An inverse Lax–Wendroff method for boundary conditions applied to Boltzmann type models

Abstract

Abstract. In this paper we present a new algorithm based on a Cartesian mesh for the numerical approximation of kinetic models on complex geometry boundary. Due to the high dimensional property, numerical algorithms based on unstructured meshes for a complex geometry are not appropriate. Here we propose to develop an inverse Lax-Wendroff procedure, which was recently introduced for conservation laws [21], to the kinetic equations. Applications in 1D × 3D and 2D × 3D of this algorithm for Boltzmann type operators (BGK, ES-BGK models) are then presented and numerical results illustrate the accuracy properties of this algorithm.

Boundary conditions for semi-Lagrangian methods for the BGK model

“…In such case, a lower order accurate but more robust extrapolation is needed to prevent oscillations. Therefore, a WENO type extrapolation [16,19] has been applied and will be described in the Appendix; however, a first order approach could be coupled with a linear extrapolation. We will denote by f E,n+1 i,j the distribution function obtained by extrapolation at time t n+1 in (x i , v j ).…”

“…A technique already available in literature to treat diffusive boundary conditions, proposed by Shu and Tan [16] and adopted by Filbet and Yang in the context of BGK model [19], is the Inverse Lax-Wendroff (ILW) method. This technique is useful when the numerical scheme requires ghost points, as in the case of IMEX schemes used in [19].…”

“…This technique is useful when the numerical scheme requires ghost points, as in the case of IMEX schemes used in [19]. In particular, the ILW method is applied to compute the values of f at the ghost points for the inflow boundary.…”

“…We suppose to have a hot wall, that will generate a flow towards the interior. We consider a bounded spatial domain [x L , x R ] and the initial data is the one used in [19], that is Table 7. Relative errors and convergence rate of the distribution function in the whole domain, in L 2 norm, using EP, CFL=4.…”

“…In Section 2 the numerical methods used and developed in [1] are recalled; in Section 3 we investigate the treatment of reflective and diffusive boundaries; two ways of adapting semi-Lagrangian schemes to diffusive boundary conditions are illustrated, using or avoiding iterative procedures. In Section 4 we report the inverse Lax-Wendroff technique proposed in [16][17][18] and adopted by Filbet and Yang in the context of diffusive boundary conditions [19]; numerical results are presented and compared in Section 5 and conclusions and future developments are discussed in Section 6.…”