Section: Research Program

Numerical schemes for nonlinear kinetic models in an arbitrary geometry

In this part, we want to focus in the numerical approximation of solutions to kinetic equations (microscopic description) set in a complex geometry with different types of boundary conditions. Many numerical schemes have been proposed to approximate the solutions of nonlinear kinetic equations, but few of them are concerned by the treatment of complex geometry and boundary conditions which have a special interest for applications. In this context, classical structured or unstructured meshes already applied in computational fluid dynamics are not appropriate due to the high dimensional property of kinetic problems. In contrast, the Cartesian mesh makes the numerical method efficient and easy to implement. Indeed, in the framework of the Inria-Calvi project, E. Sonnendrücker and his collaborators have developed several families of methods for solving transport equations in a phase space grid with specific applications to plasma physics. These methods are based on the well known semi-Lagrangian methods. The principle is to solve the equation on a phase space grid, for which the grid points are advected with the flow of the transport equation for a time step and interpolated back periodically on the initial grid. The characteristics can be solved either forward or backward in time leading to the forward semi-Lagrangian or backward semi-Lagrangian schemes. These schemes are particularly well suited for uniform Cartesian grid since they are efficient in term of accuracy (high order scheme), stability (not restricted by a CFL condition) and computational cost (fast to locate the transported grid point).

Our aim is now to use all these techniques in the context of complex geometry and for the treatment of boundary conditions. The difficulty is that obviously grid points are usually not located on the physical boundary when using a Cartesian mesh, thus a suitable numerical method to capture the boundary condition on the complex geometry is required. In order to apply numerical methods previously studied, we want to treat separately the transport equation and the boundary conditions in the complex geometry.

Several numerical methods based on Cartesian mesh have been developed in computational fluid dynamics in last decade. Among these methods, the immersed boundary method (IBM), first introduced by Peskin for the study of biological fluid mechanics problems, has attracted considerable attention because of its use of regular Cartesian grid and great simplification of tedious grid generation task. The basic idea of immersed boundary method is that the effect of the immersed boundary on the surrounding fluid is represented through the introduction of forcing terms in the momentum equations. In conservation laws, two major classes immersed boundary like methods can be distinguished on different discretization types. The first class is Cartesian cut-cell method, which is based on a finite volume method. This conceptually simple approach “cuts” solid bodies out of a background Cartesian mesh. Thus we have several polygons (cut-cells) along the boundary. Then the numerical flux at the boundary of these cut-cells are imposed by using the real boundary conditions. This method satisfies well the conservation laws, however to determine the polygons is still a delicate issue.

Here, we will consider another class of method, based on finite difference method. To achieve a high order interior scheme, several ghost points behind the boundary are added. For instance for solving hyperbolic conservations laws, an inverse Lax-Wendroff type procedure is used to impose some artificial values on the ghost points. The interest of this approach is that it preserves all the flexibility of semi-Lagrangian schemes, that is, high order accuracy, resolution in a uniform Cartesian grid and stability. The members of the project involved in this thematic pole are already studying kinetic and related models and will develop this type of numerical schemes focusing on the following goals:

• Accuracy. Achieving arbitrary high accuracy for problems with smooth solutions has been a topic of the utmost importance in the recent years and actively studied by a many researchers and groups worldwide. The project team has been investigating such methods for several years and for various PDE models, for steady and unsteady physical problems, using different formulations among which spectral and semi-Lagrangian methods, discontinuous Galerkin methods and finite volume methods. In all the cases, we consider numerical methods relying on discretization techniques that best fit to the geometrical characteristics of the problems at hand.

• Robustness. On the other hand, these methods should also be capable to accurately describe the underlying physical phenomena that may involve highly variable space and time scales. With reference to this characteristic, several strategies are possible: adaptive local refinement/coarsening of the mesh (i.e h-adaptivity) and adaptive local variation of the interpolation order (i.e. p-adaptivity). Ideally, these two strategies are combined leading to the so-called hp-adaptive methods and that will actually represent an ultimate objective of our research activities. Note that both strategies are all local in nature.

• Efficiency. Despite the ever increasing performances of microprocessors, the numerical simulation of realistic $4D$ or $5D$ kinetic problems is hardly performed on a high-end workstation and parallel computing is a mandatory path. Hence, numerical methods must be adapted to the characteristics of modern parallel computing platforms taking into account their heterogeneity and their hierarchical nature (e.g. multiple processors and multiple core systems with complex cache and memory hierarchies, possibly augmented with accelerator cards). Appropriate parallelization strategies need to be designed that combine distributed memory and shared memory paradigms i.e. MIMD (multiple instruction, multiple data) and SIMD (single instruction, multiple data) programming models.