8.12: Stability behavior of Euler’s method
We consider the so-called linear test equation
ẏ(t) = λy(t)
where λ ∈ C is a system parameter which mimics the eigenvalues of linear
systems of differential equations.
The equation is stable if Real(λ) ≤ 0. In this case the solution is
exponentially decaying. (limt→∞ y(t) = 0).
When is the numerically solution ui also decaying, limi→∞ ui = 0?
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8.12: Stability behavior of Euler’s method (Cont.)
Explicit Euler discretization of linear test equation:
ui+1 = ui + hλui
This gives ui+1 = (1 + hλ)i+1u0.
i
The solution is decaying (stable)
if |1 + hλ| ≤ 1
-2
hλ
-i
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8.13: Stability behavior of Euler’s method (Cont.)
Implicit Euler discretization of linear test equation:
ui+1 = ui + hλui+1
This gives ui+1 =
i+1
1
u0 .
1−hλ
The solution is decaying (stable)
if |1 − hλ| ≥ 1
i
-i
hλ
2
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8.14: Stability behavior of Euler’s method (Cont.)
Explicit Euler’s instability for fast decaying equations:
λ =-5
10
h=0.41
5
0
-5
-10
0
2
4
6
8
10
12
14
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8.15: Stability behavior of Euler’s method (Cont.)
Facit:
For stable ODEs with a fast decaying solution (Real(λ) << −1 )
or highly oscillatory modes (Im(λ) >> 1 )
the explicit Euler method demands small step sizes.
This makes the method inefficient for these so-called stiff systems.
Alternative: implicit Euler method.
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8.16: Implementation of implicit methods
Implicit Euler method ui+1 = ui + hif (ti+1, ui+1)
Two ways to solve for ui+1:
(k+1)
• Fixed point iteration: ui+1
k is the iteration counter, i the integration step counter
(k)
= ui + hif (ti+1, ui+1)
|
{z
}
(k)
=ϕ(ui+1 )
• Newton iteration:
ui+1 = ui + hif (ti+1, ui+1) ⇔ ui+1 − ui − hif (ti+1, ui+1) = 0
|
{z
}
=F (ui+1 )
(k)
(k)
F 0(ui+1)∆ui+1 = −F (ui+1)
(k+1)
ui+1
(k)
= ui+1 + ∆ui+1
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8.17: Implementation of implicit methods (Cont.)
These iterations are performed at every integration step!
They are started with explicit Euler method as so-called predictor:
(0)
ui+1 = ui + hif (ti, ui)
When should fixed points iteration and when Newton iteration be used?
The key is contractivity!
Let’s check the linear test equation again: ẏ = λy.
Contractivity: |ϕ0(u)| = |hλ| < 1.
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8.18: Implementation of implicit methods (Cont.)
If the differential equation is
• nonstiff: explicit Euler or
• nonstiff: implicit Euler with fixed point iteration
• stiff: implicit Euler with Newton iteration
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