Explicit Methods for ODEs

// VM is the shared utility library, loaded globally via _includes/head-scripts.html (js/**). This page uses VM.numerical.eulerSolve, VM.numerical.linearRegression, VM.expressions.makeFunction2, VM.expressions.makeNumber, VM.plotting.paddedRange, VM.ui.renderTable, VM.numerical.rk4Solve.
VM = window.VM

Given an initial value problem \(y' = f(t, y)\), \(y(0) = y_0\), an explicit method advances the solution step by step: each new value \(y_{i+1} \approx y(t_{i+1})\) is computed directly from already-known values — no equation to solve at each step, unlike an implicit method.

Euler’s method uses the slope at the start of the step:

\[ y_{i+1} = y_i + h\, f(t_i, y_i). \]

Improved Euler (the explicit trapezoidal rule) uses Euler’s method as a predictor \(\tilde y_{i+1}\), then averages the slope at both endpoints of the step:

\[ \tilde y_{i+1} = y_i + h\, f(t_i, y_i), \qquad y_{i+1} = y_i + \frac{h}{2}\Big(f(t_i, y_i) + f(t_{i+1}, \tilde y_{i+1})\Big). \]

RK4 (the classical fourth-order Runge-Kutta method) averages four slope estimates across the step:

\[ \begin{aligned} k_1 &= f(t_i, y_i), \\ k_2 &= f\!\left(t_i + \tfrac{h}{2},\ y_i + \tfrac{h}{2}k_1\right), \\ k_3 &= f\!\left(t_i + \tfrac{h}{2},\ y_i + \tfrac{h}{2}k_2\right), \\ k_4 &= f(t_i + h,\ y_i + h k_3), \\ y_{i+1} &= y_i + \frac{h}{6}\big(k_1 + 2k_2 + 2k_3 + k_4\big), \end{aligned} \]

where \(h = b/n\) is the step size for \(n\) subintervals of \([0, b]\).

viewof fText = {
  const params = new URLSearchParams(window.location.search)
  const fTextInput = Inputs.text({label: "y' = f(t, y)", value: params.get("f") ?? "y - t^2 + 1", placeholder: "Example: y - t^2 + 1, t - y, y * (1 - y)"})
  fTextInput.classList.add("ojs-fill")
  return fTextInput
}

viewof y0 = {
  const params = new URLSearchParams(window.location.search)
  const y0Input = Inputs.text({label: "Initial condition y(0)", value: params.get("y0") ?? "0.5", placeholder: "Example: 0.5, 1/2"})
  return y0Input
}

viewof bEnd = {
  const params = new URLSearchParams(window.location.search)
  const bEndInput = Inputs.text({label: "Right endpoint b", value: params.get("b") ?? "2", placeholder: "Example: 2, pi"})
  return bEndInput
}

// Plot commits the current function/initial-condition/endpoint fields —
// edits above don't take effect until this is clicked, so result/mainPlot
// don't recompute on every keystroke. The n slider and the method
// checkboxes below are unaffected and stay reactive.
viewof plotTrigger = {
  const plotButton = Inputs.button("Plot", {value: 0, reduce: v => v + 1})
  plotButton.classList.add("ojs-auto")
  return plotButton
}

viewof nControl = {
  const nSlider = Inputs.range([1, maxN], {step: 1, value: Math.min(4, maxN), label: "n (subintervals)"})
  nSlider.classList.add("ojs-fill")
  return nSlider
}

viewof maxNInput = {
  const input = Inputs.text({value: "20", label: "Max", placeholder: "Example: 20, 5*4"})
  input.classList.add("ojs-auto")
  return input
}

// maxN parses the Max field's expression, falling back to 20 for invalid
// input so the n slider always gets a usable bound.
maxN = {
  const math = window.math
  const parsed = VM.expressions.makeNumber(math, maxNInput)
  if (parsed === null || !Number.isFinite(parsed)) return 20
  return Math.max(1, Math.round(parsed))
}

viewof methods = {
  const options = ["Euler's method", "Improved Euler", "RK4"]
  const checkboxInput = Inputs.checkbox(options, {value: options})
  checkboxInput.classList.add("ojs-fill")
  return checkboxInput
}

showEuler = methods.includes("Euler's method")

showImprovedEuler = methods.includes("Improved Euler")

showRK4 = methods.includes("RK4")

