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What can and cannot be simulated

A simulation computes consequences of rules supplied by people. It can reveal behavior that is difficult to derive by hand, compare many initial conditions, and estimate a model's response where no convenient closed-form solution exists. It cannot decide that its governing assumptions are true, recover a mechanism that was omitted, or turn uncertain inputs into certain predictions.

Four layers that should not be confused

It helps to separate four things that are often called "the simulation":

  1. The physical system is the part of the world under study.
  2. The physical model selects relevant objects, properties, laws, and approximations.
  3. The numerical method approximates the model with finite arithmetic and finite steps.
  4. The program implements that method and presents selected outputs.

An error can enter at every layer. Ignoring air resistance is a modeling choice. Choosing a time step that is too large is a numerical choice. Updating the wrong array element is an implementation defect. Drawing a trajectory with a distorted axis scale is a presentation problem. A smooth curve on screen does not identify which layer is correct.

OpenStax describes a model as a representation that captures only some aspects of a physical system. It also emphasizes that calculations and computer simulations must be checked by observation and experiment. This is not a minor disclaimer; it defines what simulation results mean.

A concrete falling-body model

Suppose the state contains downward distance \(d\) and downward velocity \(v\). A simple model assumes a constant downward acceleration:

\[\frac{\mathrm{d}d}{\mathrm{d}t}=v,\qquad \frac{\mathrm{d}v}{\mathrm{d}t}=g\]

Given initial distance and velocity, these equations define future motion in the model. A numerical program might approximate them as follows:

def solve_model(time_step_seconds, end_time_seconds=2.0):
    distance_metres = 0.0
    velocity_metres_per_second = 0.0
    time_seconds = 0.0
    gravity = 9.81  # m/s^2; downward is positive
    if time_step_seconds <= 0:
        raise ValueError("The time step must be positive.")
    while time_seconds < end_time_seconds:
        dt = min(time_step_seconds, end_time_seconds - time_seconds)
        distance_metres += velocity_metres_per_second * dt
        velocity_metres_per_second += gravity * dt
        time_seconds += dt
    return distance_metres


print(f"Fall distance: {solve_model(0.1):.3f} m")

The program can answer: "What does this constant-acceleration model predict?" It cannot answer: "How will every real object fall?" A coffee filter and a steel ball have very different drag relative to their weight. Increasing the number of time steps does not add drag to an equation that omits it.

Verification: did we solve the stated model?

For release from rest, the constant-acceleration model has the analytic result

\[d_{\mathrm{exact}}(t)=\frac{1}{2}gt^2\]

This gives a reference for numerical error. At step size \(h\), define an error at a fixed comparison time as

\[e_h=\left|d_h-d_{\mathrm{exact}}\right|\]

Here \(d_h\) is the computed value. Verification asks whether the program correctly approximates the stated equations. A controlled refinement study keeps the physical inputs and final time fixed while reducing only the step size.

exact_distance_metres = 0.5 * 9.81 * 2.0**2
previous_error = float("inf")
for dt in [0.2, 0.1, 0.05, 0.025]:
    numerical_distance = solve_model(dt)
    error = abs(numerical_distance - exact_distance_metres)
    print(f"dt = {dt:.3f} s, distance = {numerical_distance:.4f} m")
    print(f"Absolute error = {error:.4f} m")
    assert error < previous_error
    previous_error = error

Passing this check is valuable, but limited. It can expose a wrong update order, a bad stopping condition, or a method outside its stable range. It cannot show that constant acceleration without drag is an adequate model of a particular real object.

When no analytic answer is available, verification can use simpler special cases, conservation laws, dimensional checks, manufactured solutions, or comparison with an independently implemented higher-accuracy method. Comparing a program only with itself at one resolution is weak evidence.

Validation: does the model represent the intended system?

Validation compares model predictions with independent observations in the conditions where the model is meant to apply. For the falling-body example, one might measure position at several times for a compact object over a modest height. Measurement uncertainty, release timing, local gravitational acceleration, and the effect of air would all belong in the comparison.

Using the same measurements both to tune a drag parameter and to announce successful validation exaggerates the evidence. A stronger procedure estimates parameters from one data set and evaluates predictions against another. Agreement in one regime does not validate extrapolation to very different speeds, sizes, or atmospheric conditions.

Verification and validation can therefore give different outcomes:

Both questions must be answered separately.

What a simulation can reveal

Within a declared scope, simulations are especially useful for systems with many coupled quantities, nonlinear feedback, irregular boundaries, or stochastic events. They can estimate trajectories, fields, probability distributions, sensitivities, and rates of change. They can compare hypotheses by changing one modeled mechanism at a time. They can also identify where a model predicts unstable or unexpected behavior that deserves analytic study or experiment.

A simulation may produce a result that was not obvious to its author, but the result is still a consequence of supplied rules and inputs. "Emergent" does not mean uncaused or independent of the model. The program can help discover implications of assumptions; it does not supply evidence that those assumptions describe nature.

What a simulation cannot supply

No amount of resolution can create missing measurements. If a material coefficient is unknown, the output should show sensitivity to plausible values or propagate a stated uncertainty. Picking one value silently creates false precision.

No amount of visual detail can establish microscopic truth. Many different models can produce similar macroscopic curves. A rendering is an encoding of output, and camera angle, color scale, interpolation, and sampling may hide or exaggerate features.

No finite run proves behavior for every possible input. Parameter sweeps sample a space; random simulations sample a distribution; neither exhausts an unbounded or continuous set. Rare events require enough samples and a justified probability model.

No simulation escapes computational limits. Finite memory restricts resolution, finite time restricts duration and sample count, and floating-point arithmetic restricts which numbers can be represented. Reporting runtime and resolution beside error helps readers judge the tradeoff rather than assuming that the most expensive run is the most valid.

What a credible result reports

A simulation result should travel with its question, equations, units, initial and boundary conditions, parameter sources, numerical method, resolution, software version, random seed policy when relevant, verification evidence, validation evidence, and known exclusions. It should distinguish measurement uncertainty, parameter uncertainty, model-form uncertainty, numerical truncation error, and floating-point effects instead of combining them into one vague caveat.

That record changes the claim from "the computer says this will happen" to "under these assumptions, inputs, and numerical checks, this model predicts this result." The second statement is narrower, but scientifically much stronger.

A worked Python notebook

A falling body provides an exact reference against which a numerical trajectory can be checked. The question for this calculation is precise: how does fall distance change when we change elapsed time, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

A match to this formula tests the solver but does not establish that drag is negligible in an experiment. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0.1 to 2 s. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for elapsed time, expressed in s. The output is fall distance, expressed in m. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 4.905 q^{2}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 1 s, the expression gives 4.905 m. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("9.81*q**2/2", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted fall distance:", reference_output, "m")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0.04905 m. The midpoint, at 1.05 s, gives 5.40776 m. The upper endpoint gives 19.62 m. Read these together with the assumptions: they describe different choices of elapsed time within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.1, 1.05, 2.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} s -> {output_value:.6g} m")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows elapsed time; the vertical axis shows fall distance. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.1, 2.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Elapsed time (s)")
ax.set_ylabel("Fall distance (m)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does fall distance change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 s is 9.81 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 9.81 q\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 1.0095
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 1.0095 s. The tangent estimate is 4.9982 m, while the full expression gives 4.99864 m. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 1.9
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of elapsed time inside the stated interval, predict whether fall distance will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: A match to this formula tests the solver but does not establish that drag is negligible in an experiment. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of m: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(2.0) - evaluate(0.1))
for points in [101, 201, 401]:
    grid = np.linspace(0.1, 2.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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