Étude de méthodes d'intégration temporelle non linéaire, A-stable et d'ordre élévé à correction différée

In fluid mechanics, problems are often stiff, such as those related to the Navier-Stokes equations, making numerical calculations generally unstable. There are, however, a few stable time integration methods, such as the 1st and 2nd order BDF (Backward Differentiation Formulae). However, the accuracy of these methods is limited. This document therefore presents a methodology for progressively increasing the order of these two methods using deferred corrections, while retaining the advantage of their unconditional stability. The strategy adopted consists in successively substituting the dominant terms of the local truncation error in the initial expression of the BDF1 and BDF2 methods. The substituted term is then the correction, interpolated by a Newton polynomial based on the solution obtained from the lower-order method, assumed to be the most accurate. By recursively of their constructions, it is shown that the order p of these methods, called 〖DC〗_p, can be increased indefinitely, thus exceeding the theoretical limit of BDF methods. Indeed, beyond order 6, BDF methods are no longer A-stable, becoming unusable. Following this theoretical development, numerical tests were carried out progressively demonstrating the ability of these methods to solve ordinary differential equations (ODEs), 1D then 2D parabolic partial differential equations (PDEs) and Navier-Stokes equations. These numerical studies were carried out with methods up to order 5, sufficient to observe the general behaviour of these methods. This limitation in order also makes it possible to reduce the complexity of implementation as well as the computational cost. Indeed, the deferred correction of order p is interpolated from solutions of order p-1. The higher the order, the greater the need to retain previous solutions, thus increasing the computational cost. In order to limit the latter, we aim to build an algorithm with adaptive time steps to minimize the number of time steps. Prior to this, the robustness of the methods with respect to the evolution of time steps is verified. For PDEs, it turns out that the most delicate configuration occurs when the evolution of time steps is abrupt. This leads to accuracy losses for methods with the highest theoretical orders. These accuracy losses are even greater when the boundary conditions are time-dependent, as errors are concentrated at the edges. In this case, Verwer's method improves the accuracy of the methods, without perfectly resolving this loss-of-order phenomenon. Finally, the importance of problem-specific parameters in revealing or inhibiting this as yet misunderstood behaviour is revealed. However, it appears for an extreme configuration in which time steps are in alternating sequence. This is the only problematic configuration. By careful attention to the evolution of the time steps, their adaptation seems viable, thus limiting the computational cost of the DC/BDF methods.