// Pressing Enter in the function/initial-condition/endpoint fields clicks
// Plot instead of submitting the field's own (invisible) form. Depends
// only on the stable views, so this wiring runs once and is never
// re-attached.
enterToPlot = {
  for (const view of [viewof fText, viewof y0, viewof bEnd]) {
    const input = view.querySelector("input")
    if (!input) continue
    input.addEventListener("keydown", (e) => {
      if (e.key !== "Enter") return
      e.preventDefault()
      viewof plotTrigger.querySelector("button").click()
    })
  }
}

// committed snapshots fText/y0/bEnd only when Plot is clicked, and writes
// them back into the URL so the address bar stays a shareable link. It
// reads `viewof fText`/`viewof y0`/`viewof bEnd` (the stable DOM views)
// rather than `fText`/`y0`/`bEnd` (the reactive values), so it does NOT
// rerun on every keystroke — only when `plotTrigger` changes. It still
// runs once on initial page load, so the first render uses the
// URL-provided (or default) values with no click.
committed = {
  plotTrigger
  const math = window.math
  const fVal = (viewof fText).value
  const y0Text = (viewof y0).value
  const bText = (viewof bEnd).value
  const y0Val = VM.expressions.makeNumber(math, y0Text)
  const bVal = VM.expressions.makeNumber(math, bText)

  const params = new URLSearchParams(window.location.search)
  params.set("f", fVal)
  params.set("y0", y0Text)
  params.set("b", bText)
  history.replaceState(null, "", `${window.location.pathname}?${params}`)

  return {fText: fVal, y0: y0Val, bEnd: bVal}
}

// improvedEulerSolve(f, t0, y0, tEnd, n) advances y' = f(t, y), y(t0) = y0
// with n steps of the improved Euler / explicit trapezoidal method: Euler's
// method as a predictor, then averages the slope at both endpoints of the
// step. Returns the full trajectory {ts, ys}.
improvedEulerSolve = (f, t0, y0, tEnd, n) => {
  const h = (tEnd - t0) / n
  const ts = [t0]
  const ys = [y0]
  let t = t0, y = y0
  for (let i = 0; i < n; i++) {
    const k1 = f(t, y)
    const predictor = y + h * k1
    const tNext = t0 + (i + 1) * h
    const k2 = f(tNext, predictor)
    y = y + (h / 2) * (k1 + k2)
    t = tNext
    ts.push(t)
    ys.push(y)
  }
  return {ts, ys}
}

// result: for the current committed function/initial condition/endpoint,
// builds the endpoint-value history used by the convergence plots and
// table.
//   rows — array of { n, eulerValue, improvedEulerValue, rk4Value, trueValue,
//     eulerError, improvedEulerError, rk4Error }, one entry per subinterval
//     count n = 1..maxN — the computed y(b) for each method against a
//     reference value.
//   f — parsed right-hand side, reused for the trajectories in setup
//   trueValue — reference y(b), shared by every row
result = {
  const math = window.math

  const f = VM.expressions.makeFunction2(math, committed.fText)
  const fallback = (t, y) => NaN

  if (!f) return {rows: [], f: fallback, trueValue: NaN}

  const t0 = 0
  const y0 = committed.y0
  const b = committed.bEnd
  if (!Number.isFinite(y0) || !Number.isFinite(b) || b === t0) return {rows: [], f, trueValue: NaN}

  // Reference value: RK4 with a very fine step count, far finer than any n
  // tested below.
  const refSteps = 2000
  const trueValue = VM.numerical.rk4Solve(f, t0, y0, b, refSteps).ys[refSteps]

  const rows = []
  let n = 1
  while (n <= maxN) {
    const eulerValue = VM.numerical.eulerSolve(f, t0, y0, b, n).ys[n]
    const improvedEulerValue = improvedEulerSolve(f, t0, y0, b, n).ys[n]
    const rk4Value = VM.numerical.rk4Solve(f, t0, y0, b, n).ys[n]

    const eulerError = Math.abs(eulerValue - trueValue)
    const improvedEulerError = Math.abs(improvedEulerValue - trueValue)
    const rk4Error = Math.abs(rk4Value - trueValue)

    rows.push({n, eulerValue, improvedEulerValue, rk4Value, trueValue, eulerError, improvedEulerError, rk4Error})
    n++
  }

