{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Numerical electromagnetic field evolution\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/numerical-field-evolution). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "Maxwell's equations describe continuous fields, but a computer stores a finite collection\n", "of samples. Numerical field evolution replaces spatial and temporal derivatives with\n", "discrete approximations, advances the samples, and measures how well the result satisfies\n", "the original equations.\n", "\n", "This page explains the method without embedding an interactive simulation. The central\n", "example is a one-dimensional finite-difference time-domain update. It is simple enough to\n", "inspect line by line while retaining the essential coupling, staggering, stability,\n", "boundary, and conservation issues of larger electromagnetic solvers.\n", "\n", "## Reduce Maxwell's equations to one dimension\n", "\n", "Consider a source-free plane wave traveling along x with electric field along y and\n", "magnetic field along z. The relevant Maxwell equations in a uniform medium are\n", "\n", "\n", "$$\n", "\\frac{\\partial E_y}{\\partial x}\n", " =-\\frac{\\partial B_z}{\\partial t}\n", "$$\n", "\n", "\n", "and\n", "\n", "\n", "$$\n", "\\frac{\\partial B_z}{\\partial x}\n", " =-\\mu\\varepsilon\n", " \\frac{\\partial E_y}{\\partial t}\n", "$$\n", "\n", "\n", "Equivalently, using\n", "$H_z=B_z/\\mu$,\n", "\n", "\n", "$$\n", "\\frac{\\partial H_z}{\\partial t}\n", " =-\\frac{1}{\\mu}\n", " \\frac{\\partial E_y}{\\partial x},\n", " \\qquad\n", " \\frac{\\partial E_y}{\\partial t}\n", " =-\\frac{1}{\\varepsilon}\n", " \\frac{\\partial H_z}{\\partial x}\n", "$$\n", "\n", "\n", "Each field's spatial change drives the other's time change. The signs depend on the stated\n", "component and propagation orientation; changing a component direction changes them.\n", "\n", "## Stagger fields in space and time\n", "\n", "The Yee finite-difference idea places electric and magnetic samples at interleaved\n", "positions. In one dimension, electric samples can lie at integer grid indices and\n", "magnetic samples halfway between them. Their times are also staggered by half a step.\n", "\n", "This arrangement lets a centered difference use neighboring samples naturally. It also\n", "avoids pretending that all quantities occupy the same point and time when they do not.\n", "A record should distinguish\n", "$E_i^n$ from\n", "$H_{i+1/2}^{n+1/2}$.\n", "\n", "The magnetic update is\n", "\n", "\n", "$$\n", "H_{i+1/2}^{n+1/2}\n", " =H_{i+1/2}^{n-1/2}\n", " -\\frac{\\Delta t}{\\mu\\Delta x}\n", " \\left(E_{i+1}^n-E_i^n\\right)\n", "$$\n", "\n", "\n", "Then the electric update is\n", "\n", "\n", "$$\n", "E_i^{n+1}\n", " =E_i^n\n", " -\\frac{\\Delta t}{\\varepsilon\\Delta x}\n", " \\left(H_{i+1/2}^{n+1/2}-H_{i-1/2}^{n+1/2}\\right)\n", "$$\n", "\n", "\n", "The order is meaningful: the new magnetic half-step is used to advance the electric field\n", "to the next whole step." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "# Periodic 1D Maxwell system in units epsilon=mu=c=1.\n", "N, length = 200, 1.0\n", "dx = length / N\n", "dt = 0.5 * dx\n", "x = np.arange(N) * dx\n", "E = np.sin(2 * np.pi * x)\n", "B = np.sin(2 * np.pi * (x + dx / 2 + dt / 2)) # B at space half-steps, time -dt/2.\n", "for _ in range(100):\n", " B -= dt / dx * (np.roll(E, -1) - E)\n", " E -= dt / dx * (B - np.roll(B, 1))\n", "print(\"Electric field at x=0:\", E[0])\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Use separate arrays or a carefully documented time-staggered storage scheme. Traversal\n", "order must not alter the result.\n", "\n", "## Courant stability condition\n", "\n", "The physical wave speed in a uniform medium is\n", "\n", "\n", "$$\n", "v=\\frac{1}{\\sqrt{\\mu\\varepsilon}}\n", "$$\n", "\n", "Define the one-dimensional Courant number\n", "\n", "\n", "$$\n", "S=\\frac{v\\Delta t}{\\Delta x}\n", "$$\n", "\n", "\n", "For the basic one-dimensional explicit scheme, stability requires\n", "\n", "\n", "$$\n", "S\\le1\n", "$$\n", "\n", "\n", "This says the physical wave must not cross more than one grid cell per time step in this\n", "scheme. The limit changes with dimension, material variation, and algorithm. In a\n", "three-dimensional uniform Cartesian grid, a common bound is\n", "\n", "\n", "$$\n", "v\\Delta t\n", " \\sqrt{\\frac{1}{\\Delta x^2}\n", " +\\frac{1}{\\Delta y^2}\n", " +\\frac{1}{\\Delta z^2}}\n", " \\le1\n", "$$\n", "\n", "\n", "Meeting the bound prevents a major instability but does not guarantee accurate phase\n", "speed. A coarse stable grid can still propagate different wavelengths at incorrect\n", "speeds." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "spacings, c = np.array([0.01, 0.02, 0.02]), 299792458.0\n", "dt = 0.9 / (c * np.sqrt(np.sum(1 / spacings**2)))\n", "assert c * dt * np.sqrt(np.sum(1 / spacings**2)) <= 1\n", "print(\"Stable 3D time step (s):\", dt)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "Use the maximum wave speed present in the domain, which usually occurs where\n", "$\\mu\\varepsilon$ is\n", "smallest.