{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Fields as partial differential equations\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/fields-partial-differential-equations). 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": [ "A field assigns a quantity to every position and time. Temperature is a scalar field.\n", "Fluid velocity, electric field, and magnetic field are vector fields. A partial\n", "differential equation, or PDE, states how local spatial and temporal changes in those\n", "fields are related to sources and to one another.\n", "\n", "Maxwell's equations are field PDEs. They do not merely calculate a force after fields\n", "are known; they describe how charge and current source electromagnetic fields and how\n", "changing electric and magnetic fields generate each other. Initial data and boundary\n", "conditions are therefore part of a complete electromagnetic problem.\n", "\n", "## Scalar and vector fields\n", "\n", "A scalar field returns one value:\n", "\n", "\n", "$$\n", "u=u(x,y,z,t)\n", "$$\n", "\n", "\n", "A vector field returns components in a chosen basis:\n", "\n", "\n", "$$\n", "\\mathbf E(x,y,z,t)\n", " =E_x\\hat{\\mathbf x}\n", " +E_y\\hat{\\mathbf y}\n", " +E_z\\hat{\\mathbf z}\n", "$$\n", "\n", "\n", "The basis and coordinates matter. A vector is not merely three unrelated numbers, and a\n", "screen coordinate system with downward-positive vertical position must not silently\n", "replace a right-handed physical coordinate system.\n", "\n", "## Partial derivatives\n", "\n", "A partial derivative changes one independent variable while holding the others fixed.\n", "For example,\n", "\n", "\n", "$$\n", "\\frac{\\partial E_x}{\\partial t}\n", "$$\n", "\n", "\n", "measures the local time rate of change at a fixed position, while\n", "\n", "\n", "$$\n", "\\frac{\\partial E_x}{\\partial x}\n", "$$\n", "\n", "\n", "measures how that component changes from one nearby position to another at a fixed time.\n", "Confusing these derivatives can turn wave propagation into motion of sample points\n", "through a frozen field.\n", "\n", "## Gradient, divergence, and curl\n", "\n", "The gradient of a scalar field points in the direction of steepest increase:\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla u\n", " =\\left(\n", " \\frac{\\partial u}{\\partial x},\n", " \\frac{\\partial u}{\\partial y},\n", " \\frac{\\partial u}{\\partial z}\n", " \\right)\n", "$$\n", "\n", "\n", "Electrostatic potential and electric field are related by\n", "\n", "\n", "$$\n", "\\mathbf E=-\\boldsymbol\\nabla V\n", "$$\n", "\n", "\n", "Divergence measures local net outward flux per unit volume:\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla\\cdot\\mathbf E\n", " =\\frac{\\partial E_x}{\\partial x}\n", " +\\frac{\\partial E_y}{\\partial y}\n", " +\\frac{\\partial E_z}{\\partial z}\n", "$$\n", "\n", "\n", "Curl measures local circulation:\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla\\times\\mathbf E\n", " =\\begin{pmatrix}\n", " \\partial E_z/\\partial y-\\partial E_y/\\partial z\\\\\n", " \\partial E_x/\\partial z-\\partial E_z/\\partial x\\\\\n", " \\partial E_y/\\partial x-\\partial E_x/\\partial y\n", " \\end{pmatrix}\n", "$$\n", "\n", "\n", "These operators connect local differential equations with integral conservation laws\n", "through the divergence theorem and Stokes' theorem.\n", "\n", "## Maxwell's equations as local PDEs\n", "\n", "In vacuum with charge density $\\rho$\n", "and current density $\\mathbf J$, the\n", "differential equations are\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla\\cdot\\mathbf E\n", " =\\frac{\\rho}{\\varepsilon_0}\n", "$$\n", "\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla\\cdot\\mathbf B=0\n", "$$\n", "\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla\\times\\mathbf E\n", " =-\\frac{\\partial\\mathbf B}{\\partial t}\n", "$$\n", "\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla\\times\\mathbf B\n", " =\\mu_0\\mathbf J\n", " +\\mu_0\\varepsilon_0\n", " \\frac{\\partial\\mathbf E}{\\partial t}\n", "$$\n", "\n", "\n", "The two divergence equations constrain allowed field configurations. The two curl\n", "equations describe how fields change and circulate. They are coupled: a changing electric\n", "field contributes to magnetic curl, and a changing magnetic field contributes to electric\n", "curl.\n", "\n", "## A worked Python notebook\n", "\n", "Check one spatial derivative of a smooth field before assembling a multidimensional PDE update. The question for this calculation is precise: how does **scalar field along y=1**\n", "change when we change **horizontal coordinate**, 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", "The other spatial and time derivatives remain separate parts of the field equation. 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", "-2 to 2 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 horizontal coordinate, expressed in m. The output is\n", "scalar field along y=1, expressed in scaled field units. 