Numerical Treatment of the Navier-Stokes Equations: Proceedings of the Fifth GAMM-Seminar, Kiel, January 20–22, 1989(30 5 Notes on Numerical Fluid Mechanics and Multidisciplinary Design)

The most frequently used method for the numerical integration of parabolic differential equa- tions is the method of lines, where one first uses a discretization of space derivatives by finite differences or finite elements and then uses some time-stepping method for the the solution of resulting system of ordinary differential equations. Such methods are, at least conceptually, easy to perform. However, they can be expensive if steep gradients occur in the solution, stability must be controlled, and the global error control can be troublesome. This paper considers a simultaneaus discretization of space and time variables for a one-dimensional parabolic equation on a relatively long time interval, called 'time-slab'. The discretization is repeated or adjusted for following 'time-slabs' using continuous finite element approximations. In such a method we utilize the efficiency of finite elements by choosing a finite element mesh in the time-space domain where the finite element mesh has been adjusted to steep gradients of the solution both with respect to the space and the time variables. In this way we solve all the difficulties with the classical approach since stability, discretization error estimates and global error control are automatically satisfied. Such a method has been discussed previously in [3] and [4]. The related boundary value techniques or global time integration for systems of ordinary differential equations have been discussed in several papers, see [12] and the references quoted therein.

Table of Contents:
Extension of an Abstract Theory of Discretization Algorithms to Problems with only Weak and Non-Unique Solutions.- A Time-Space Finite Element Method for Nonlinear Convection Diffusion Problems.- Parallelization of Robust Multi-Grid Methods: ILU Factorization and Frequency Decomposition Method.- The Influence of Reentrant Corners in the Numerical Approximation of Viscous Flow Problems.- A Finite Discretization with Improved Accuracy for the Compressible Navier-Stokes Equations.- Calculation of Viscous Incompressible Flows in Time-Dependent Domains.- Two-Dimensional Wind Flow over Buildings.- Laminar Shock/Boundary-Layer Interaction — A Numerical Test Problem.- Comparison of Upwind and Central Finite-Difference Methods for the Compressible Navier-Stokes Equations.- A Comparison of Finite-Difference Approximations for the Stream Function Formulation of the Incompressible Navier-Stokes Equations.- NSFLEX — An Implicit Relaxation Method for the Navier-Stokes Equations for a Wide Range of Mach Numbers.- A Multigrid Algorithm for the Incompressible Navier-Stokes Equations.- Analysis and Application of a Line Solver for the Recirculating Flows Using Multigrid Methods.- A Posteriori Error Estimators and Adaptive Mesh-Refinement for a Mixed Finite Element Discretization of the Navier-Stokes Equations.- R-Transforming Smoothers for the Incompressible Navier-Stokes Equations.- List of Participants.