Dynamic Multilevel Methods and Turbulence

Abstract

In this chapter, we describe numerical multilevel methods applied to the Navier–Stokes equations (homogeneous isotropic turbulence and channel flow problem), to the shallow water equations (geophysical problems), and to the renormalization of small eddies. The first difficulties with multilevel methods is to separate the scales. In this work, various cases are considered: the case of spectral methods, the case of finite difference methods, and the case of finite element methods. Then the numerical treatment is adapted to the size of the computed scales. The aim is to obtain new schemes, with better modeling properties and improved numerical stability. The spectral case is studied in detail, and the results of numerical simulations are reported in this case. Multilevel numerical methods are indispensable for the approximation of phenomena involving a large number of scales, such as turbulence. Although the methods described in this chapter are generally limited to two levels, they can, as explained in the text, be extended to a more important number of levels, thus extending significantly their performances.

Multilevel Preconditioning of Polynomial Chaos Method for Quantifying Uncertainties in a Blood Pump

“…Note that, the subscript k indicates the iteration number within the smoothing procedure, which is not the same for Newton step in Equation (11) and Equation (12). A pseudo algorithm about our stochastic Multilevel solver/preconditioner can be found in Algorithm 1.…”

“…By this setting, the local linear algebra structure remains the same, hence, only few values demand update. We employ the Variational Multi-Scale method [11,22], which inherits the consideration of two separated scales from the large eddy simulation (LES) model and the concept of stabilized finite element method [28], it provides the feasibility of modeling the blood flow within the high rotation speed instrument.…”