  return {rows, f, trueValue}
}

// setup computes the data needed by the main chart at the current n: each
// method's step-by-step trajectory, and a fine reference trajectory for
// the smooth "true" solution curve.
setup = {
  if (result.rows.length === 0) {
    const empty = {lo: -3, hi: 3}
    return {n: 0, eulerTs: [], eulerYs: [], improvedEulerTs: [], improvedEulerYs: [], rk4Ts: [], rk4Ys: [], refTs: [], refYs: [], tRange: empty, yRange: {lo: -1, hi: 1}}
  }

  const n = Math.min(Math.max(Math.round(Number(nControl)), 1), result.rows.length)

  const t0 = 0
  const y0 = committed.y0
  const b = committed.bEnd

  const euler = VM.numerical.eulerSolve(result.f, t0, y0, b, n)
  const improvedEuler = improvedEulerSolve(result.f, t0, y0, b, n)
  const rk4 = VM.numerical.rk4Solve(result.f, t0, y0, b, n)

  // A fine reference trajectory for the smooth "true" solution curve.
  const reference = VM.numerical.rk4Solve(result.f, t0, y0, b, 500)

  const tRange = VM.plotting.paddedRange([t0, b], {emptyRange: [-3, 3], relativePadding: 0.1, minPadding: 0.25})

  // Cap at 1e8 so a diverging method doesn't blow up the axis range.
  const yValues = []
  for (const y of reference.ys) if (Number.isFinite(y) && Math.abs(y) <= 1e8) yValues.push(y)
  for (const y of euler.ys) if (Number.isFinite(y) && Math.abs(y) <= 1e8) yValues.push(y)
  for (const y of improvedEuler.ys) if (Number.isFinite(y) && Math.abs(y) <= 1e8) yValues.push(y)
  for (const y of rk4.ys) if (Number.isFinite(y) && Math.abs(y) <= 1e8) yValues.push(y)
  const yRange = VM.plotting.paddedRange(yValues, {emptyRange: [-1, 1], relativePadding: 0.25, minPadding: 0.5})

  return {
    n,
    eulerTs: euler.ts, eulerYs: euler.ys,
    improvedEulerTs: improvedEuler.ts, improvedEulerYs: improvedEuler.ys,
    rk4Ts: rk4.ts, rk4Ys: rk4.ys,
    refTs: reference.ts, refYs: reference.ys,
    tRange, yRange
  }
}

// mainPlot draws the fine reference solution curve and, depending on the
// checkboxes, the piecewise-linear trajectory computed by each method at
// the current n.
//
// _plotDiv is kept in a closure so Plotly reuses the same DOM node across
// reactive updates — this preserves zoom and pan when n or the checkboxes
// change.
mainPlot = {
  const Plotly = window.Plotly
  let _plotDiv = null

  return (data, showEuler, showImprovedEuler, showRK4) => {
    const traces = [
      // The fine reference solution curve
      {x: data.refTs, y: data.refYs, type: "scatter", mode: "lines", name: "Reference solution", showlegend: true, line: {color: "#2563eb", width: 3}}
    ]

    if (showEuler && data.eulerTs.length > 0) {
      traces.push({
        x: data.eulerTs, y: data.eulerYs, type: "scatter", mode: "lines+markers", name: "Euler's method", showlegend: true,
        line: {color: "#dc2626", width: 2}, marker: {color: "#dc2626", size: 6}
      })
    }
    if (showImprovedEuler && data.improvedEulerTs.length > 0) {
      traces.push({
        x: data.improvedEulerTs, y: data.improvedEulerYs, type: "scatter", mode: "lines+markers", name: "Improved Euler", showlegend: true,
        line: {color: "#f59e0b", width: 2}, marker: {color: "#f59e0b", size: 6}
      })
    }
    if (showRK4 && data.rk4Ts.length > 0) {
      traces.push({
        x: data.rk4Ts, y: data.rk4Ys, type: "scatter", mode: "lines+markers", name: "RK4", showlegend: true,
        line: {color: "#16a34a", width: 2}, marker: {color: "#16a34a", size: 6}
      })
    }

    const layout = {
      margin: {l: 0, r: 0, t: 0, b: 0},
      xaxis: {title: "t", range: [data.tRange.lo, data.tRange.hi], zeroline: true},
      yaxis: {title: "y", range: [data.yRange.lo, data.yRange.hi], zeroline: true},
      legend: {orientation: "h", y: -0.08},
      hovermode: "closest",
      uirevision: "static",  // keeps zoom/pan state across reactive updates
      autosize: true
    }
    const config = {responsive: true, displaylogo: false}

