Étude de méthodes d'intégration temporelle basées sur des schémas de corrections différées d'ordres élevés et A-stables

There are two main aspects of fluid numerical simulations: spatial discretization and temporal discretization. These two aspects go hand in hand and have an influence on the accuracy and the cost of a computation. We will focus our work on the temporal discretization and more specifically on the BDF – Backward Differentiation Formulae – methods because it allows stable computations for stiff problems. These problems are very sensitive to fluctuations of parameters and so can be unstable, Navier-Stokes problems correspond to this description. But the BDF methods are stable only for order one and two. So we build a new method that use a deferred correction to increase the order of convergence of the BDF. This correction is based on the truncation error of each BDF method. Thus we develop a method, named DC5/BDF1, where we successively correct a BDF1 method to get a stable method of order 5. By the same process we build a DC5/BDF2, also of order 5 but based on the BDF2. Besides we think these schemes in a way that it also works with alternate time-step. Indeed we open the possibility to improve our method with a adaptative time-step algorithm. We test our methods with simple ODE and system of ODE. We check that we reach the expected convergence order. And we discuss about the impact of the stiffness and the variability of the time-step. We also test with PDE to understand the influence of the boundary conditions and of the time-step on the result. We note that alternate time step and specific boundary conditions can lead to an order reduction. So we also develop a different way to implement the boundary conditions inspired by the Verwer correction. Previously we used Dirichlet boundary condition, now the idea is to impose a differential equation of order one on the boundary. It is possible to do it directly into the weak form of the problem using a Lagrange multiplier. Using the Lagrange multiplier has the double benefit to solve the order reduction and to provide more information during the computation. The Lagrange multiplier correspond to the heat flux in a heat and mass transfer problem and it gives us the sum of forces on a bound in a Navier-Stokes problem. This information is very useful in Fluid Structure Interaction problem because it will avoid the difficult calculation of forces in post treatment. To have a full set test we simulate a heat and mass transfer problem and Navier Stokes problem in two dimensions. It is the occasion to talk about the effect of Verwer's correction. Finally, to validate our scheme we simulate Von Karman vortex. For this particular situation, the DC5/BDF2 method will require the DC5/BDF1 method for the initialization. By comparing our result to others in the literature, we can validate our methods.