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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 CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published methods
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published methods
Assurance
Core
Declared partition coverage
PASS · 7/7 declared partitions (exponential-range, shifted-time, scaled-results, numerical-range, euler, rk4, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
y(t1) of a scalar first-order IVP by classical RK4, with an engine-linked Live graph of numerical y(t).
Scope
The Result Card includes an engine-linked Live graph of numerical y(t) using the same Overview / Focus template as Log (Overview window t0…t1; current point is published y(t1); same RK4 stepper as REST/MCP). Same engine as /ordinary-differential-equation. Not rk45, not systems, not PDE, not CAS.
Verification
Engine tested · Specification checked · v1.0.4 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.4 · CVP protocol 1.0.0-proposed · Evidence 2026-10-07.ode-input-summary
Verification revision
2026-10-07.ode-input-summary · 1/1 property · digest 4e8cee39192f
Legacy regression
9/9 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production STALE · Overall CVP state: VERIFIED_STALE under CVP 1.0.0-proposed
CVP status
STALE · Capability production binding: STALE · Site report: STALE · Evidence changed after the last successful production attestation. Re-attestation required.
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 2/2 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
27/27 golden · 2/2 CVP boundary · 15/15 invalid · 1/1 property · 8/8 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 27/27 oracle-backed golden · 15/15 invalid · Artifact integrity PASS
STALE · Build schema 1.0.4 matches page · production binding STALE · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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 signed error (y − reference)

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 1.0.4 · 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, with an engine-linked Live graph of numerical y(t).
Inputs
  • rhs
  • y0
  • t0
  • t1
  • n?
  • k?
  • coeffs?
Outputs
  • y
  • exact
  • abs_err
  • h
  • n
Formula
classical RK4
Assumptions
  • The Result Card includes an engine-linked Live graph of numerical y(t) using the same Overview / Focus template as Log (Overview window t0…t1; current point is published y(t1); same RK4 stepper as REST/MCP). Same engine as /ordinary-differential-equation. Not rk45, not systems, not PDE, not CAS.
  • Scalar first-order only; method locked to rk4
Units
  • dimensionless
Boundary conditions
  • Unrepresentable time grid, zero/nonfinite h, nonfinite step/reference/error or nonzero analytic underflow → NUMERICAL_RANGE_ERROR
  • 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 6/6 · 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
1.0.4

Other calculators in this family: Brent Method Calculator, Definite Integral Calculator, Euler Method Calculator, Gauss Quadrature Calculator, Golden Section Search Calculator, Nonlinear Equation Calculator, Numerical Derivative Calculator, Numerical Interpolation 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. › → Nonlinear Equation Calculator Find one approximate real zero of a polynomial or sin x with Newton, bisection, or secant. Uses the Numerical Root engine. Runs locally. › → Numerical Derivative Calculator Finite-difference derivative of a polynomial, sin, or exp, plus a numeric two-sided limit. Central, forward, backward, or limit. Discovery /two-sided-limit. 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. ›

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.