{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Numerical stability in trajectory calculations\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/numerical-stability-trajectory-simulation). 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 numerical trajectory replaces continuous equations of motion with a sequence of\n", "finite time updates. A method can be programmed correctly and still produce an\n", "unphysical result when the time step is too large, the problem is poorly scaled, or the\n", "method does not preserve important structure. Stability and convergence must be tested,\n", "not inferred from a smooth-looking path.\n", "\n", "## Turn Newton's equation into a state equation\n", "\n", "For a particle of constant mass,\n", "\n", "\n", "$$\n", "\\frac{d\\mathbf r}{dt}=\\mathbf v,\\qquad \\frac{d\\mathbf v}{dt}=\\mathbf a(\\mathbf r,\\mathbf v,t)=\\frac{\\mathbf F(\\mathbf r,\\mathbf v,t)}{m}\n", "$$\n", "\n", "\n", "Combining position and velocity into a state\n", "$\\mathbf y=(\\mathbf r,\\mathbf v)$\n", "gives the general first-order form\n", "\n", "\n", "$$\n", "\\frac{d\\mathbf y}{dt}=\\mathbf f(t,\\mathbf y)\n", "$$\n", "\n", "\n", "The numerical method approximates the flow of this equation over a finite step\n", "$h$. Position and velocity have different\n", "units, so a single raw error tolerance applied to every state component is rarely\n", "meaningful. Absolute and relative tolerances should be scaled per quantity.\n", "\n", "## Explicit Euler\n", "\n", "The forward Euler method uses the derivative at the start of a step:\n", "\n", "\n", "$$\n", "\\mathbf y_{n+1}=\\mathbf y_n+h\\mathbf f(t_n,\\mathbf y_n)\n", "$$\n", "\n", "\n", "For mechanics this becomes\n", "\n", "\n", "$$\n", "\\mathbf r_{n+1}=\\mathbf r_n+h\\mathbf v_n,\\qquad \\mathbf v_{n+1}=\\mathbf v_n+h\\mathbf a_n\n", "$$\n", "\n", "\n", "Both right-hand sides use the old state. Updating velocity in memory and then using the\n", "new value for position is a different method, often called semi-implicit or symplectic\n", "Euler." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "\n", "def forward_euler(state, dt):\n", " x, v = state\n", " return np.array([x + dt * v, v - dt * x])\n", "\n", "\n", "print(forward_euler(np.array([1.0, 0.0]), 0.1))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Forward Euler has local truncation error proportional to\n", "$h^2$ and global error proportional\n", "to $h$ over a fixed interval under suitable\n", "smoothness and stability conditions. First-order accuracy does not guarantee stability\n", "for an arbitrary step.\n", "\n", "## An oscillator exposes instability\n", "\n", "Use the ideal harmonic oscillator\n", "\n", "\n", "$$\n", "\\ddot x=-\\omega^2x\n", "$$\n", "\n", "\n", "with conserved energy\n", "\n", "\n", "$$\n", "E=\\frac12mv^2+\\frac12m\\omega^2x^2\n", "$$\n", "\n", "\n", "The exact motion is bounded. Applying forward Euler gives\n", "\n", "\n", "$$\n", "x_{n+1}=x_n+hv_n,\\qquad v_{n+1}=v_n-h\\omega^2x_n\n", "$$\n", "\n", "\n", "Substituting into the energy shows\n", "\n", "\n", "$$\n", "E_{n+1}=(1+h^2\\omega^2)E_n\n", "$$\n", "\n", "\n", "for this oscillator. Because the multiplier exceeds one for every nonzero step, energy\n", "grows repeatedly and the trajectory spirals outward in phase space. A smaller step slows\n", "the growth but does not turn forward Euler into an energy-conserving method over\n", "arbitrarily long times.\n", "\n", "This is numerical instability, not physical resonance. The force law and initial state\n", "describe bounded motion; the update rule injects artificial energy.\n", "\n", "## Semi-implicit Euler\n", "\n", "For acceleration depending on position, one common symplectic Euler ordering is\n", "\n", "\n", "$$\n", "\\mathbf v_{n+1}=\\mathbf v_n+h\\mathbf a(\\mathbf r_n),\\qquad \\mathbf r_{n+1}=\\mathbf r_n+h\\mathbf v_{n+1}\n", "$$" ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "def semi_implicit_euler(state, dt):\n", " x, v = state\n", " new_v = v - dt * x\n", " return np.array([x + dt * new_v, new_v])\n", "\n", "\n", "print(semi_implicit_euler(np.array([1.0, 0.0]), 0.1))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "This method is still first-order accurate, but for many Hamiltonian systems it has\n", "better long-term qualitative behavior. Oscillator energy typically oscillates around a\n", "nearby value instead of drifting monotonically without bound when the step is within\n", "the method's stable regime. It does not conserve the exact physical energy at every\n", "step, and it is not automatically appropriate for strong damping or velocity-dependent\n", "forces.\n", "\n", "## Velocity Verlet\n", "\n", "For acceleration depending on position, velocity Verlet uses acceleration at both ends:\n", "\n", "\n", "$$\n", "\\mathbf r_{n+1}=\\mathbf r_n+h\\mathbf v_n+\\frac12h^2\\mathbf a_n\n", "$$\n", "\n", "\n", "then evaluates\n", "$\\mathbf a_{n+1}=\\mathbf a(\\mathbf r_{n+1})$\n", "and updates\n", "\n", "\n", "$$\n", "\\mathbf v_{n+1}=\\mathbf v_n+\\frac h2(\\mathbf a_n+\\mathbf a_{n+1})\n", "$$" ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "def verlet(state, dt):\n", " x, v = state\n", " a = -x\n", " new_x = x + dt * v + 0.5 * dt**2 * a\n", " new_a = -new_x\n", " new_v = v + 0.5 * dt * (a + new_a)\n", " return np.array([new_x, new_v])\n", "\n", "\n", "print(verlet(np.array([1.0, 0.0]), 0.1))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Velocity Verlet is second-order accurate and time-reversible for the intended class of\n", "problems. It is widely useful in conservative particle mechanics because its symplectic\n", "structure gives good long-term phase-space behavior. If acceleration depends on\n", "velocity, the simple form above may require modification or an implicit solve.