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Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: An eigenvalue analysisA hyperbolic initial-boundary-value problem can be approximated by a system of ordinary differential equations (ODEs) by replacing the spatial derivatives by finite-difference approximations. The resulting system of ODEs is called a semidiscrete approximation. A complication is the fact that more boundary conditions are required for the spatially discrete approximation than are specified for the partial differential equation. Consequently, additional numerical boundary conditions are required and improper treatment of these additional conditions can lead to instability. For a linear initial-boundary-value problem (IBVP) with homogeneous analytical boundary conditions, the semidiscrete approximation results in a system of ODEs of the form du/dt = Au whose solution can be written as u(t) = exp(At)u(O). Lax-Richtmyer stability requires that the matrix norm of exp(At) be uniformly bounded for O less than or = t less than or = T independent of the spatial mesh size. Although the classical Lax-Richtmyer stability definition involves a conventional vector norm, there is no known algebraic test for the uniform boundedness of the matrix norm of exp(At) for hyperbolic IBVPs. An alternative but more complicated stability definition is used in the theory developed by Gustafsson, Kreiss, and Sundstrom (GKS). The two methods are compared.
Document ID
19880002052
Acquisition Source
Legacy CDMS
Document Type
Technical Memorandum (TM)
Authors
Warming, Robert F.
(NASA Ames Research Center Moffett Field, CA, United States)
Beam, Richard M.
(NASA Ames Research Center Moffett Field, CA, United States)
Date Acquired
September 5, 2013
Publication Date
September 1, 1986
Subject Category
Numerical Analysis
Report/Patent Number
NASA-TM-88328
A-86318
NAS 1.15:88328
Accession Number
88N11434
Funding Number(s)
PROJECT: RTOP 505-60-01
Distribution Limits
Public
Copyright
Work of the US Gov. Public Use Permitted.
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