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Numerical electromagnetic field evolution

Maxwell's equations describe continuous fields, but a computer stores a finite collection of samples. Numerical field evolution replaces spatial and temporal derivatives with discrete approximations, advances the samples, and measures how well the result satisfies the original equations.

This page explains the method without embedding an interactive simulation. The central example is a one-dimensional finite-difference time-domain update. It is simple enough to inspect line by line while retaining the essential coupling, staggering, stability, boundary, and conservation issues of larger electromagnetic solvers.

Reduce Maxwell's equations to one dimension

Consider a source-free plane wave traveling along x with electric field along y and magnetic field along z. The relevant Maxwell equations in a uniform medium are

\[\frac{\partial E_y}{\partial x} =-\frac{\partial B_z}{\partial t}\]

and

\[\frac{\partial B_z}{\partial x} =-\mu\varepsilon \frac{\partial E_y}{\partial t}\]

Equivalently, using \(H_z=B_z/\mu\),

\[\frac{\partial H_z}{\partial t} =-\frac{1}{\mu} \frac{\partial E_y}{\partial x}, \qquad \frac{\partial E_y}{\partial t} =-\frac{1}{\varepsilon} \frac{\partial H_z}{\partial x}\]

Each field's spatial change drives the other's time change. The signs depend on the stated component and propagation orientation; changing a component direction changes them.

Stagger fields in space and time

The Yee finite-difference idea places electric and magnetic samples at interleaved positions. In one dimension, electric samples can lie at integer grid indices and magnetic samples halfway between them. Their times are also staggered by half a step.

This arrangement lets a centered difference use neighboring samples naturally. It also avoids pretending that all quantities occupy the same point and time when they do not. A record should distinguish \(E_i^n\) from \(H_{i+1/2}^{n+1/2}\).

The magnetic update is

\[H_{i+1/2}^{n+1/2} =H_{i+1/2}^{n-1/2} -\frac{\Delta t}{\mu\Delta x} \left(E_{i+1}^n-E_i^n\right)\]

Then the electric update is

\[E_i^{n+1} =E_i^n -\frac{\Delta t}{\varepsilon\Delta x} \left(H_{i+1/2}^{n+1/2}-H_{i-1/2}^{n+1/2}\right)\]

The order is meaningful: the new magnetic half-step is used to advance the electric field to the next whole step.

import numpy as np

# Periodic 1D Maxwell system in units epsilon=mu=c=1.
N, length = 200, 1.0
dx = length / N
dt = 0.5 * dx
x = np.arange(N) * dx
E = np.sin(2 * np.pi * x)
B = np.sin(2 * np.pi * (x + dx / 2 + dt / 2))  # B at space half-steps, time -dt/2.
for _ in range(100):
    B -= dt / dx * (np.roll(E, -1) - E)
    E -= dt / dx * (B - np.roll(B, 1))
print("Electric field at x=0:", E[0])

Use separate arrays or a carefully documented time-staggered storage scheme. Traversal order must not alter the result.

Courant stability condition

The physical wave speed in a uniform medium is

\[v=\frac{1}{\sqrt{\mu\varepsilon}}\]

Define the one-dimensional Courant number

\[S=\frac{v\Delta t}{\Delta x}\]

For the basic one-dimensional explicit scheme, stability requires

\[S\le1\]

This says the physical wave must not cross more than one grid cell per time step in this scheme. The limit changes with dimension, material variation, and algorithm. In a three-dimensional uniform Cartesian grid, a common bound is

\[v\Delta t \sqrt{\frac{1}{\Delta x^2} +\frac{1}{\Delta y^2} +\frac{1}{\Delta z^2}} \le1\]

Meeting the bound prevents a major instability but does not guarantee accurate phase speed. A coarse stable grid can still propagate different wavelengths at incorrect speeds.

spacings, c = np.array([0.01, 0.02, 0.02]), 299792458.0
dt = 0.9 / (c * np.sqrt(np.sum(1 / spacings**2)))
assert c * dt * np.sqrt(np.sum(1 / spacings**2)) <= 1
print("Stable 3D time step (s):", dt)

Use the maximum wave speed present in the domain, which usually occurs where \(\mu\varepsilon\) is smallest.

Numerical dispersion and resolution

Finite differences do not reproduce the continuum dispersion relation exactly. The numerical phase speed depends on wavelength relative to grid spacing and, in multiple dimensions, propagation direction relative to the grid. Waves represented by only a few cells per wavelength can be badly distorted even while the calculation remains finite.

Choose a resolution based on an error study rather than one universal cells-per-wavelength rule. Compare the phase at a fixed physical probe after a fixed travel distance while refining the grid and time step together. A pulse also contains a range of wavelengths; its shortest important wavelength usually controls resolution.

A worked Python notebook

Choose the time step for a one-dimensional vacuum electromagnetic update. The question for this calculation is precise: how does stable 1D time step at Courant 0.5 change when we change grid spacing, 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.

