Undergraduate · Euler stability
Euler stability: Fast decay with a safe small step
Euler stability: investigate fast decay with a safe small step with decay rate a = 2; step size h = 0.2.
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An exact decaying differential equation can produce a growing numerical approximation when the time step is too large. Euler's amplification factor is 1−ah, and its magnitude determines stability for this test equation. Negative factors create alternating signs that the positive exact solution never has. This investigation starts with Decay rate a = 2; Step size h = 0.2. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.
A relationship to keep
y′=−ay; yₙ=(1−ah)ⁿ; exact y(nh)=e^(−anh)
Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.
- STEP 1
Set up the mathematical model
Compute ah and the amplification factor before running the discrete updates. The starting case is “Fast decay with a safe small step.”
- STEP 2
Follow the changing quantity
Compare ten Euler values with the exact exponential at the same times nh. Lines connect discrete samples only to guide the eye.
- STEP 3
Explain and test the result
Distinguish positive decay, alternating decay, a nondecaying boundary case, and unstable growth. Stability alone does not guarantee small approximation error.
Your turn to explain
Make a prediction. Test your reasoning.
Keep Decay rate a = 2; Step size h = 0.2. Pause the timeline at 40%. Given euler step = 4, calculate computed value, exact value, amplification |1−ah|. Show the substitution into the displayed formula.
Use the values specified in the question. Reset restores the initial values for this investigation.
Compare your explanation
One Euler update multiplies by 1−(2)(0.2)=0.6. After n=4 steps, yₙ=(0.6)^4=0.1296; the exact solution at time nh=0.8 is exp(−2(0.8))=0.201897. Results: Computed value: 0.13; Exact value: 0.202; Amplification |1−ah|: 0.6: decays. Decimal values are rounded; retain the original parameters when checking.
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