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Algebraic fractional-step schemes with spectral methods for the incompressible Navier-Stokes equations
No full textThe numerical investigation of a recent family of algebraic fractional-step methods for the solution of the incompressible
time-dependent Navier–Stokes equations is presented. These methods are improved versions of the Yosida
method proposed in [A. Quarteroni, F. Saleri, A. Veneziani, Factorization methods for the numerical approximation
of Navier–Stokes equations Comput. Methods Appl. Mech. Engrg. 188(1–3) (2000) 505–526; A. Quarteroni, F. Saleri,
A. Veneziani, J. Math. Pures Appl. (9), 78(5) (1999) 473–503] and one of them (the Yosida4 method) is proposed in this
paper for the first time. They rely on an approximate LU block factorization of the matrix obtained after the discretization
in time and space of the Navier–Stokes system, yielding a splitting in the velocity and pressure computation. In
this paper, we analyze the numerical performances of these schemes when the space discretization is carried out with a
spectral element method, with the aim of investigating the impact of the splitting on the global accuracy of the
computation
Algebraic fractional-step schemes for time-dependent incompressible Navier-Stokes equations.
No full textThe numerical investigation of a recent family of algebraic fractional-step methods (the so called Yosida methods) for the solution of the incompressible time-dependent Navier–Stokes equations is presented. A comparison with the Karniadakis–Israeli–Orszag method Karniadakis et al. (1991, J. Comput. Phys. 97, 414–443) is carried out. The high accuracy in time of these schemes well combines with the high accuracy in space of spectral methods
Numerische Mathematik
No full textVol. 1 --> pp. 368
Reihe: Springer-Lehrbuch
Übersetzung des 1. Teils der engl. Ausgabe "Numerical Mathematics", erschienen als TAM 37, SV New York, 2000.
2002, XIV, 368 S., Softcover
ISBN: 978-3-540-67878-6
Vol. 2 --> pp. 328
Übersetzung des 2. Teils der engl. Ausgabe "Numerical Mathematics", erschienen als TAM 37, SV New York, 2000.
2002, XIV, 328 S., Softcover
ISBN: 978-3-540-43616-
Factorization methods for the numerical approximation of Navier-Stokes equations
No full textWe investigate a general approach for the numerical approximation of incompressible Navier–Stokes equations based on splitting the original problem into successive subproblems cheaper to solve. The splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations. Several schemes based on approximate factorization are investigated. For some of these methods a formal analogy with well known time advancing schemes, such as the projection Chorin–Temam's, can be pointed out. Features and limits of this analogy (that was earlier introduced in B. Perot, J. Comp. Phys. 108 (1993) 51–58) are addressed. Other, new methods can also be formulated starting from this approach: in particular, we introduce here the so called Yosida method, which can be investigated in the framework of quasi-compressibility schemes. Numerical results illustrating the different performances of the different methods here addressed are presented for a couple of test cases
Méthodes numériques pour le Calcul ScientifiqueProgrammes en MATLAB
No full textLe livre a pour but de présenter les fondements théoriques et méthodologiques de l'analyse numérique. Une attention toute particulière est portée sur les concepts de stabilité, précision et complexité des algorithmes.Les méthodes modernes relatives aux thèmes suivants sont présentées et analysées en détails: résolution des systèmes linéaires et non linéaires, approximation polynômiale, optimisation, intégration numérique, polynôme orthogonaux, transformations rapides, équations différentielles ordinaires.Les techniques présentées sont illustrées par de nombreux tableaux et figures. Beaucoup d'exemples et de contre-exemples sont proposés pour permettre au lecteur de développer son sens critique.Une caractéristique principale du livre réside dans l'abondance des programmes MATLAB qui accompagnent toutes les méthodes numériques proposées et qui les illustrent par des applications concrètes. Le lecteur détient ainsi tous les outils pour acquérir de solides connaissances théoriques et les appliquer directement sur ordinateur.Cet ouvrage s'adresse aux étudiants du second cycle des universités, aux élèves des écoles d'ingénieurs, et plus généralement, à toutes les personnes qui pratiquent le calcul scientifique
Analysis of the Yosida method for the incompressible Navier–Stokes equations
No full textThe Yosida method was introduced in (Quarteroni et al., to appear) for the numerical approximation of the incompressible unsteady Navier–Stokes equations. From the algebraic viewpoint, it can be regarded as an inexact factorization of the matrix arising from the space and time discretization of the problem. However, its differential interpretation resides on an elliptic stabilization of the continuity equation through the Yosida regularization of the Laplacian (see (Brezis, 1983, Ciarlet and Lions, 1991)). The motivation of this method as well as an extensive numerical validation were given in (Quarteroni et al., to appear).
In this paper we carry out the analysis of this scheme. In particular, we consider a first-order time advancing unsplit method. In the case of the Stokes problem, we prove unconditional stability and moreover that the splitting error introduced by the Yosida scheme does not affect the overall accuracy of the solution, which remains linear with respect to the time step. Some numerical experiments, for both the Stokes and Navier–Stokes equations, are presented in order to substantiate our theoretical results
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