Locally refined discrete velocity grids for stationary rarefied flow simulations

Abstract

Revised version (new material: a section on the numerical boundary conditions, some minor modifications)International audienceMost of deterministic solvers for rarefied gas dynamics use discrete velocity (or discrete ordinate) approximations of the distribution function on a Cartesian grid. This grid must be sufficiently large and fine to describe the distribution functions at every space position in the computational domain. For 3-dimensional hypersonic flows, like in re-entry problems, this induces much too dense velocity grids that cannot be practically used, for memory storage requirements. In this article, we present an approach to generate automatically a locally refined velocity grid adapted to a given simulation. This grid contains much less points than a standard Cartesian grid and allows us to make realistic 3-dimensional simulations at a reduced cost, with a comparable accuracy

Numerical boundary conditions in Finite Volume and Discontinuous Galerkin schemes for the simulation of rarefied flows along solid boundaries

“…The numerical resolution of the BGK equation is done by a deterministic method using a discrete velocity approach (see [15,16,17], and the extension to polyatomic gases [10]). The velocity variable is replaced by discrete values v k of a Cartesian grid (see [1] for the use of locally refined velocity grids). The continuous distribution f is then replaced by its approximation at each point v k , and we get the following discrete velocity BGK equation…”

A multiscale fast semi-Lagrangian method for rarefied gas dynamics

“…For all tests, the choice of the bounds in velocity space has been done ad hoc in order to guarantee that during all the simulation the distribution function is well represented. This however cannot be done for general problem and one should think to some techniques which permit to adapt the velocity grid to the solution during the time evolution of the problem [9,2]. This problem has not been considered in this paper and we remind to future works for adapting the method proposed in this paper to adaptive in time velocity grids.…”

“…(2) This is the so-called Boltzmann-BGK equation describing the distribution, denoted by f = f (X, V , t) and always positive, of particles at position X ∈ ⊂ R d x , at time t > 0 and which move with velocity V ∈ R d v . We consider the general case in which we have d x = d v = d = 3 dimensions in space and velocity, namely position and velocity have the three following components:…”

Solving kinetic equations with adaptive mesh in phase space for rarefied gas dynamics and plasma physics (Invited)

“…Attempts to use adaptive mesh in velocity space for solving kinetic equations can be found in the recent literature [6]. AMR has been used to solve Vlasov equations with reduced numberof cells in phase space [ 7 ].…”

“…Here, we have developedaconservative scheme for solving kinetic equations with AMPSfollowing the idea of [6].We illustrate this approach for the BGK-type kinetic equation, which can be written in a semi-implicit form as:…”