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Undergraduate · Euler stability

Euler stability: Moderate positive decay

Euler stability: investigate moderate positive decay with decay rate a = 1; step size h = 0.5.

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Euler stability: Moderate positive decay. Euler step: 0. Computed value: 1. Exact value: 1. Amplification |1−ah|: 0.5: decaysDiscrete steps versus exact decay010510computed amountTeal: Euler · dashed: exact decay at time nh
The equation is scalar linear decay with y₀=1 and constant positive a and h. Stable decay requires |1−ah|<1. This threshold is specific to explicit Euler on this equation and is not a universal time-step rule.

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Euler step
0
Computed value
1
Exact value
1
Amplification |1−ah|
0.5: decays

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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 = 1; Step size h = 0.5. 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.

  1. STEP 1

    Set up the mathematical model

    Compute ah and the amplification factor before running the discrete updates. The starting case is “Moderate positive decay.”

  2. 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.

  3. 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 = 1; Step size h = 0.5. Pause the timeline at 40%. Given euler step = 4, calculate computed value, exact value, amplification |1−ah|. Show the substitution into the displayed formula.

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Compare your explanation

One Euler update multiplies by 1−(1)(0.5)=0.5. After n=4 steps, yₙ=(0.5)^4=0.0625; the exact solution at time nh=2 is exp(−1(2))=0.135335. Results: Computed value: 0.063; Exact value: 0.135; Amplification |1−ah|: 0.5: decays. Decimal values are rounded; retain the original parameters when checking.

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Undergraduate · Calculus

Check whether a numerical decay method actually decays

Derive the step-size stability condition for forward Euler.