Conception and Analysis of Very High Order Residual Distribution Schemes. Application to Fluid Mechanics.

Numerical simulations are nowadays a major tool in aerodynamic design in aeronautic, automotive, naval industry, etc... One of the main challenges to push further the limits of the simulation codes is to increase their accuracy within a fixed set of resources (computational power and/or time). Two possible approaches to deal with this issue are either to construct discretization yielding, on a given mesh, very high order accurate solutions, or to construct compact, massively parallelizable schemes to minimize the computational time by means of a high performance parallel implementation. In this thesis, we try to combine both approaches by investigating the construction and implementation of very high order Residual Distribution Schemes (RDS) with the most possible compact stencil. The manuscript starts with a review of the mathematical theory of hyperbolic Conservation Laws (CLs). The aim of this initial part is to highlight the properties of the analytical solutions we are trying to approximate, in order to be able to link these properties with the ones of the sought discrete solutions. Next, we describe the three main steps toward the construction of a very high order RDS: \begin{itemize} \item The definition of higher order polynomial representations of the solution over polygons and polyhedra; \item The design of low order compact conservative RD schemes consistent with a given (high degree) polynomial representation. Among these, particular accent is put on the simplest, given by a generalization of the Lax-Friedrich's (LxF) scheme; \item The design of a positivity preserving nonlinear transformation, mapping first-order linear schemes onto nonlinear very high order schemes. \end{itemize} In the manuscript, we show formally that the schemes obtained following this procedure are consistent with the initial CL, that they are stable in $L^{\infty}$ norm, and that they have the proper truncation error. Even though all the theoretical developments are carried out for scalar CLs, remarks on the extension to systems are given whenever possible. Unfortunately, when employing the first order LxF scheme as a basis for the construction of the nonlinear discretization, the final nonlinear algebraic equation is not well-posed in general. In particular, for smoothly varying solutions one observes the appearance of high frequency spurious modes. In order to kill these modes, a streamline dissipation term is added to the scheme. The analytical implications of this modifications, as well as its practical computation, are thoroughly studied. Lastly, we focus on a correct discretization of the boundary conditions for the very high order RDS proposed. The theory is then extensively verified on a variety of scalar two dimensional test cases. Both triangular, and hybrid triangular-quadrilateral meshes are used to show the generality of the approach. The results obtained in these tests confirm all the theoretical expectations in terms of accuracy and stability and underline some advantages of the hybrid grids. Next, we consider solutions of the two dimensional Euler equations of gas dynamics. The results obtained are quite satisfactory and yet, we are not able to obtain the desired convergence rates on problems involving solid wall boundaries. Further investigation of this problem is under way. We then discuss the parallel implementation of the schemes, and analyze and illustrate the performance of this implementation on large three dimensional problems. Due to the preliminary character and the complexity of these three dimensional problems, a rather qualitative discussion is made for these tests cases: the overall behavior seems to be the correct one, but more work is necessary to assess the properties of the schemes in three dimensions. In the last chapter, we consider one possible extension to the Navier-Stokes equations in which the viscous terms are discretized by a standard Galerkin approach. We formally show that the overall discretization is consistent with the Navier-Stokes equations. However some accuracy issues are highlighted and discussed. The method is tested on a flat plate laminar boundary layer flow. The results are satisfactory. The work presented in this thesis allows a better understanding of the general properties of very high order RDS, and contributes substantially to bring forward a number of open issues for future improvement. These improvements should make RD discretizations a very appealing alternative to now classical high order and very high order FV ENO/WENO schemes, and to the increasingly popular class of Discontinuous Galerkin schemes.