LETTER TO THE EDITOR: There Is No Error in the Kleiser–Schumann Influence Matrix Method

✍ Scribed by Leonhard Kleiser; Carlos Härtel; Torsten Wintergerste

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 41 KB
• Volume: 141
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1998.5924

✦ Synopsis

In 1980 Kleiser and Schumann [1] introduced the influence matrix method for solving the coupled velocity-pressure equations of the Stokes problem which arises when the incompressible Navier-Stokes equations are discretized in time. The influence matrix method is based on a suitable decomposition of the desired solution into a sum of solutions of subproblems which can be solved efficiently in a sequential manner. The technique guarantees exact fulfillment of the continuity equation by employing the proper boundary conditions for the pressure Poisson equation, which are obtained in the solution algorithm by enforcing the divergence-free condition on the boundary. While the original paper [1] treated the case of plane channel flow, a more general formulation of the influence matrix method was given in [2]. Numerous authors have since been using this method, or variants thereof, as well as its obvious extension to the analogous problem of the treatment of the vorticity boundary condition when the streamfunction-vorticity formulation of the basic equations is used.

In two recent papers [3,4] the treatment of truncation errors in the original formulation of the influence matrix method was criticized. The claim was that the algorithm described in [1] contains some wrong boundary conditions which eventually lead to numerical solutions with significant divergence errors. It is the purpose of the present Note to clarify that the method given in [1] is indeed correct, and that the criticism in [3,4] originates from a misunderstanding of the original algorithm. To make the point of misunderstanding clear, we will discuss the key points in the following. For a more detailed description of the influencematrix technique, we refer the reader to the references cited and to the presentation given in Chapter 7.3.1. of the book by Canuto et al. [5].

Kleiser and Schumann [1] employed the influence matrix method within a spectral scheme developed for plane channel flow. A mixed explicit / implicit time discretization was utilized