\n", "\n", "## Numerical dispersion and resolution\n", "\n", "Finite differences do not reproduce the continuum dispersion relation exactly. The\n", "numerical phase speed depends on wavelength relative to grid spacing and, in multiple\n", "dimensions, propagation direction relative to the grid. Waves represented by only a few\n", "cells per wavelength can be badly distorted even while the calculation remains finite.\n", "\n", "Choose a resolution based on an error study rather than one universal cells-per-wavelength\n", "rule. Compare the phase at a fixed physical probe after a fixed travel distance while\n", "refining the grid and time step together. A pulse also contains a range of wavelengths;\n", "its shortest important wavelength usually controls resolution.\n", "\n", "## A worked Python notebook\n", "\n", "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**\n", "change when we change **grid spacing**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "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\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "0.001 to 0.05 m. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for grid spacing, expressed in m. The output is\n", "stable 1D time step at Courant 0.5, expressed in s. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = 1.66782047599076 \\cdot 10^{-9} q\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 0.01 m, the expression gives 1.66782e-11 s. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"0.5*q/299792458\", locals={\"q\": q})\n", "reference_input = 0.01\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted stable 1D time step at Courant 0.5:\", reference_output, \"s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "1.66782e-12 s. The midpoint, at 0.0255 m, gives\n", "4.25294e-11 s. The upper endpoint gives 8.3391e-11 s.\n", "Read these together with the assumptions: they describe different choices of grid spacing\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.001, 0.025500000000000002, 0.05])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} m -> {output_value:.6g} s\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows grid spacing; the vertical axis shows stable 1D time step at Courant 0.5. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "input_grid = np.linspace(0.001, 0.05, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Grid spacing (m)\")\n", "ax.set_ylabel(\"Stable 1d time step at courant 0.5 (s)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "stable 1D time step at Courant 0.5 change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.01 m is\n", "1.66782e-09 in those ratio units. Locally, the output increases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = 1.66782047599076 \\cdot 10^{-9}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 0.010245\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"s\")\n", "print(\"Full expression:\", full_prediction, \"s\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Here the changed input is 0.010245 m. The tangent estimate is\n", "1.70868e-11 s, while the full expression gives\n", "1.70868e-11 s. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "input_span = 0.049\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "grid spacing inside the stated interval, predict whether stable 1D time step at Courant 0.5 will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "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\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of s:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(0.05) - evaluate(0.001))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.001, 0.05, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources and initial conditions\n", "\n", "A source can be imposed as a current density in Ampère–Maxwell's equation or as a\n", "carefully defined field injection. Abruptly turning on a sinusoid introduces a broad\n", "frequency spectrum. A smooth envelope reduces that startup transient but changes the\n", "source waveform and must be recorded.\n", "\n", "Initial electric and magnetic arrays must satisfy the divergence constraints and describe\n", "compatible wave directions. An isolated electric pulse with zero magnetic field usually\n", "splits into waves traveling in both directions. To initialize one traveling wave, its\n", "magnetic field must have the correct amplitude, phase, and orientation.\n", "\n", "## Boundary conditions\n", "\n", "The finite domain needs an edge policy:\n", "\n", "- periodic boundaries make a repeating domain and return outgoing waves from the opposite side;\n", "- perfect-conductor boundaries reflect waves with component-dependent sign;\n", "- absorbing boundaries attempt to approximate an unbounded exterior;\n", "- prescribed boundaries inject or constrain fields.