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Numerical stability in trajectory calculations

A numerical trajectory replaces continuous equations of motion with a sequence of finite time updates. A method can be programmed correctly and still produce an unphysical result when the time step is too large, the problem is poorly scaled, or the method does not preserve important structure. Stability and convergence must be tested, not inferred from a smooth-looking path.

Turn Newton's equation into a state equation

For a particle of constant mass,

\[\frac{d\mathbf r}{dt}=\mathbf v,\qquad \frac{d\mathbf v}{dt}=\mathbf a(\mathbf r,\mathbf v,t)=\frac{\mathbf F(\mathbf r,\mathbf v,t)}{m}\]

Combining position and velocity into a state

\(\mathbf y=(\mathbf r,\mathbf v)\)

gives the general first-order form

\[\frac{d\mathbf y}{dt}=\mathbf f(t,\mathbf y)\]

The numerical method approximates the flow of this equation over a finite step \(h\). Position and velocity have different units, so a single raw error tolerance applied to every state component is rarely meaningful. Absolute and relative tolerances should be scaled per quantity.

Explicit Euler

The forward Euler method uses the derivative at the start of a step:

\[\mathbf y_{n+1}=\mathbf y_n+h\mathbf f(t_n,\mathbf y_n)\]

For mechanics this becomes

\[\mathbf r_{n+1}=\mathbf r_n+h\mathbf v_n,\qquad \mathbf v_{n+1}=\mathbf v_n+h\mathbf a_n\]

Both right-hand sides use the old state. Updating velocity in memory and then using the new value for position is a different method, often called semi-implicit or symplectic Euler.

import numpy as np


def forward_euler(state, dt):
    x, v = state
    return np.array([x + dt * v, v - dt * x])


print(forward_euler(np.array([1.0, 0.0]), 0.1))

Forward Euler has local truncation error proportional to \(h^2\) and global error proportional to \(h\) over a fixed interval under suitable smoothness and stability conditions. First-order accuracy does not guarantee stability for an arbitrary step.

An oscillator exposes instability

Use the ideal harmonic oscillator

\[\ddot x=-\omega^2x\]

with conserved energy

\[E=\frac12mv^2+\frac12m\omega^2x^2\]

The exact motion is bounded. Applying forward Euler gives

\[x_{n+1}=x_n+hv_n,\qquad v_{n+1}=v_n-h\omega^2x_n\]

Substituting into the energy shows

\[E_{n+1}=(1+h^2\omega^2)E_n\]

for this oscillator. Because the multiplier exceeds one for every nonzero step, energy grows repeatedly and the trajectory spirals outward in phase space. A smaller step slows the growth but does not turn forward Euler into an energy-conserving method over arbitrarily long times.

This is numerical instability, not physical resonance. The force law and initial state describe bounded motion; the update rule injects artificial energy.

Semi-implicit Euler

For acceleration depending on position, one common symplectic Euler ordering is

\[\mathbf v_{n+1}=\mathbf v_n+h\mathbf a(\mathbf r_n),\qquad \mathbf r_{n+1}=\mathbf r_n+h\mathbf v_{n+1}\] 
def semi_implicit_euler(state, dt):
    x, v = state
    new_v = v - dt * x
    return np.array([x + dt * new_v, new_v])


print(semi_implicit_euler(np.array([1.0, 0.0]), 0.1))

This method is still first-order accurate, but for many Hamiltonian systems it has better long-term qualitative behavior. Oscillator energy typically oscillates around a nearby value instead of drifting monotonically without bound when the step is within the method's stable regime. It does not conserve the exact physical energy at every step, and it is not automatically appropriate for strong damping or velocity-dependent forces.

Velocity Verlet

For acceleration depending on position, velocity Verlet uses acceleration at both ends:

\[\mathbf r_{n+1}=\mathbf r_n+h\mathbf v_n+\frac12h^2\mathbf a_n\]

then evaluates

\(\mathbf a_{n+1}=\mathbf a(\mathbf r_{n+1})\)

and updates

\[\mathbf v_{n+1}=\mathbf v_n+\frac h2(\mathbf a_n+\mathbf a_{n+1})\] 
def verlet(state, dt):
    x, v = state
    a = -x
    new_x = x + dt * v + 0.5 * dt**2 * a
    new_a = -new_x
    new_v = v + 0.5 * dt * (a + new_a)
    return np.array([new_x, new_v])


print(verlet(np.array([1.0, 0.0]), 0.1))

Velocity Verlet is second-order accurate and time-reversible for the intended class of problems. It is widely useful in conservative particle mechanics because its symplectic structure gives good long-term phase-space behavior. If acceleration depends on velocity, the simple form above may require modification or an implicit solve.

