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Runge–Kutta Calculator

Classical fourth-order Runge–Kutta for a scalar first-order IVP y′=f(t,y). Same engine as Ordinary Differential Equation. Not Euler, not rk45, not a system. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 9/9 tests · Production surface contract 4/4 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
y(t1) of a scalar first-order IVP by classical RK4.
Scope
Scalar first-order only; method locked to rk4
Verification
Engine tested · 9/9 tests · Production surface contract 4/4 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 2 boundary · 4 property · Production surface contract 4/4 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Slopesk1=f(t,y), k2=f(t+h/2, y+h k1/2), …
Updatey ← y + (h/6)(k1+2k2+2k3+k4)
iFourth-order accurate. The public slug is runge-kutta, not rk4. Linear rhs y′=k y has exact y0 e^{k(t1−t0)}. Polynomial rhs y′=p(t) has exact y0+∫p.

How to use

1

Enter rhs, y0, t0, t1

Linear needs k. Poly needs coeffs. t1 must differ from t0.

2

Read y(t1) and |err|

The exact check is shown for linear and poly. Euler lives on the sibling discovery URL.

Example calculations

Common configurations with formula and result.

ϟ

y′=y, y(0)=1 at t=1

n=40, k=1

y = e
2.718…
ϟ

y′=2t, y(0)=0 at t=1

poly coeffs 0,2

y = 1
1

Runge–Kutta calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Classical RK4 takes four slope samples per step. This page is method=rk4 on math.numerical.ode. Not a second ODE engine. Not Euler. Not adaptive RK45.
What it calculates
y(t1) of a scalar first-order IVP by classical RK4.
Inputs
  • rhs
  • y0
  • t0
  • t1
  • n?
  • k?
  • coeffs?
Outputs
  • y
  • exact
  • abs_err
  • h
  • n
Formula
classical RK4
Assumptions
  • Scalar first-order only; method locked to rk4
  • Not rk45, not systems, not PDE, not CAS
Units
  • dimensionless
Boundary conditions
  • t1 = t0 → INVALID_INPUT
  • n < 1 or n > 10000 → INVALID_INPUT
Example
rk4 linear k=1 y0=1 t0=0 t1=1 n=40 → exact=e
Validation cases

2 published on this page · 9/9 tests · Production surface contract 4/4 · View evidence

  • rhs=linear k=1 y0=1 t0=0 t1=1 n=40 → method=rk4, exact=e
  • method=rk45 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Brent Method Calculator, Definite Integral Calculator, Euler Method Calculator, Gauss Quadrature Calculator, Golden Section Search Calculator, Numerical Derivative Calculator, Numerical Interpolation Calculator, Numerical Root Calculator . Explore all Numerical Calculus.

→ Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → Euler Method Calculator Forward Euler for a scalar first-order IVP y′=f(t,y). Same engine as Ordinary Differential Equation. Not RK4, not rk45, not a system. Runs locally. › → Gauss Quadrature Calculator n-point Gauss–Legendre quadrature of a polynomial, sin, or exp on [a, b]. Same engine as Definite Integral. Not indefinite CAS. Runs locally. › → Golden Section Search Calculator Golden-section search for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not Brent, not multivariate. Runs locally. › → Numerical Derivative Calculator Finite-difference derivative of a polynomial, sin, or exp. Central, forward, or backward. Not symbolic d/dx. Runs locally. › → Numerical Interpolation Calculator Piecewise-linear, Lagrange, or natural cubic spline of tabulated (x, y). Not regression, not CAS. Runs locally. › → Numerical Root Calculator Find a real root of a polynomial with Newton, bisection, or secant. One capability, three methods. Not the nth-root page. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this adaptive RK45 or a second ODE engine?

No. It is method=rk4 on math.numerical.ode, the same engine as /calc/math/ordinary-differential-equation. rk45 is INVALID_MODE.

Where does this run?

Locally in the browser by default. REST and MCP call the same ordinary-differential-equation engine.