The exact solution of y′=−10y decays rapidly. Explicit Euler with h=0.3 multiplies each value by 1−3=−2, causing alternating growth instead. The local slope formula is consistent as h tends to zero, but this chosen step is unstable. Stability asks how repeated numerical updates amplify perturbations and modes, a different question from whether a one-step Taylor expansion has the right leading terms.
Predict
Predict what happens at h=0.2. The factor is −1, so numerical magnitudes do not decay at all. At h=0.1 the factor is zero, killing the mode immediately; this is stable but not an exact reproduction of exponential decay. A stable method can still have substantial approximation error. Likewise a result that stays bounded does not necessarily preserve positivity or oscillation phase.
Experiment
Compare methods for the same scalar decay equation at h=0.3.
Method
Amplification
Qualitative outcome
Explicit Euler
−2
Alternating growth
Implicit Euler
1/4
Positive decay
Implicit trapezoid
−1/5
Alternating decay
Compute the exact factor e^(−3) for comparison using a reference evaluator. The table diagnoses amplification, not an equal-cost accuracy contest; implicit steps require solving equations.
Understand
Apply a one-step method to y′=λy and write yₙ₊₁=R(z)yₙ with z=hλ. Its absolute-stability region usually means |R(z)|≤1; strict decay requires a strict inequality. Explicit Euler has R(z)=1+z, giving a disk centered at −1. For real λ<0, strict decay requires 0<h<2/|λ|. Implicit Euler solves yₙ₊₁=yₙ+hλyₙ₊₁, so R(z)=1/(1−z). If Re z≤0 then |1−z|≥1, proving A-stability: the whole closed left half-plane is nonamplifying. Its factor tends to zero for very negative real z, giving strong damping of stiff modes.
Apply
Implicit trapezoid has R(z)=(1+z/2)/(1−z/2), also A-stable because the denominator magnitude dominates when Re z≤0. But R(z) tends to −1 along the negative real axis, so very stiff components may persist as alternating numerical oscillations. This distinguishes A-stability from the stronger stiff-damping behavior associated with L-stability. For systems, eigenvalues give important information, but nonnormal transient growth and nonlinear dynamics can require more than independent scalar-mode reasoning. Stability regions are not universal guarantees for every nonlinear problem at arbitrary step size.
An oscillatory test also exposes a limitation. For λ=iω with real nonzero ω, exact solutions preserve magnitude. Explicit Euler has |1+ihω|=√(1+h²ω²)>1 for every positive h, so it introduces gradual amplitude growth. Implicit Euler damps that oscillation, while implicit trapezoid preserves magnitude on the imaginary axis. None of these statements alone controls phase error. Calculate one factor for hω=1 and distinguish amplitude fidelity from phase fidelity before selecting a method for periodic motion.
Continue
Stiffness often means fast stable dynamics impose explicit step restrictions far smaller than those needed to resolve the slow behavior of interest. Implicit methods can ease that restriction but introduce nonlinear solves, Jacobian work and their own tolerances. Accuracy, event resolution and invariant preservation still constrain the step. Next, convergence estimates combine local defects with stable propagation over a fixed interval, and adaptive controllers use error estimates while acknowledging that local acceptance does not automatically certify global accuracy.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For y′=−8y and explicit Euler h=1/4, find the amplification factor.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For the same equation and h, find implicit Euler’s factor.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Derive explicit Euler’s absolute-stability region for complex z=hλ.
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Prove implicit Euler is A-stable.
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Show implicit trapezoid is A-stable but does not strongly damp very negative real z.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain why stability can demand much smaller explicit steps than accuracy of the slow solution suggests.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
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