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", "$$\n", "f(q) = q^{2} + 1\n", "$$\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 1 m, the expression gives 2 scaled field units. 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-003", "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(\"q**2+1\", locals={\"q\": q})\n", "reference_input = 1.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted scalar field along y=1:\", reference_output, \"scaled field units\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "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", "5 scaled field units. The midpoint, at 0 m, gives\n", "1 scaled field units. The upper endpoint gives 5 scaled field units.\n", "Read these together with the assumptions: they describe different choices of horizontal coordinate\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-005", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([-2.0, 0.0, 2.0])\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} scaled field units\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "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 horizontal coordinate; the vertical axis shows scalar field along y=1. 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-007", "metadata": {}, "source": [ "input_grid = np.linspace(-2.0, 2.0, 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(\"Horizontal coordinate (m)\")\n", "ax.set_ylabel(\"Scalar field along y=1 (scaled field units)\")\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-008", "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", "scalar field along y=1 change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 1 m is\n", "2 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", "$$\n", "f'(q) = 2 q\n", "$$\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-009", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 1.02\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, \"scaled field units\")\n", "print(\"Full expression:\", full_prediction, \"scaled field units\")\n", "print(\n", " \"Approximation difference:\",\n", " full_prediction - linear_prediction,\n", " \"scaled field units\",\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "Here the changed input is 1.02 m. The tangent estimate is\n", "2.04 scaled field units, while the full expression gives\n", "2.0404 scaled field units. 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-011", "metadata": {}, "source": [ "input_span = 4.0\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-012", "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", "horizontal coordinate inside the stated interval, predict whether scalar field along y=1 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: The other spatial and time derivatives remain separate parts of the field equation. 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 scaled field units:\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-013", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(2.0) - evaluate(-2.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(-2.0, 2.0, 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-014", "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 must conserve charge\n", "\n", "Taking the divergence of the Ampère–Maxwell equation and using the identity that the\n", "divergence of a curl is zero gives\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla\\cdot\\mathbf J\n", " +\\frac{\\partial\\rho}{\\partial t}=0\n", "$$\n", "\n", "\n", "This is local charge conservation. A prescribed current density and charge density\n", "cannot be chosen independently if they violate this equation. In computation, a field\n", "update may look smooth while inconsistent source data steadily create constraint error.\n", "\n", "## Initial and boundary conditions\n", "\n", "A PDE alone rarely specifies one unique solution. Initial electric and magnetic fields\n", "must satisfy the divergence constraints. Boundaries must state what happens at conductors,\n", "material interfaces, periodic edges, or an open exterior.\n", "\n", "At a perfect conductor in electrostatic equilibrium, the tangential electric field at\n", "the surface is zero. Across material interfaces, field components obey boundary relations\n", "derived from Maxwell's integral equations. An artificial outer boundary that simply sets\n", "fields to zero can reflect waves back into the domain. Absorbing or radiating boundary\n", "conditions are approximations designed to reduce that reflection.\n", "\n", "The selected boundary changes the physical problem. Periodic boundaries describe a\n", "repeating domain, not empty space. Fixed boundaries can represent driven or conducting\n", "surfaces only when their values match the intended model.\n", "\n", "## Finite differences\n", "\n", "A computer stores samples rather than a continuum. On a one-dimensional uniform grid,\n", "a centered first derivative is approximated by\n", "\n", "\n", "$$\n", "\\left.