\n", "\n", "No finite absorbing condition is perfect for every frequency and angle. A reflected wave\n", "can return after many steps, so short runs may conceal boundary error. Measure reflection\n", "at probes and include boundary energy flux in the ledger.\n", "\n", "## Material interfaces\n", "\n", "At a material interface, permittivity, permeability, and conductivity can change. Maxwell\n", "boundary conditions determine which field components are continuous and which respond to\n", "surface charge or current. Placing material parameters at staggered locations requires an\n", "averaging rule. Different averages can affect accuracy near sharp interfaces.\n", "\n", "For an ohmic conductor with conductivity\n", "$\\sigma$, current density\n", "$\\mathbf J=\\sigma\\mathbf E$ adds\n", "a loss term. Field energy becomes internal energy in the material. A falling field-energy\n", "total is physical only when the work term accounts for it.\n", "\n", "## Constraint and energy diagnostics\n", "\n", "The update should report more than finite values. Check discrete Gauss laws,\n", "\n", "\n", "$$\n", "r_E=\\boldsymbol\\nabla_h\\cdot\\mathbf E\n", " -\\frac{\\rho}{\\varepsilon},\n", " \\qquad\n", " r_B=\\boldsymbol\\nabla_h\\cdot\\mathbf B\n", "$$\n", "\n", "\n", "and compute stored field energy and boundary Poynting flux. In a source-free periodic\n", "domain, total energy should remain close to constant for an energy-compatible scheme. In\n", "an open domain, interior energy can decrease as outward flux carries it away.\n", "\n", "Staggered fields may require interpolation before forming products such as\n", "$\\mathbf E\\times\\mathbf H$.\n", "State the interpolation and time level or the energy comparison cannot be reproduced.\n", "\n", "## Convergence study\n", "\n", "Keep the physical domain, materials, sources, initial state, boundary locations, and final\n", "time fixed. Refine the grid. Reduce the time step to keep the Courant number fixed or\n", "otherwise controlled. Compare the same observables at the same physical coordinates and\n", "times." ] }, { "cell_type": "code", "id": "code-019", "metadata": {}, "source": [ "for N in [100, 200, 400]:\n", " dx, duration = 1 / N, 0.25\n", " steps = N // 2\n", " dt = duration / steps\n", " x = np.arange(N) * dx\n", " E = np.sin(2 * np.pi * x)\n", " B = np.sin(2 * np.pi * (x + dx / 2 + dt / 2))\n", " for _ in range(steps):\n", " B -= dt / dx * (np.roll(E, -1) - E)\n", " E -= dt / dx * (B - np.roll(B, 1))\n", " error = np.max(abs(E - np.sin(2 * np.pi * (x - duration))))\n", " print(N, \"maximum field error:\", error)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "Convergence toward a stable answer is necessary but does not prove that the physical\n", "model is complete. Two refined runs can agree while sharing an incorrect source,\n", "boundary, material law, or sign convention.\n", "\n", "## Failure reporting\n", "\n", "Reject nonfinite material properties, nonpositive grid spacing, invalid time steps, and\n", "source arrays with incompatible dimensions. Stop if fields become nonfinite. Report the\n", "first location and time of failure along with the Courant number and largest field values.\n", "Do not rescale an exploding result to fit a color map and continue.\n", "\n", "## Model limits\n", "\n", "The example assumes classical macroscopic Maxwell equations on a regular grid. Fine\n", "geometric features, curved conductors, point sources, dispersive media, nonlinear response,\n", "plasma kinetics, and multiscale structures require additional methods or material state.\n", "Very small scales can require quantum physics rather than a classical continuum model.\n", "\n", "Numerical error has several sources: spatial truncation, time stepping, staircased\n", "geometry, source injection, material averaging, boundary reflection, interpolation, and\n", "roundoff. A responsible result reports enough configuration and diagnostics to separate\n", "these effects from genuine physical behavior.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 2*, Section 16.1, “Maxwell’s Equations and Electromagnetic Waves”](https://openstax.org/books/university-physics-volume-2/pages/16-1-maxwells-equations-and-electromagnetic-waves).\n", "- [OpenStax, *University Physics Volume 2*, Section 16.2, “Plane Electromagnetic Waves”](https://openstax.org/books/university-physics-volume-2/pages/16-2-plane-electromagnetic-waves).\n", "- [OpenStax, *University Physics Volume 2*, Section 16.3, “Energy Carried by Electromagnetic Waves”](https://openstax.org/books/university-physics-volume-2/pages/16-3-energy-carried-by-electromagnetic-waves).\n", "- [Benjamin Crowell, *Light and Matter*](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [Physics LibreTexts, “Electricity and Magnetism”](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "numerical-field-evolution.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/numerical-field-evolution", "source_sha256": "412cf90b8dbd600d4096ddf848928b911a9f3ee143babcc3da66dc24f46fdc5c" } }, "nbformat": 4, "nbformat_minor": 5 }