\n", "\n", "## Fourth-order Runge-Kutta\n", "\n", "Classical RK4 estimates the derivative at the start, two midpoints, and the end of a\n", "step:\n", "\n", "\n", "$$\n", "\\mathbf y_{n+1}=\\mathbf y_n+\\frac h6(\\mathbf k_1+2\\mathbf k_2+2\\mathbf k_3+\\mathbf k_4)\n", "$$\n", "\n", "\n", "where the stage derivatives are evaluated from staged trial states. Written vertically:" ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "def derivative(state):\n", " x, v = state\n", " return np.array([v, -x])\n", "\n", "\n", "def rk4(state, dt):\n", " k1 = derivative(state)\n", " k2 = derivative(state + dt * k1 / 2)\n", " k3 = derivative(state + dt * k2 / 2)\n", " k4 = derivative(state + dt * k3)\n", " return state + dt * (k1 + 2 * k2 + 2 * k3 + k4) / 6\n", "\n", "\n", "print(rk4(np.array([1.0, 0.0]), 0.1))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "RK4 has fourth-order global accuracy for sufficiently smooth problems in its stable\n", "regime. It can be far more accurate than Euler over a moderate fixed interval. Standard\n", "RK4 is not symplectic, however, so very small short-term error does not guarantee ideal\n", "long-term conservation in orbital or oscillator calculations.\n", "\n", "## Stability is not the same as accuracy\n", "\n", "Accuracy asks how close a numerical result is to the exact solution. Stability asks\n", "whether errors remain controlled as steps are repeated. A result may be stable but\n", "inaccurate, settling onto the wrong trajectory without exploding. It may also be highly\n", "accurate for a short interval yet accumulate secular drift over many periods.\n", "\n", "For an oscillatory problem, the dimensionless product\n", "$h\\omega$ measures\n", "how much phase elapses per step. A step must resolve the fastest relevant time scale,\n", "not merely produce a visually smooth line between points.\n", "\n", "Stiff systems contain very different time scales. An explicit method can require a tiny\n", "step for stability even when the slow behavior is the main interest. Implicit methods\n", "can have larger stability regions, but each step may require solving equations. No\n", "integrator is best for every force model and goal.\n", "\n", "## Step refinement and observed convergence\n", "\n", "Run the same physical problem with step sizes\n", "$h$,\n", "$h/2$, and\n", "$h/4$. Compare states at the same physical\n", "times. For a method of order\n", "$p$, error in the asymptotic regime is\n", "expected to scale roughly as\n", "$h^p$." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "def solve(stepper, steps, duration=2.0):\n", " state = np.array([1.0, 0.0])\n", " for _ in range(steps):\n", " state = stepper(state, duration / steps)\n", " return state\n", "\n", "\n", "exact = np.array([np.cos(2.0), -np.sin(2.0)])\n", "for method in [forward_euler, semi_implicit_euler, verlet, rk4]:\n", " for steps in [20, 40, 80]:\n", " error = np.linalg.norm(solve(method, steps) - exact)\n", " print(method.__name__, steps, f\"scaled state error: {error:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "If refinement does not reduce differences, possible causes include a coding error,\n", "unstable regime, discontinuity, inaccurate event handling, roundoff, or an unresolved\n", "spatial calculation inside the force function. Agreement between two coarse runs is\n", "not enough.\n", "\n", "## Conservation laws as diagnostics\n", "\n", "For a closed conservative model, track energy, linear momentum, and angular momentum\n", "when the corresponding symmetries apply. Report relative drift using a meaningful\n", "reference scale. Dividing by an initial value near zero produces an unhelpful enormous\n", "ratio, so absolute error or another characteristic scale may be needed.\n", "\n", "Exact numerical conservation of one quantity does not prove the trajectory is correct.\n", "A method could conserve energy while accumulating phase error. Conversely, dissipative\n", "or driven systems should not conserve mechanical energy; compare against the correct\n", "work or dissipation balance instead.\n", "\n", "## Event and boundary errors\n", "\n", "Collision, landing, and switching events usually occur between grid times. Advancing\n", "past a wall and clamping the endpoint changes both event time and subsequent state.\n", "Locate the crossing within the step, apply the physical event rule, and advance through\n", "any remaining time. Refine both the integration step and event tolerance.\n", "\n", "Adaptive methods estimate local error and change step size. They should not step across\n", "events unnoticed, and their accepted error estimate is not a guarantee of global\n", "physical accuracy.\n", "\n", "## Minimum validation record\n", "\n", "- Compare against an exact solution where one exists.\n", "- Repeat with multiple smaller time steps at shared physical times.