\\frac{\\partial u}{\\partial x}\\right|_i\n", " \\approx\\frac{u_{i+1}-u_{i-1}}{2\\Delta x}\n", "$$\n", "\n", "\n", "and a centered second derivative by\n", "\n", "\n", "$$\n", "\\left.\\frac{\\partial^2u}{\\partial x^2}\\right|_i\n", " \\approx\\frac{u_{i+1}-2u_i+u_{i-1}}{(\\Delta x)^2}\n", "$$\n", "\n", "\n", "These approximations become accurate as the grid is refined for a sufficiently smooth\n", "field, but they are not exact at finite spacing. Boundaries require separate stencils or\n", "ghost values, and sharp material interfaces need special care." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "import numpy as np\n", "\n", "coordinates = np.linspace(-1, 1, 101)\n", "x, y = np.meshgrid(coordinates, coordinates, indexing=\"ij\")\n", "field = x * x + y * y\n", "dx, dy = np.gradient(field, coordinates, coordinates, edge_order=2)\n", "assert np.allclose(dx, 2 * x)\n", "assert np.allclose(dy, 2 * y)\n", "print(\"Gradient at (0.5, 0.5):\", dx[75, 75], dy[75, 75])\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "All derivative calculations should read the same old array. Updating in place makes the\n", "answer depend on traversal order and silently changes the numerical method.\n", "\n", "## Time evolution and the Courant scale\n", "\n", "Wave equations propagate information at a finite speed. An explicit grid update must not\n", "take a time step so large that a wave crosses too many cells in one update. In a simple\n", "one-dimensional wave scheme, a typical stability condition is\n", "\n", "\n", "$$\n", "C=\\frac{c\\Delta t}{\\Delta x}\\le1\n", "$$\n", "\n", "\n", "The exact bound depends on dimension and discretization. In multiple dimensions the\n", "allowed step is usually smaller. Satisfying a stability condition prevents a common\n", "numerical blow-up but does not guarantee small phase error or correct boundary behavior." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "wave_speed, spacing, dt = 1.0, 0.01, 0.005\n", "courant = wave_speed * dt / spacing\n", "assert courant <= 1.0\n", "print(\"1D wave Courant number:\", courant)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "## Constraint versus evolution equations\n", "\n", "In electromagnetism, curl equations can advance fields through time, while divergence\n", "equations act as constraints. A discretization should preserve those constraints or\n", "control their error. If numerical divergence of magnetic field grows, the calculation\n", "has created an unphysical magnetic source. If electric divergence disagrees with charge\n", "density, Gauss's law has been violated.\n", "\n", "Define diagnostic residuals such as\n", "\n", "\n", "$$\n", "r_E=\\boldsymbol\\nabla_h\\cdot\\mathbf E\n", " -\\frac{\\rho}{\\varepsilon_0},\n", " \\qquad\n", " r_B=\\boldsymbol\\nabla_h\\cdot\\mathbf B\n", "$$\n", "\n", "\n", "and report norms and spatial locations, not only a pass/fail flag. A small global average\n", "can hide large positive and negative local errors.\n", "\n", "## Verification ladder\n", "\n", "Start with a constant field: every discrete spatial derivative should vanish away from\n", "boundaries. Use a linear scalar field to test that centered differences recover a constant\n", "gradient. Use a vector field with known divergence and curl to test component signs.\n", "Then compare a propagating wave with its analytic speed and phase.\n", "\n", "Refine space and time together while keeping the physical domain and output times fixed.\n", "Compare field components, conserved energy, and constraint residuals. Numerical\n", "convergence shows that the discrete method approaches some continuum solution; comparison\n", "with an analytic case or an independent formulation helps show it is the intended one.\n", "\n", "## Model limits\n", "\n", "A grid resolves neither features much smaller than its spacing nor time variation much\n", "faster than its step. Point charges and abrupt material interfaces can be singular or\n", "under-resolved. Open boundaries, curved conductors, dispersive media, and nonlinear\n", "materials need more sophisticated discretizations than the simple centered examples.\n", "\n", "Classical Maxwell equations also omit quantum electrodynamics and microscopic material\n", "structure. Within their domain, the equations are extraordinarily accurate, but a field\n", "calculation still inherits assumptions about sources, constitutive relations, boundaries,\n", "resolution, and numerical precision.\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", "- [Benjamin Crowell, *Light and Matter*](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [Ralph A. Llewellyn and Paul A. Tipler, *Modern Physics*](https://web.pdx.edu/~pmoeck/books/Tipler_Llewellyn.pdf).\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": "fields-partial-differential-equations.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/fields-partial-differential-equations", "source_sha256": "ca14e530593fcd84808dd06ea5d76bc11dcbb55556767530a76f75694451de5e" } }, "nbformat": 4, "nbformat_minor": 5 }