Institut für Allgemeine Elektrotechnik, Uni Rostock - Universität ...

Central Difference Quotient - Example
(absolute) largest eigenvalue (dimensionless):
± 2 j ± 2 jc0
9
b
4
λAmax
,
= = = ± j ⋅0.59958 ⋅10
µ
0ε
Δ
0Δ
e 1
b 3
⇒ maximal time step :
2 Δ
−9
b Δ tmax
= = = 3.33564 ⋅10
2
λ c
Amax ,
0
(Identical to experimentally found value)
Yet, CFL condition yields only the estimate
Δ
Δt max
≈
3 c0
which, in this case, is a limit too small by a factor of 3 ≈ 1.792
b 1
Prof. Dr. Ursula van Rienen, Universität Rostock, Fakultät für Informatik und Elektrotechnik (IEF), Institut für Allgemeine Elektrotechnik (IAE)
The largest absolute eigenvalue can be computed and is given above.
From that, the maximal time step can be computed as well.
It coincides with the value which was experimentally found.
Comparing that value with the estimate given by the CFL condition we see
that the CFL condition asks for a maximal time step which is smaller by a
factor of 1.792.
Generally, the CFL asks for too small limits but guarentees stabilty.
In realistic cases (much more than just 4 grid cells) the CFL stimate is very
close to the actual stability limit!
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