Runge-Kutta RK4 Solver – ODE Integration

RK4 weighs four slope probes across each step — at the left edge, twice at the midpoint, and at the right edge — and combines them with weights 1:2:2:1. For the test equation y′ = y, y(0) = 1, h = 0.1: the four slopes evaluate to k₁ = 1, k₂ = 1.05, k₃ = 1.0525, k₄ = 1.10525, giving y(0.1) = 1 + (0.1/6)(1 + 2.1 + 2.105 + 1.10525) = 1.1051702. The exact solution e⁰·¹ = 1.1051709 differs only in the 7th digit — RK4’s local error is O(h⁵). Euler’s method on the same step returns 1.1, off by 0.005. The calculator prints every k, the weighted update, a step table, and the global error against known solutions (exponential growth, decay, logistics). Halving h divides global error by 16, the signature of a 4th-order method.

1. 1

   Enter f(x, y) and initial values

   The ODE right-hand side plus x₀, y₀

2. 2

   Set step h and target

   e.g. h = 0.1 out to x = 1

3. 3

   Compute the four slopes

   k₁…k₄ are listed per step

4. 4

   Advance and report error

   The weighted update and exact-solution comparison appear

• 1

  Local error O(h⁵), global O(h⁴) — halve h, cut error 16×

• 2

  Weights 1:2:2:1 on the four slope samples

• 3

  One RK4 step often beats ten Euler steps

• Evaluating all four slopes at the left endpoint instead of staggering them

• Using weights 1:1:1:1 — that collapses to a crude average

• Believing RK4 handles stiff equations well; implicit methods are needed there

• Skipping the error check against a known solution during development