    // First render: create the div. Later renders: update in place (preserves zoom).
    if (!_plotDiv) {
      _plotDiv = document.createElement("div")
      _plotDiv.className = "plotly-box-large"
      Plotly.newPlot(_plotDiv, traces, layout, config)
      // newPlot fires before the div is in the DOM, so Plotly measures 0 width.
      // A ResizeObserver catches the real size once it's inserted and laid out,
      // and keeps correcting it on any later layout/viewport change.
      new ResizeObserver(() => Plotly.Plots.resize(_plotDiv)).observe(_plotDiv)
    } else {
      Plotly.react(_plotDiv, traces, layout, config)
    }
    return _plotDiv
  }
}

mainPlot(setup, showEuler, showImprovedEuler, showRK4)

NoteConvergence plots

// iteratesPlot draws each method's computed y(b) vs n for the entire run,
// against a reference line at the reference value of y(b) — it doesn't
// depend on the current step, so it's just plotted once per run.
iteratesPlot = {
  const eulerName = "Euler's method"
  const improvedEulerName = "Improved Euler"
  const rk4Name = "RK4"
  const trueName = "Reference value"

  const marks = [
    Plot.ruleY([result.trueValue], {stroke: () => trueName, strokeDasharray: "4,3"}),
    Plot.lineY(result.rows, {x: "n", y: "eulerValue", stroke: () => eulerName}),
    Plot.dot(result.rows, {x: "n", y: "eulerValue", fill: () => eulerName, r: 3}),
    Plot.lineY(result.rows, {x: "n", y: "improvedEulerValue", stroke: () => improvedEulerName}),
    Plot.dot(result.rows, {x: "n", y: "improvedEulerValue", fill: () => improvedEulerName, r: 3}),
    Plot.lineY(result.rows, {x: "n", y: "rk4Value", stroke: () => rk4Name}),
    Plot.dot(result.rows, {x: "n", y: "rk4Value", fill: () => rk4Name, r: 3})
  ]
  const colorDomain = [eulerName, improvedEulerName, rk4Name, trueName]
  const colorRange = ["#dc2626", "#f59e0b", "#16a34a", "#111827"]

  const chart = Plot.plot({
    width: 340, height: 300, marginLeft: 46,
    x: {label: "n"},
    y: {label: "y(b)"},
    color: {domain: colorDomain, range: colorRange},
    marks
  })
  const legend = Plot.legend({color: {domain: colorDomain, range: colorRange}})

  const container = document.createElement("div")
  container.append(chart, legend)
  return container
}
// convergencePlot draws log₁₀|error| vs log₁₀ n for all three methods, with
// a linear regression line for each. The slope of each line estimates the
// order of convergence of that method (negated, since error shrinks as n
// grows) — for the entire run, so it's just plotted once per run.
convergencePlot = {

  const eulerPts = []
  const improvedEulerPts = []
  const rk4Pts = []
  for (const row of result.rows) {
    if (Number.isFinite(row.eulerError) && row.eulerError > 0) {
      eulerPts.push({x: Math.log10(row.n), y: Math.log10(row.eulerError)})
    }
    if (Number.isFinite(row.improvedEulerError) && row.improvedEulerError > 0) {
      improvedEulerPts.push({x: Math.log10(row.n), y: Math.log10(row.improvedEulerError)})
    }
    if (Number.isFinite(row.rk4Error) && row.rk4Error > 0) {
      rk4Pts.push({x: Math.log10(row.n), y: Math.log10(row.rk4Error)})
    }
  }

  const eulerName = "Euler error"
  const improvedEulerName = "Improved Euler error"
  const rk4Name = "RK4 error"
  const marks = [
    Plot.dot(eulerPts, {x: "x", y: "y", fill: () => eulerName, r: 4}),
    Plot.dot(improvedEulerPts, {x: "x", y: "y", fill: () => improvedEulerName, r: 4}),
    Plot.dot(rk4Pts, {x: "x", y: "y", fill: () => rk4Name, r: 4})
  ]
  const colorDomain = [eulerName, improvedEulerName, rk4Name]
  const colorRange = ["#dc2626", "#f59e0b", "#16a34a"]
  // legendDomain/legendRange hold only the regression lines — the legend
  // shown below the chart displays just the fitted orders, not the raw
  // point series, to keep it uncluttered.
  const legendDomain = [], legendRange = []

