SCIENCE · INVESTIGATE AND EXPLAIN

Derive the unstable mode of an otherwise decaying equation

The exact equation dv/dt=−v/tau with tau1 s and v0=10 m/s has positive exponential solution10exp(−t). Forward Euler gives vn+1=vn−dt vn/tau=(1−r)vn, where r=dt/tau. Repeating a scalar multiplication makes its numerical behavior transparent.

Learning outcome: Compare r below1, between1 and2, at2 and above2. Derive the amplification inequalities and retain intermediate signs, endpoint error and exact decay.

Starting recap

Differentiate/integrate with explicit initial conditions; identify each force recipient and the inertial frame. Preserve force dimensions, geometric constraints and the distinction between model and solver error.

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At r2 and four steps, Euler returns10 m/s at t8 s rather than10e⁻⁸. At r2.5, each step multiplies the magnitude by1.5 and alternates sign.

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Predict

For forward Euler on this relaxation equation, r=2 gives:

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Experiment

Compare r below1, between1 and2, at2 and above2. Derive the amplification inequalities and retain intermediate signs, endpoint error and exact decay.

Derive numerical amplification rather than judging a smooth plot

Worked teaching model · synthetic values · Exact equation dv/dt=−v/tau with tau1 s and initial v10 m/s. Forward Euler has amplification1−dt/tau. Finite12-step workload intentionally includes oscillatory and unstable numerical results; sign changes are method errors, not new physical forces.

Step divided by relaxation time: 1.5 ratio Number of steps: 6 steps

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Numerical amplification against physical decay

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: Euler iterates (solid).2: exact relaxation (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Numerical amplification against physical decay

Euler iterates

time (s)	velocity (m/s)
0	10
1.5	-5
3	2.5
4.5	-1.25
6	0.625
7.5	-0.3125
9	0.15625

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

exact relaxation

time (s)	velocity (m/s)
0	10
0.1125	8.935973
0.225	7.985162
0.3375	7.13552
0.45	6.376282
0.5625	5.697828
0.675	5.091564
0.7875	4.549808
0.9	4.065697
1.0125	3.633096
1.125	3.246525
1.2375	2.901086
1.35	2.592403
1.4625	2.316564
1.575	2.070076
1.6875	1.849814
1.8	1.652989
1.9125	1.477106
2.025	1.319938
2.1375	1.179493
2.25	1.053992
2.3625	0.9418447
2.475	0.8416299
2.5875	0.7520782
2.7	0.6720551
2.8125	0.6005467
2.925	0.5366469
3.0375	0.4795463
3.15	0.4285213
3.2625	0.3829255
3.375	0.3421812
3.4875	0.3057722
3.6	0.2732372
3.7125	0.2441641
3.825	0.2181844
3.9375	0.194969
4.05	0.1742237
4.1625	0.1556859
4.275	0.1391205
4.3875	0.1243177
4.5	0.11109
4.6125	0.0992697
4.725	0.08870714
4.8375	0.07926846
4.95	0.07083409
5.0625	0.06329715
5.175	0.05656217
5.2875	0.0505438
5.4	0.04516581
5.5125	0.04036005
5.625	0.03606563
5.7375	0.03222815
5.85	0.02879899
5.9625	0.0257347
6.075	0.02299646
6.1875	0.02054958
6.3	0.01836305
6.4125	0.01640917
6.525	0.01466319
6.6375	0.01310299
6.75	0.0117088
6.8625	0.01046295
6.975	0.009349664
7.0875	0.008354835
7.2	0.007465858
7.3125	0.006671471
7.425	0.005961609
7.5375	0.005327278
7.65	0.004760441
7.7625	0.004253918
7.875	0.00380129
7.9875	0.003396822
8.1	0.003035391
8.2125	0.002712418
8.325	0.002423809
8.4375	0.00216591
8.55	0.001935451
8.6625	0.001729514
8.775	0.001545489
8.8875	0.001381045
9	0.001234098

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Euler amplification per step	-0.5	ratio
Numerical endpoint velocity	0.15625	m/s
Exact endpoint velocity	0.001234098	m/s
Endpoint signed error	0.1550159	m/s
Absolute amplification below 1	1	flag

The numerical sequence decays with alternating signs, unlike the positive exact solution.

Euler gives vn+1=(1−dt/tau)vn. Absolute amplification below 1 requires0<dt/tau<2; nonnegative monotone values require0<dt/tau≤1. At2 signs alternate without decay; above2 magnitudes grow. Stability is not accuracy: even a decaying numerical trajectory may poorly match the exact exponential.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

The numerical sequence decays with alternating signs, unlike the positive exact solution.

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Understand

Derive the unstable mode of an otherwise decaying equation

For0 < r < 1 the factor is positive and less than one, so values decay without alternating signs. At r1 the numerical value becomes zero after one step, whereas the exact solution is still positive. For1 < r < 2, values alternate signs while their magnitudes decay. The physical exponential does not reverse direction in either case.

At r2 the factor is−1 and the magnitude never decays. Above2 its magnitude exceeds one and the method grows an unphysical mode. Stability requires absolute amplification below one; accuracy requires a separate comparison with the exact solution. A stable but coarse grid can still be very inaccurate.

At r1.5 and six steps, Euler gives10(−.5)^6=.15625 m/s at t9 s, while exact velocity is10e⁻⁹≈.00123410 m/s. The positive final sign hides alternating intermediate signs. Inspecting only the endpoint or a smooth line connecting sparse values can conceal that error.

Changed case: predict before checking

At r2 and four steps, Euler returns10 m/s at t8 s rather than10e⁻⁸. At r2.5, each step multiplies the magnitude by1.5 and alternates sign.

Retained investigation

Compare r below1, between1 and2, at2 and above2. Derive the amplification inequalities and retain intermediate signs, endpoint error and exact decay. Why is numerical stability a necessary but insufficient accuracy check?

Sources and boundaries

Primary conceptual reference checked30 October2026: OpenStax University Physics Volume1. Original mathematical cases and synthetic records; no performed experiments, practical certification or actual equipment safety claims. Numerical method/error relations are derived explicitly for the supplied equations. Preserve independent checks, conditional regimes and first-response evidence. No physical release, launch, loading or pulley procedure is requested.

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Why is numerical stability a necessary but insufficient accuracy check?

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