An explicit numerical stability check
A fictional numerical experiment solves y'=−20y with y(0)=1using explicit Euler. The exact solution decays, but one trial with step h=0.15alternates and grows in magnitude. A programmer suspects that every growing numerical sequence proves the physical model is unstable. Another proposes displaying more digits without changing the step. The exercise asks learners to derive the update factor and separate stability from accuracy and floating-point presentation. They should compare the problematic step with h=0.05and identify what still needs checking even if the smaller step satisfies the stability inequality. The test equation and parameters are supplied mathematical data, not an actual control-system operating recommendation.
1. Why is h=0.15unstable for decay?
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Euler gives y_next=(1−20h)y. At0.15the factor is−2, whose modulus exceeds1. The discrete values alternate and grow although the exact solution decays.
2. What does h=0.05establish and what does it not?
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The factor becomes0and satisfies the strict decay stability condition0<h<0.1. This alone does not establish a desired accuracy; a convergence and error check is still needed.