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

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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! 32
Properties of Explicite Leap Frog Scheme In realistic applications, the limit determined by the CFL condition is not too small but close to the stability limit. i The leap frog recursion scheme is suitable to compute loss-free systems. i The time step is limited for stability reasons. i The numerical effort per step is one matrix-vector-multiplication. i The integration with time steps fulfilling the CFL condition is sufficiently accurate. (Generally, spatial and time discretization error are equal.) Prof. Dr. Ursula van Rienen, Universität Rostock, Fakultät für Informatik und Elektrotechnik (IEF), Institut für Allgemeine Elektrotechnik (IAE) 33
Page 1 and 2: Limitations of the Leap-Frog Scheme
Page 3 and 4: TESLA Cavity Photo/graphic: DESY Ha
Page 9 and 10: Properties of the System Matrix (
Page 11 and 12: Properties of the System Matrix Gen
Page 13 and 14: Stability Stability is one of the m
Page 15 and 16: Stability of Some Recursion Methods
Page 17 and 18: Forward Difference Quotient Im ( λ
Page 19 and 20: Backward-Difference Quotient Im (
Page 21 and 22: Leap-Frog Update Equations ( m ( )
Page 23 and 24: Central Difference Quotient ⇒ y '
Page 25 and 26: Central Difference Quotient (Leap F
Page 31: Central Difference Quotient - Examp
Page 35 and 36: Example: Rectangular Waveguide Prof
Page 39 and 40: Example: Rectangular Waveguide - Ju
Page 41 and 42: Example: Rectangular Waveguide - Ju
Page 51 and 52: Example: Circular Waveguide TE11, b
Page 53 and 54: Example: Circular Waveguide TE21, b
Page 55 and 56: Waveguide Junction CST-Demo Port 1
Page 57 and 58: HUGO and RF-Waves of a WLAN-Card Ma
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Page 63 and 64: Parasitic Effects in IC‘s Current
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Page 75 and 76: Sim Prof.D MQS:M Fin Ri Dis 2 (c r.
Page 77: Sim Prof.D •M Moving- v Eis -joc

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