SCIENCE · INVESTIGATE AND EXPLAIN

Test numerical decay against passivity and exact time

An RC discharge with normalized initial voltage one satisfies dv/du=−v, where u=t/(RC). Its exact response e^(−u) remains positive and decreases, and normalized stored energy is v². A numerical update can violate those physical features without the underlying circuit becoming active.

Learning outcome: Compare h=0.5 with four steps and h=0.25 with eight steps, both at u=2. Retain signed voltage error and energy ratio. Then test h=1.5,2,2.5 to separate alternating decay, neutral alternation and growth; do not clip the rejected behavior.

Starting recap

Review charge continuity, definite integrals, first-order differential equations, initial conditions and signed source work. Distinguish displayed time from physical time.

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Try

At h=1.5 and four steps, final normalized time is 6. Explicit Euler gives (−0.5)⁴=0.0625 after alternating signs, backward Euler gives (1/2.5)⁴=0.0256, and the exact value is e^(−6)≈0.00247875. Both numerical endpoint values overestimate the true remaining voltage even though their magnitudes decrease.

Open the experiment

Predict

For explicit Euler with h=1.5, which statement is correct?

Your prediction and reason

Experiment

Compare h=0.5 with four steps and h=0.25 with eight steps, both at u=2. Retain signed voltage error and energy ratio. Then test h=1.5,2,2.5 to separate alternating decay, neutral alternation and growth; do not clip the rejected behavior.

Distinguish discrete stability from physical decay

Worked teaching model · synthetic values · Normalized RC discharge vprime=−v, v(0)=1, time in exact time constants. Compare explicit Euler factor 1−h, backward Euler factor 1/(1+h), and exact exp(−t). Algorithms are not new physical circuits. No hidden adaptive step or clipping of unstable values.

Step divided by time constant: 1.5 ratio Number of steps: 4 count

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Actual integration nodes versus the exact solution

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: explicit Euler nodes (solid).2: backward Euler nodes (dashed).3: exact exponential (dotted). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Actual integration nodes versus the exact solution

explicit Euler nodes

time divided by tau	normalized voltage
0	1
1.5	-0.5
3	0.25
4.5	-0.125
6	0.0625

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.

backward Euler nodes

time divided by tau	normalized voltage
0	1
1.5	0.4
3	0.16
4.5	0.064
6	0.0256

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 exponential

time divided by tau	normalized voltage
0	1
0.075	0.9277435
0.15	0.860708
0.225	0.7985162
0.3	0.7408182
0.375	0.6872893
0.45	0.6376282
0.525	0.5915554
0.6	0.5488116
0.675	0.5091564
0.75	0.4723666
0.825	0.438235
0.9	0.4065697
0.975	0.3771924
1.05	0.3499377
1.125	0.3246525
1.2	0.3011942
1.275	0.279431
1.35	0.2592403
1.425	0.2405085
1.5	0.2231302
1.575	0.2070076
1.65	0.1920499
1.725	0.1781731
1.8	0.1652989
1.875	0.153355
1.95	0.1422741
2.025	0.1319938
2.1	0.1224564
2.175	0.1136082
2.25	0.1053992
2.325	0.09778344
2.4	0.09071795
2.475	0.08416299
2.55	0.07808167
2.625	0.07243976
2.7	0.06720551
2.775	0.06234948
2.85	0.05784432
2.925	0.05366469
3	0.04978707
3.075	0.04618963
3.15	0.04285213
3.225	0.03975578
3.3	0.03688317
3.375	0.03421812
3.45	0.03174564
3.525	0.02945181
3.6	0.02732372
3.675	0.02534941
3.75	0.02351775
3.825	0.02181844
3.9	0.02024191
3.975	0.0187793
4.05	0.01742237
4.125	0.01616349
4.2	0.01499558
4.275	0.01391205
4.35	0.01290681
4.425	0.01197421
4.5	0.011109
4.575	0.0103063
4.65	0.009561602
4.725	0.008870714
4.8	0.008229747
4.875	0.007635094
4.95	0.007083409
5.025	0.006571586
5.1	0.006096747
5.175	0.005656217
5.25	0.005247518
5.325	0.004868351
5.4	0.004516581
5.475	0.004190229
5.55	0.003887457
5.625	0.003606563
5.7	0.003345965
5.775	0.003104198
5.85	0.002879899
5.925	0.002671808
6	0.002478752

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
Final normalized time	6	tau
Explicit Euler voltage	0.0625	ratio
Backward Euler voltage	0.0256	ratio
Exact voltage	0.002478752	ratio
Explicit signed error	0.06002125	ratio
Backward signed error	0.02312125	ratio
Explicit energy relative to initial	0.00390625	ratio
Exact energy relative to initial	6.14421e-6	ratio
Absolute explicit amplification below one	1	boolean

A stable algorithm can still oscillate or be inaccurate; passive physical decay does neither here.

Explicit Euler gives vn+1=(1−h)vn. Its magnitude decays for 0<h<2, but for 1<h<2 its sign alternates unphysically. At h=2 the sign alternates without decay; above 2 it grows. Backward Euler gives vn+1=vn/(1+h), positive and decaying for every positive h, yet it generally decays too slowly at coarse steps. Compare at identical physical final time before claiming convergence; stability alone is not accuracy.

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

A stable algorithm can still oscillate or be inaccurate; passive physical decay does neither here.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

State the current, source and measurement model

Derive the integral or initial-value solution

Explicit Euler uses the old slope: vn+1=vn−hvn=(1−h)vn. Magnitude decreases for h strictly between zero and two, but h above one produces sign alternation. At h=2 it alternates without magnitude decay; above two it grows. Backward Euler instead evaluates the slope at the new value: vn+1=vn−hvn+1, so vn+1=vn/(1+h). This algebra directly derives its positive amplification factor.

Test conservation, uncertainty and limits

Backward Euler is positive and decaying for every positive h here, but coarse steps decay too slowly. Stability is not accuracy, and energy decrease alone does not rule out unphysical voltage sign oscillation. Compare algorithms at the same final normalized time by changing the step count together with the step size; merely keeping the count fixed changes the physical question.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For backward Euler with h=0.5 and four steps, find normalized final voltage.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic evidence checked 31 October 2026 against the primary explanation. No live circuit construction, energized switching, component rating or electrical handling instruction is requested. Preserve first predictions, critique, revision and fresh assigned work separately from public practice. Software agreement does not certify measurement technique or learning.

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