Additional dimensions tighten the Courant condition; this expression is not a general three-dimensional limit. 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 0.001 to 0.05 m. 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 grid spacing, expressed in m. The output is stable 1D time step at Courant 0.5, expressed in s. 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.

\[f(q) = 1.66782047599076 \cdot 10^{-9} q\]

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 0.01 m, the expression gives 1.66782e-11 s. 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("0.5*q/299792458", locals={"q": q})
reference_input = 0.01
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted stable 1D time step at Courant 0.5:", reference_output, "s")

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 1.66782e-12 s. The midpoint, at 0.0255 m, gives 4.25294e-11 s. The upper endpoint gives 8.3391e-11 s. Read these together with the assumptions: they describe different choices of grid spacing 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([0.001, 0.025500000000000002, 0.05])
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} m -> {output_value:.6g} s")
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 grid spacing; the vertical axis shows stable 1D time step at Courant 0.5. 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(0.001, 0.05, 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("Grid spacing (m)")
ax.set_ylabel("Stable 1d time step at courant 0.5 (s)")
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 stable 1D time step at Courant 0.5 change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.01 m is 1.66782e-09 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.

\[f'(q) = 1.66782047599076 \cdot 10^{-9}\]

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 = 0.010245
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, "s")
print("Full expression:", full_prediction, "s")
print("Approximation difference:", full_prediction - linear_prediction, "s")

Here the changed input is 0.010245 m. The tangent estimate is 1.70868e-11 s, while the full expression gives 1.70868e-11 s. 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 = 0.049
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 grid spacing inside the stated interval, predict whether stable 1D time step at Courant 0.5 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: Additional dimensions tighten the Courant condition; this expression is not a general three-dimensional limit. 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 s: 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(0.05) - evaluate(0.001))
for points in [101, 201, 401]:
    grid = np.linspace(0.001, 0.05, 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 and initial conditions

A source can be imposed as a current density in Ampère–Maxwell's equation or as a carefully defined field injection. Abruptly turning on a sinusoid introduces a broad frequency spectrum. A smooth envelope reduces that startup transient but changes the source waveform and must be recorded.

Initial electric and magnetic arrays must satisfy the divergence constraints and describe compatible wave directions. An isolated electric pulse with zero magnetic field usually splits into waves traveling in both directions. To initialize one traveling wave, its magnetic field must have the correct amplitude, phase, and orientation.

Boundary conditions

The finite domain needs an edge policy:

No finite absorbing condition is perfect for every frequency and angle. A reflected wave can return after many steps, so short runs may conceal boundary error. Measure reflection at probes and include boundary energy flux in the ledger.

Material interfaces

At a material interface, permittivity, permeability, and conductivity can change. Maxwell boundary conditions determine which field components are continuous and which respond to surface charge or current. Placing material parameters at staggered locations requires an averaging rule. Different averages can affect accuracy near sharp interfaces.

For an ohmic conductor with conductivity \(\sigma\), current density \(\mathbf J=\sigma\mathbf E\) adds a loss term. Field energy becomes internal energy in the material. A falling field-energy total is physical only when the work term accounts for it.

Constraint and energy diagnostics

The update should report more than finite values. Check discrete Gauss laws,

\[r_E=\boldsymbol\nabla_h\cdot\mathbf E -\frac{\rho}{\varepsilon}, \qquad r_B=\boldsymbol\nabla_h\cdot\mathbf B\]

and compute stored field energy and boundary Poynting flux. In a source-free periodic domain, total energy should remain close to constant for an energy-compatible scheme. In an open domain, interior energy can decrease as outward flux carries it away.

Staggered fields may require interpolation before forming products such as \(\mathbf E\times\mathbf H\). State the interpolation and time level or the energy comparison cannot be reproduced.

Convergence study

Keep the physical domain, materials, sources, initial state, boundary locations, and final time fixed. Refine the grid. Reduce the time step to keep the Courant number fixed or otherwise controlled. Compare the same observables at the same physical coordinates and times.

for N in [100, 200, 400]:
    dx, duration = 1 / N, 0.25
    steps = N // 2
    dt = duration / steps
    x = np.arange(N) * dx
    E = np.sin(2 * np.pi * x)
    B = np.sin(2 * np.pi * (x + dx / 2 + dt / 2))
    for _ in range(steps):
        B -= dt / dx * (np.roll(E, -1) - E)
        E -= dt / dx * (B - np.roll(B, 1))
    error = np.max(abs(E - np.sin(2 * np.pi * (x - duration))))
    print(N, "maximum field error:", error)

Convergence toward a stable answer is necessary but does not prove that the physical model is complete. Two refined runs can agree while sharing an incorrect source, boundary, material law, or sign convention.

Failure reporting

Reject nonfinite material properties, nonpositive grid spacing, invalid time steps, and source arrays with incompatible dimensions. Stop if fields become nonfinite. Report the first location and time of failure along with the Courant number and largest field values. Do not rescale an exploding result to fit a color map and continue.

Model limits

The example assumes classical macroscopic Maxwell equations on a regular grid. Fine geometric features, curved conductors, point sources, dispersive media, nonlinear response, plasma kinetics, and multiscale structures require additional methods or material state. Very small scales can require quantum physics rather than a classical continuum model.

Numerical error has several sources: spatial truncation, time stepping, staircased geometry, source injection, material averaging, boundary reflection, interpolation, and roundoff. A responsible result reports enough configuration and diagnostics to separate these effects from genuine physical behavior.

Sources

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