK scheme admits more variables than the original one and the boundary conditions for these additional variables are not defined. The classical choice is to impose the equilibrium at the boundary. In this case we obtain instabilities and only the first order convergence. After an analysis of the symmetric modified Strang splitting method, we have identified the dynamic for the non-physical variables and proposed boundary conditions compatible with this dynamic. We obtain stable and second order boundary conditions.

#### Palindromic BGK scheme for Low-Mach models

**Participants**: ClĂ©mentine CourtĂ¨s (IRMA), Emmanuel Franck, Philippe Helluy, Laurent Navoret.

Another drawback of the method is the application for "two-scale" problems like Low-Mach flows. Indeed, in this case the BGK representation used generate an large error on the slow scale which is homogeneous to the fast scale. Consequently the slow scale is not well resolved. This problem comes from the fact that the BGK approximation uses a linearization with a constant fast scale to approximate all the systems. We have proposed a new method where we also introduce a slow scale in the BGK approximation. Using this, we obtain accurate results for the Euler equation in the low-Mach regime in 1D. The method gives interesting results also for other applications. In the future we must extend the method in 2D.

#### Palindromic BGK scheme for diffusion models

**Participants**: Laura Mendoza, Emmanuel Franck, Laurent Navoret.

In MHD simulations for ITER, we must also discretize with an implicit scheme the anisotropic diffusion. Firstly, we have proposed to extend the previous Palindromic BGK method to the parabolic problems. For that we must use a different Palindromic BGK model with specific parameters. We obtain a second order scheme without CFL for the Heat equation in 1D and 2D. In the future we will consider the high-order schemes and the extension to the anisotropic case.

#### Semi-Lagrangian on complex geometries for Palindromic BGK scheme

**Participants**: Laura Mendoza, Emmanuel Franck, Philippe Helluy.

To apply the Palindromic BGK method we must have an advection solver without CFL. In the code Slappy we propose a 3D high-order Semi-Lagrangian solver able to treat blocks-structured meshes with overlapping and non-conformity. This allows to treat complex geometries easily. The solver is written in PYOpenCL and can be used on GPU. In the code the relaxation step is also implemented, which allows to use the Palindromic BGK method on some PDE (Euler, Diffusion etc).

#### Lattice Boltzmann scheme with PyOpenCL

**Participants**: Florence Drui, Emmanuel Franck, Philippe Helluy.

In the same idea, another code has been developed to treat hyperbolic systems with the BGK approach. In this case the transport is exact and consequently the method is equivalent to the Lattice Boltzmann scheme. The parallel part is similar and also based on PyOpenCL. This version is less accurate than the previous code, can be used only in Cartesian grids but is more stable and can run more complex problems. The main result is the simulation of 2D resistive MHD instabilities which have the same structure than Tokamak instabilities.