Fourth-order Runge-Kutta

Classical RK4 estimates the derivative at the start, two midpoints, and the end of a step:

\[\mathbf y_{n+1}=\mathbf y_n+\frac h6(\mathbf k_1+2\mathbf k_2+2\mathbf k_3+\mathbf k_4)\]

where the stage derivatives are evaluated from staged trial states. Written vertically:

def derivative(state):
    x, v = state
    return np.array([v, -x])


def rk4(state, dt):
    k1 = derivative(state)
    k2 = derivative(state + dt * k1 / 2)
    k3 = derivative(state + dt * k2 / 2)
    k4 = derivative(state + dt * k3)
    return state + dt * (k1 + 2 * k2 + 2 * k3 + k4) / 6


print(rk4(np.array([1.0, 0.0]), 0.1))

RK4 has fourth-order global accuracy for sufficiently smooth problems in its stable regime. It can be far more accurate than Euler over a moderate fixed interval. Standard RK4 is not symplectic, however, so very small short-term error does not guarantee ideal long-term conservation in orbital or oscillator calculations.

Stability is not the same as accuracy

Accuracy asks how close a numerical result is to the exact solution. Stability asks whether errors remain controlled as steps are repeated. A result may be stable but inaccurate, settling onto the wrong trajectory without exploding. It may also be highly accurate for a short interval yet accumulate secular drift over many periods.

For an oscillatory problem, the dimensionless product \(h\omega\) measures how much phase elapses per step. A step must resolve the fastest relevant time scale, not merely produce a visually smooth line between points.

Stiff systems contain very different time scales. An explicit method can require a tiny step for stability even when the slow behavior is the main interest. Implicit methods can have larger stability regions, but each step may require solving equations. No integrator is best for every force model and goal.

Step refinement and observed convergence

Run the same physical problem with step sizes \(h\), \(h/2\), and \(h/4\). Compare states at the same physical times. For a method of order \(p\), error in the asymptotic regime is expected to scale roughly as \(h^p\).

def solve(stepper, steps, duration=2.0):
    state = np.array([1.0, 0.0])
    for _ in range(steps):
        state = stepper(state, duration / steps)
    return state


exact = np.array([np.cos(2.0), -np.sin(2.0)])
for method in [forward_euler, semi_implicit_euler, verlet, rk4]:
    for steps in [20, 40, 80]:
        error = np.linalg.norm(solve(method, steps) - exact)
        print(method.__name__, steps, f"scaled state error: {error:.6g}")

If refinement does not reduce differences, possible causes include a coding error, unstable regime, discontinuity, inaccurate event handling, roundoff, or an unresolved spatial calculation inside the force function. Agreement between two coarse runs is not enough.

Conservation laws as diagnostics

For a closed conservative model, track energy, linear momentum, and angular momentum when the corresponding symmetries apply. Report relative drift using a meaningful reference scale. Dividing by an initial value near zero produces an unhelpful enormous ratio, so absolute error or another characteristic scale may be needed.

Exact numerical conservation of one quantity does not prove the trajectory is correct. A method could conserve energy while accumulating phase error. Conversely, dissipative or driven systems should not conserve mechanical energy; compare against the correct work or dissipation balance instead.

Event and boundary errors

Collision, landing, and switching events usually occur between grid times. Advancing past a wall and clamping the endpoint changes both event time and subsequent state. Locate the crossing within the step, apply the physical event rule, and advance through any remaining time. Refine both the integration step and event tolerance.

Adaptive methods estimate local error and change step size. They should not step across events unnoticed, and their accepted error estimate is not a guarantee of global physical accuracy.

Minimum validation record

A trajectory calculation becomes trustworthy through convergence evidence and physical diagnostics. Method names and smooth plots are not substitutes for those checks.

A worked Python notebook

For a unit-frequency oscillator, forward Euler multiplies the quadratic energy by a known factor each step. The question for this calculation is precise: how does Euler energy amplification per step change when we change time step, 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.

This factor describes the numerical method, not energy supplied by a physical force. 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 0.01 to 0.5 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 time step, expressed in s. The output is Euler energy amplification per step, expressed in dimensionless. 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.

\[f(q) = q^{2} + 1\]

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 0.1 s, the expression gives 1.01 dimensionless. 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("1+q**2", locals={"q": q})
reference_input = 0.1
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print(
    "Predicted Euler energy amplification per step:", reference_output, "dimensionless"
)

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 1.0001 dimensionless. The midpoint, at 0.255 s, gives 1.06502 dimensionless. The upper endpoint gives 1.25 dimensionless. Read these together with the assumptions: they describe different choices of time step 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([0.01, 0.255, 0.5])
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} dimensionless")
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 time step; the vertical axis shows Euler energy amplification per step. 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(0.01, 0.5, 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("Time step (s)")
ax.set_ylabel("Euler energy amplification per step (dimensionless)")
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 Euler energy amplification per step change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.1 s is 0.2 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.

\[f'(q) = 2 q\]

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 = 0.10245
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, "dimensionless")
print("Full expression:", full_prediction, "dimensionless")
print("Approximation difference:", full_prediction - linear_prediction, "dimensionless")

Here the changed input is 0.10245 s. The tangent estimate is 1.01049 dimensionless, while the full expression gives 1.0105 dimensionless. 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 = 0.49
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 time step inside the stated interval, predict whether Euler energy amplification per step 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: This factor describes the numerical method, not energy supplied by a physical force. 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 dimensionless: 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(0.5) - evaluate(0.01))
for points in [101, 201, 401]:
    grid = np.linspace(0.01, 0.5, 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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