The nonlinear Galerkin method in computational fluid dynamics
The numerical integration of the Navier-Stokes equations on large intervals of time yields new problems and new challenges with which we shall be faced in the coming years. Indeed the considerable increase in the computing power during the last years make it thinkable to solve these equations and similar ones in dynamically nontrivial situations, i.e., when the flow does not converge to a steady state as time goes to infinity. In such a case the flow remains permanently time-dependent even if the driving forces or velocities are time-independent. From the computational point of view we are faced with the problem of large time integration of the equations; hence we need algorithms which are well suited for large time integration, robust and economical. As an attempt to address these new computational problems we discuss here the nonlinear Galerkin method which corresponds to the projection of the equations under consideration on a nonlinear manifold close to the attractor. This method stems from the theory of inertial manifolds and approximate inertial manifolds which has proved to be a powerful tool of investigation for dynamical systems. In this paper, we intend to briefly review some results concerning the attractors for the Navier-Stokes equations and their approximations. Then we discuss the implementation of the nonlinear Galerkin method in the case of spectral methods. The implementation of the method in the case of finite elements or finite differences in relation with multigrid methods will be discussed elsewhere. We do think that this new class of algorithms is well adapted to the large time integration of dissipative evolution equations, in particular those arising in computational fluid dynamics.
scientific article; zbMATH DE number 4152036
scientific article