  const eulerFit = VM.numerical.linearRegression(eulerPts)
  if (eulerFit) {
    const lineName = `Euler order ≈ ${(-eulerFit.slope).toFixed(2)}`
    marks.push(Plot.line(
      [{x: eulerFit.xlo, y: eulerFit.slope * eulerFit.xlo + eulerFit.intercept}, {x: eulerFit.xhi, y: eulerFit.slope * eulerFit.xhi + eulerFit.intercept}],
      {x: "x", y: "y", stroke: () => lineName, strokeDasharray: "6,4"}
    ))
    colorDomain.push(lineName)
    colorRange.push("#f87171")
    legendDomain.push(lineName)
    legendRange.push("#f87171")
  }

  const improvedEulerFit = VM.numerical.linearRegression(improvedEulerPts)
  if (improvedEulerFit) {
    const lineName = `Improved Euler order ≈ ${(-improvedEulerFit.slope).toFixed(2)}`
    marks.push(Plot.line(
      [{x: improvedEulerFit.xlo, y: improvedEulerFit.slope * improvedEulerFit.xlo + improvedEulerFit.intercept}, {x: improvedEulerFit.xhi, y: improvedEulerFit.slope * improvedEulerFit.xhi + improvedEulerFit.intercept}],
      {x: "x", y: "y", stroke: () => lineName, strokeDasharray: "6,4"}
    ))
    colorDomain.push(lineName)
    colorRange.push("#fbbf24")
    legendDomain.push(lineName)
    legendRange.push("#fbbf24")
  }

  const rk4Fit = VM.numerical.linearRegression(rk4Pts)
  if (rk4Fit) {
    const lineName = `RK4 order ≈ ${(-rk4Fit.slope).toFixed(2)}`
    marks.push(Plot.line(
      [{x: rk4Fit.xlo, y: rk4Fit.slope * rk4Fit.xlo + rk4Fit.intercept}, {x: rk4Fit.xhi, y: rk4Fit.slope * rk4Fit.xhi + rk4Fit.intercept}],
      {x: "x", y: "y", stroke: () => lineName, strokeDasharray: "6,4"}
    ))
    colorDomain.push(lineName)
    colorRange.push("#4ade80")
    legendDomain.push(lineName)
    legendRange.push("#4ade80")
  }

  const chart = Plot.plot({
    width: 340, height: 300, marginLeft: 46,
    x: {label: "log₁₀ n"},
    y: {label: "log₁₀ |error|"},
    color: {domain: colorDomain, range: colorRange},
    marks
  })
  const legend = Plot.legend({color: {domain: legendDomain, range: legendRange}})

  const container = document.createElement("div")
  container.append(chart, legend)
  return container
}

NoteValue table

// Value table — shows the entire run, not just the current n, with all
// three method estimates of y(b) and their errors against the reference
// value.
valueTable = {
  if (result.rows.length === 0) {
    // Always return a Node so Quarto's OJS runtime never latches this
    // declaration cell as hidden.
    return document.createElement("div")
  }

  const formattedRows = []
  for (const row of result.rows) {
    formattedRows.push([
      row.n,
      row.eulerValue.toPrecision(10),
      row.improvedEulerValue.toPrecision(10),
      row.rk4Value.toPrecision(10),
      row.trueValue.toPrecision(10),
      row.eulerError.toExponential(4),
      row.improvedEulerError.toExponential(4),
      row.rk4Error.toExponential(4)
    ])
  }

  return VM.ui.renderTable({
    html,
    headers: ["n", tex`y_{\text{Euler}}(b)`, tex`y_{\text{Imp. Euler}}(b)`, tex`y_{\text{RK4}}(b)`, tex`y_{\text{ref}}(b)`, tex`|y_{\text{Euler}} - y_{\text{ref}}|`, tex`|y_{\text{Imp. Euler}} - y_{\text{ref}}|`, tex`|y_{\text{RK4}} - y_{\text{ref}}|`],
    csvHeaders: ["n", "y_Euler(b)", "y_ImprovedEuler(b)", "y_RK4(b)", "y_ref(b)", "|y_Euler - y_ref|", "|y_ImprovedEuler - y_ref|", "|y_RK4 - y_ref|"],
    filename: "explicit-methods-values.csv",
    rows: formattedRows
  })
}