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src/engine/numerics.js

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+// Numerical methods for transcendental equations
+
+/**
+ * Bisection method — finds root of f in [lo, hi] within tolerance.
+ * Returns null if no sign change is found.
+ */
+export function bisect(f, lo, hi, tol = 1e-10, maxIter = 200) {
+  let flo = f(lo)
+  let fhi = f(hi)
+  if (!isFinite(flo) || !isFinite(fhi)) return null
+  if (flo * fhi > 0) return null // no sign change
+  for (let i = 0; i < maxIter; i++) {
+    const mid = (lo + hi) / 2
+    if ((hi - lo) / 2 < tol) return mid
+    const fmid = f(mid)
+    if (fmid === 0) return mid
+    if (flo * fmid < 0) { hi = mid; fhi = fmid }
+    else { lo = mid; flo = fmid }
+  }
+  return (lo + hi) / 2
+}
+
+/**
+ * Isentropic area-ratio function A/A* as a function of Mach M and gamma.
+ *   A/A* = (1/M) * [(2/(γ+1)) * (1 + (γ-1)/2 * M²)]^((γ+1)/(2(γ-1)))
+ */
+export function areaRatioFromMach(M, gamma) {
+  const exp = (gamma + 1) / (2 * (gamma - 1))
+  const base = (2 / (gamma + 1)) * (1 + (gamma - 1) / 2 * M * M)
+  return (1 / M) * Math.pow(base, exp)
+}
+
+/**
+ * Solve Mach number from area ratio and gamma using bisection.
+ * supersonic=true searches M > 1, false searches M < 1.
+ * Returns null if unsolvable.
+ */
+export function machFromAreaRatio(eps, gamma, supersonic = true) {
+  const f = (M) => areaRatioFromMach(M, gamma) - eps
+  if (supersonic) return bisect(f, 1.0001, 200, 1e-9)
+  else return bisect(f, 0.0001, 0.9999, 1e-9)
+}