The Navier-Stokes equations governing the three-dimensional flows of viscous, compressible, heat-conducting gas and augmented by turbulence modeling present the most realistic model for gas flows around the elements of aircraft configurations. We study the stability of one of the Jameson's schemes of 1981, which approximates the set of five Navier-Stokes equations completed by the turbulence model of Baldwin and Lomax. The analysis procedure implements the check-up of the necessary von Neumann stability criterion. It is shown with the aid of the proposed symbolic-numeric strategy that the physical viscosity terms in the Navier-Stokes equations have a dominant effect on the sizes of the stability region in comparison with the heat conduction terms. It turns out that the consideration of turbulence with the aid of eddy viscosity model of Baldwin and Lomax has an insignificant effect on the size of the necessary stability region.

We present a new symbolic-numerical method for an automatic stability analysis of difference schemes approximating scalar linear or nonlinear partial differential equations (PDEs) of hyperbolic or parabolic type.In this method the grid values of the numerical solution for any fixed moment of time are considered aa random correlated variables obeying the normal distribution law.Therefore, one can apply the notion of the Shannon's entropy to characterize the stability of a difference scheme.The reduction of this entropy, or uncertainty, is taken as a stability criterion.It is shown at a number of examples that this criterion yields the same stability regions in the cases of linear difference initialvalue problems, as the Fourier method.In the case of two spatial variables the present probabilistic method is computationally by two orders of magnitude faster than the Fourier method.

Article Free Access Share on A new symbolic-numeric approach to stability analysis of difference schemes Authors: V. G. Ganzha View Profile , E. V. Vorozhtsov View Profile , J. A. van Hulzen View Profile Authors Info & Claims ISSAC '92: Papers from the international symposium on Symbolic and algebraic computationAugust 1992 Pages 9–15https://doi.org/10.1145/143242.143254Online:01 August 1992Publication History 2citation347DownloadsMetricsTotal Citations2Total Downloads347Last 12 Months1Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my Alerts New Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

We propose a number of symbolic-numeric approaches to the computer aided construction of the stability domains of difference schemes approximating the partial differential equations with constant coefficients. We use the Fourier method, the algebraic methods of the Routh-Hurwitz and Schur-Cohn theories for the localization of the polynomial zeros, the methods of the optimization theory as well as the means of computer algebra, digital image processing and computer graphics. The efficiency of the approaches is demonstrated at the practical examples of difference schemes for the fluid dynamics problems.