Institut für Allgemeine Elektrotechnik, Uni Rostock - Universität ...
Institut für Allgemeine Elektrotechnik, Uni Rostock - Universität ...
Institut für Allgemeine Elektrotechnik, Uni Rostock - Universität ...
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Central Difference Quotient (Leap Frog)<br />
To determine the eigenvalues of G we first study the<br />
eigenvalue equation of the system matrix A :<br />
<br />
⎛h<br />
⎞<br />
i<br />
Ayi = λA , i<br />
yi with yi<br />
=⎜ ⎜<br />
⎟<br />
⎝ei<br />
⎠<br />
<br />
<br />
→ A e = λ h and A h = λ e<br />
12 i A, i i 21 i A,<br />
i i<br />
<br />
⎛ h ⎞<br />
i<br />
Let y' i<br />
= ⎜ <br />
be an eigenvalue of .<br />
α ⎟<br />
G<br />
⎝ ei<br />
⎠<br />
Multiply inserting the last 2 equations yields<br />
<br />
⎛<br />
i ( 1 α t λA,<br />
i)<br />
⎞<br />
⎛ ⎞<br />
h + Δ ⋅<br />
hi<br />
G<br />
⎜ ⎟<br />
⎜ <br />
1<br />
2<br />
.<br />
−<br />
α ⎟<br />
=<br />
<br />
i α<br />
i ( 1+ α Δt⋅ λA , i<br />
+ ( Δt⋅λ<br />
⎝ e ⎠ ⎜ e<br />
A,<br />
i)<br />
) ⎟<br />
⎝<br />
⎠<br />
Prof. Dr. Ursula van Rienen, <strong>Uni</strong>versität <strong>Rostock</strong>, Fakultät <strong>für</strong> Informatik und <strong>Elektrotechnik</strong> (IEF), <strong>Institut</strong> <strong>für</strong> <strong>Allgemeine</strong> <strong>Elektrotechnik</strong> (IAE)<br />
22
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