Knowledge card
Classical Fourth-order Runge–Kutta
Mathematical Modelling — Fluid Dynamics & Heat TransferNumerical analysis; boundary-layer similarity theory
Symbol
RK4The workhorse one-step integrator marching the IVP across [0, η_max]: four staged slope evaluations per step, combined in Simpson-like weights, giving local error O(h⁵) and global error O(h⁴).
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In words:equalsplus.
Open full derivation chain: 2 stepsDerivation
- 1
Open Step 1: Given / definition · Four staged slope evaluations per step: k₁ = F(ηₙ, yₙ), k₂ = F(ηₙ+h/2, yₙ+(h/2)k₁), k₃ = F(ηₙ+h…
Four staged slope evaluations per step: k₁ = F(ηₙ, yₙ), k₂ = F(ηₙ+h/2, yₙ+(h/2)k₁), k₃ = F(ηₙ+h/2, yₙ+(h/2)k₂), k₄ = F(ηₙ+h, yₙ+h·k₃).Given / definition Defining ruleThe starting relation of a derivation — the definition or established equation every later step transforms; it justifies no change itself. - ★
Open Step 2: Advance one integration step · Combine the stages with Simpson-like weights (1, 2, 2, 1)/6 — the update matches the Taylor ser…
Combine the stages with Simpson-like weights (1, 2, 2, 1)/6 — the update matches the Taylor series of the true solution through h⁴, so the local error is O(h⁵) and the global error O(h⁴).Advance one integration step Defining ruleyₙ₊₁ = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄)Advancing the state one mesh step with a one-step integrator (classical RK4 shown).Open term-change ledger: 1 records
Open record 1: ∅ → (h/6)(k₁+2k₂+2k₃+k₄)
Consumes:∅Produces:L0The one-step update introduces the increment (h/6)(k₁ + 2k₂ + 2k₃ + k₄) — the Simpson-weighted average of the four staged slopes defined in the Given step — so that yₙ₊₁ = yₙ + increment matches the true solution’s Taylor series through h⁴.(h/6)(k₁+2k₂+2k₃+k₄)Resulting expressionState IncrementThe full RK4 update increment.Level 0 · more availableBrief — named quantitiesAbout levels
Why the levels load one at a time
Only level 0 travels with this article. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them — and how deep it goes is not known until you get there.
To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.
How many levels remain is not yet known — the depth is discovered one level at a time.
Equation Workspace
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Equation Workspace
Set known values, calculate unknowns, inspect dependencies, and run numerical solves.
Equation workspace
Set the values you know, then calculate the remaining quantities in Classical Fourth-order Runge–Kutta.
Domain Analysis
Unit status, dimensional trails, per-step checks, and custom unit expressions.
Domain Analysis
Unit status, dimensional trails, per-step checks, and custom unit expressions.
Step-by-step unit check (2 steps)
Step 1 given
Expand checked this step.Step 2 integrate_step
Expand checked this step.
Try your own — the unit calculator
Ever wondered whether a formula “adds up”? This tool answers one plain question: do the units match? Pick a couple of quantities (distance, time, a speed…), join them with ×, ÷, +, or =, and press Run. It works out the resulting unit — for example distance ÷ time gives a speed, LT⁻¹ (metres per second) — and flags anything that can’t be right, like adding a length to a time.
Reads as: RK4
“RK4” is analysed by expanding its full defining equation (its children and their relations), not as a bare symbol. Use Check this concept’s units above for the same result.
Mathematical Analysis
Object type, lawful operations, conditions, comparisons, and derivation-step checks.
Mathematical Analysis
Object type, lawful operations, conditions, comparisons, and derivation-step checks.
Check one step at a time (2 steps)
Step 1 — given
Step 2 — integrate step
Dimension I
Definition
What it is — and what it is not
- What it is
- S1The classical fourth-order Runge–Kutta method: four slope stages per step combined with weights (1, 2, 2, 1)/6. [1.1]
- S2The four RK4 stages sample the slope field at the start, twice at midstep, and at the end of each step.
- Wisdoms
- S3The RK4 update: the state advances by the weighted average of the four staged slopes.
- S3.1essenceA composed relation of this step — see the built expression.
- S3.2essenceA composed relation of this step — see the built expression.
- S3The RK4 update: the state advances by the weighted average of the four staged slopes.
- Attributes
- S4Local truncation error O(h⁵), global error O(h⁴): halving h should reduce profile errors ≈16×— the convergence check of the marching stage. [1.1]
Dimension II
In practice
How to deal with it
- In context
- S5MATLAB’s ode45 (Dormand–Prince) serves the same role with adaptive step control; classical RK4 with fixed h is the transparent reference implementation. [1.1]