Bulletin of the American Physical Society
76th Annual Meeting of the Division of Fluid Dynamics
Sunday–Tuesday, November 19–21, 2023; Washington, DC
Session L13: CFD: General I
8:00 AM–10:36 AM,
Monday, November 20, 2023
Room: 143C
Chair: Peter Brady, Los Alamos National Laboratory
Abstract: L13.00002 : A surprising failure of the classical von Neumann analysis*
8:13 AM–8:26 AM
Presenter:
Diego A Donzis
(Texas A&M University)
Authors:
Diego A Donzis
(Texas A&M University)
Komal Kumari
(Dassault Systemes)
to assess stability of numerical schemes for PDEs.
In this analysis, the so-called amplification factor,
defined as the ratio of the amplitude of the error at
two successive steps, must be bounded by unity in order
for the scheme to be stable.
Using a simple leap-frog scheme for the wave equation,
we show how the von Neumann analysis gives, surprisingly,
incorrect results. According to the analysis,
the amplification factor approaches a constant less than
or equal to unity when parameters are inside the stability
region. We show this is actually not the case.
The surprising failure of the approach, even for very simple cases,
can be traced to an implicit assumption in the
analysis of multi-step methods
which we investigate analytically. We show how and when
the von Neumann analysis gives correct results and when
alternative approaches are necessary.
We generalize the approach and provide an alternative
stability criterion that applies more broadly to
general multi-step methods.
*NSF Grant CBET-1605914
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