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stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: stationary modesSpatially discrete difference approximations for hyperbolic initial-boundary-value problems (IBVPs) require numerical boundary conditions in addition to the analytical boundary conditions specified for the differential equations. Improper treatment of a numerical boundary condition can cause instability of the discrete IBVP even though the approximation is stable for the pure initial-value or Cauchy problem. In the discrete IBVP stability literature there exists a small class of discrete approximations called borderline cases. For nondissipative approximations, borderline cases are unstable according to the theory of the Gustafsson, Kreiss, and Sundstrom (GKS) but they may be Lax-Richtmyer stable or unstable in the L sub 2 norm on a finite domain. It is shown that borderline approximation can be characterized by the presence of a stationary mode for the finite-domain problem. A stationary mode has the property that it does not decay with time and a nontrivial stationary mode leads to algebraic growth of the solution norm with mesh refinement. An analytical condition is given which makes it easy to detect a stationary mode; several examples of numerical boundary conditions are investigated corresponding to borderline cases.
Document ID
19880020071
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
July 1, 1988
Subject Category
NUMERICAL ANALYSIS
Report/Patent Number
NAS 1.15:100994
A-88165
NASA-TM-100994
Distribution Limits
Public
Copyright
Work of the US Gov. Public Use Permitted.

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