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Instability of difference models for hyperbolic initial boundary value problems
Author(s) -
Trefethen Lloyd N.
Publication year - 1984
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160370305
Subject(s) - mathematics , instability , mathematical analysis , boundary value problem , boundary (topology) , finite difference , reflection (computer programming) , hyperbolic partial differential equation , finite difference method , dimension (graph theory) , wave equation , space (punctuation) , partial differential equation , physics , pure mathematics , mechanics , linguistics , philosophy , computer science , programming language
A th00 eory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary. According to this theory, instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity C ≥ 0. To make this point of view precise, we first develop a rigorous description of group velocity for difference schemes and of reflection of waves at boundaries. From these results we then obtain lower bounds for growth rates of unstable finite difference solution operators in l 2 norms, which extend earlier results due to Osher and to Gustafsson, Kreiss, and Sundström. In particular we investigate l 2 ‐instability with respect to both initial and boundary data and show how they are affected by (a) finite versus infinite reflection coefficients and (b) wave radiation with C = 0 versus C > 0.