Stability of finite-difference models containing two boundaries or interfaces

Trefethen, Lloyd N.

doi:10.1090/S0025-5718-1985-0804924-2

Stability of finite-difference models containing two boundaries or interfaces
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Math. Comp. 45 (1985), 279-300 Request permission

Abstract:

It is known that the stability of finite-difference models of hyperbolic initial-boundary value problems is connected with the propagation and reflection of parasitic waves. Here the waves point of view is applied to models containing two boundaries or interfaces, where repeated reflection of trapped wave packets is a potential new source of instability. Our analysis accounts for various known instability phenomena in a unified way and leads to several new results, three of which are as follows. (1) Dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface. (2) Algebraic "GKS instabilities" can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present. (3) "GKS-stability" and "P-stability" can be established in certain problems by showing that the numerical reflection coefficient matrices have norm less than one.

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