Numerical Analysis of Nonstationary Fluid Flow

This lecture serveys several aspects of the numerical approximation of the nonstationary Navier-Stokes equations. In particular, some essential criteria are discussed for choosing appropriate time stepping schemes. These are the “smoothing property” and the “global regularity property”. The importance of these principles for computing flows, at least in the range of moderate Reynolds-numbers, can be supported by a rigorous theoretical analysis. For flows with higher Reynolds-numbers, additional numerical damping is necessary.

[1]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

[2]  Roger Temam,et al.  On the theory and numerical analysis of the Navier-Stokes equations , 1973 .

[3]  R. Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization , 1982 .

[4]  O. Pironneau.  On the transport-diffusion algorithm and its applications to the Navier-Stokes equations , 1982 .

[5]  Eckart Gekeler.  Discretization Methods for Stable Initial Value Problems , 1984 .

[6]  R. Rannacher.  Finite element solution of diffusion problems with irregular data , 1984 .

[7]  R. Rannacher,et al.  An analysis of stability concepts for the Navier-Stokes equations. , 1986 .

[8]  Rolf Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem, part II: Stability of solutions and error estimates uniform in time , 1986 .

[9]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[10]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier—Stokes equations and the second law of thermodynamics , 1986 .

[11]  Jukka Saranen,et al.  Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations , 1986 .

[12]  R. Glowinski,et al.  Numerical methods for the navier-stokes equations. Applications to the simulation of compressible and incompressible viscous flows , 1987 .

[13]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems , 1987 .

[14]  Claes Johnson.  Numerical solution of partial differential equations by the finite element method , 1988 .

[15]  Rolf Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem, part III. Smoothing property and higher order error estimates for spatial discretization , 1988 .

[16]  R. Rannacher,et al.  Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization , 1990 .