\n", "- Test initial conditions and zero-force constant-velocity motion.\n", "- Track applicable conserved quantities or known balance laws.\n", "- Report phase error as well as amplitude or position error for periodic motion.\n", "- Verify that output changes appropriately with integrator order.\n", "- Test collisions and boundaries independently of free-flight integration.\n", "- State tolerances with units and scales.\n", "- Reject nonpositive time steps and nonfinite state values.\n", "- Keep numerical error separate from uncertainty in the physical model.\n", "\n", "A trajectory calculation becomes trustworthy through convergence evidence and physical\n", "diagnostics. Method names and smooth plots are not substitutes for those checks.\n", "\n", "## A worked Python notebook\n", "\n", "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**\n", "change when we change **time step**, 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", "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\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", "0.01 to 0.5 s. 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 time step, expressed in s. The output is\n", "Euler energy amplification per step, expressed in dimensionless. 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 0.1 s, the expression gives 1.01 dimensionless. 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-013", "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(\"1+q**2\", locals={\"q\": q})\n", "reference_input = 0.1\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"s\")\n", "print(\n", " \"Predicted Euler energy amplification per step:\", reference_output, \"dimensionless\"\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "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", "1.0001 dimensionless. The midpoint, at 0.255 s, gives\n", "1.06502 dimensionless. The upper endpoint gives 1.25 dimensionless.\n", "Read these together with the assumptions: they describe different choices of time step\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-015", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.01, 0.255, 0.5])\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} s -> {output_value:.6g} dimensionless\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "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 time step; the vertical axis shows Euler energy amplification per step. 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-017", "metadata": {}, "source": [ "input_grid = np.linspace(0.01, 0.5, 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(\"Time step (s)\")\n", "ax.set_ylabel(\"Euler energy amplification per step (dimensionless)\")\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-018", "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", "Euler energy amplification per step change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.1 s is\n", "0.2 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-019", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 0.10245\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, \"dimensionless\")\n", "print(\"Full expression:\", full_prediction, \"dimensionless\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"dimensionless\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "Here the changed input is 0.10245 s. The tangent estimate is\n", "1.01049 dimensionless, while the full expression gives\n", "1.0105 dimensionless. 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-021", "metadata": {}, "source": [ "input_span = 0.49\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-022", "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", "time step inside the stated interval, predict whether Euler energy amplification per step 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: This factor describes the numerical method, not energy supplied by a physical force. 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 dimensionless:\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-023", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(0.5) - evaluate(0.01))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.01, 0.5, 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-024", "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\n", "\n", "- [Physics LibreTexts, \"Numerical Approaches\"](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Essential_Graduate_Physics_-_Classical_Mechanics_%28Likharev%29/05%3A_Oscillations/5.07%3A_Numerical_Approaches).\n", "- [OpenStax, *University Physics Volume 1*, Section 15.2, \"Energy in Simple Harmonic Motion\"](https://openstax.org/books/university-physics-volume-1/pages/15-2-energy-in-simple-harmonic-motion).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 23, \"Simple Harmonic Motion\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter23/).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 14, \"Conservation of Energy\"](https://open.umn.edu/opentextbooks/textbooks/light-and-matter)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "numerical-stability-trajectory-simulation.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/numerical-stability-trajectory-simulation", "source_sha256": "e404ce0609bbd7557b38629dea47cbf35ce895cb574cccc7607b619970a31a6c" } }, "nbformat": 4, "nbformat_minor": 5 }