Mathematics · Numerical methods

Classical Fourth-Order Runge-Kutta Method (RK4)

Classical RK4 approximates one step of an initial-value problem using four weighted slope evaluations.

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Formula

k_{1} = f (t_{n}, y_{n}), k_{2} = f (t_{n} + h /2, y_{n} + h k_{1} /2), k_{3} = f (t_{n} + h /2, y_{n} + h k_{2} /2), k_{4} = f (t_{n} + h, y_{n} + h k_{3}), y_{n + 1} = y_{n} + \frac{h}{6} (k_{1} + 2 k_{2} + 2 k_{3} + k_{4})

t,h in time units; y in state units; k_i=f(t,y) in y per time unit; every stage argument must be dimensionally consistent.

Variables & units

Variables & units
Symbol	Meaning	Unit
$tn$	Time at the start of the current step	s
$yn$	Approximate solution at the step start	unit of the state variable
$h$	Step size	s
$f(t,y)$	Right-hand side of the ODE y′=f(t,y)	unit of y / time
k1…k4	Slopes at the four Runge-Kutta stages	unit of y / time

LaTeX

k_1=f(t_n,y_n),\ k_2=f(t_n+h/2,y_n+hk_1/2),\ k_3=f(t_n+h/2,y_n+hk_2/2),\ k_4=f(t_n+h,y_n+hk_3),\ y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4)

QUANTA STUDYquanta-study.de

The smooth reference curve y=e to the t shows one step from t_n to t_n+h. Samples are k1 at (t_n,y_n), k2 and k3 at the same midpoint time with different trial values, and k4 at the end time using the third trial value. The internal points are RK4 trial states and need not lie exactly on the exact curve. — For y′=y and h=1, k₁…k₄ sample the start, two midpoint trial states and the end time; trial values need not lie exactly on y=eᵗ.

Derivation & background

Exam blueprint

Validity range

Derivation steps

Classical RK4 approximates one step of an initial-value problem using four weighted slope evaluations.

1Identify the quantities in the equation and check its assumptions: t,h in time units; y in state units; k_i=f(t,y) in y per time unit; every stage argument must be dimensionally consistent.
2Set up the model for “Classical Fourth-Order Runge-Kutta Method (RK4)” and assign each value to a symbol.
3Work through the example and verify it with an independent relation or limiting-case check: For y′=y, y(0)=1 and h=0.1: k₁=1, k₂=1.05, k₃=1.0525, k₄=1.10525; y₁=1+0.1(1+2·1.05+2·1.0525+1.10525)/6≈1.105170833. Exact: e^0.1≈1.105170186, so the step error is about 6.47×10⁻⁷.

Rearrangements

A complete RK4 step

y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4)

The k_i are slopes; multiply by h only in the weighted sum.

Task variant

Rework the example for “Classical Fourth-Order Runge-Kutta Method (RK4)” and explain why the check is valid.

For y′=y, y(0)=1 and h=0.1: k₁=1, k₂=1.05, k₃=1.0525, k₄=1.10525; y₁=1+0.1(1+2·1.05+2·1.0525+1.10525)/6≈1.105170833.
Exact: e^0.1≈1.105170186, so the step error is about 6.47×10⁻⁷.

Common mistakes

Ignoring an assumption of the formula: For sufficiently smooth solutions, RK4 has global error O(h⁴) over a fixed interval and local one-step error O(h⁵)

Check units and model conditions before substitution; then verify the result in the original relation or a limiting case.

Exam context

Numerical initial-value problems in mechanics, circuits, population models and dynamical systems.

Formula cluster

Numerical methods

Related relations in the formula sheet: euler-verfahren-differentialgleichung, trennbare-differentialgleichung, lineare-differentialgleichung-zweiter-ordnung.

y_n+1=y_n+h·f(t_n,y_n)

Forward Euler Method

dy/h(y)=g(x)dx

Separable Differential Equation

y″+ay′+by=0; charakteristisches Polynom r²+ar+b

Second-Order Linear ODE with Constant Coefficients

Worked example

For y′=y, y(0)=1 and h=0.1: k₁=1, k₂=1.05, k₃=1.0525, k₄=1.10525; y₁=1+0.1(1+2·1.05+2·1.0525+1.10525)/6≈1.105170833. Exact: e^0.1≈1.105170186, so the step error is about 6.47×10⁻⁷.

Applications

Numerical initial-value problems in mechanics, circuits, population models and dynamical systems.

How do you calculate with Classical Fourth-Order Runge-Kutta Method (RK4)?

Here is how to work through a typical Classical Fourth-Order Runge-Kutta Method (RK4) (yₙ₊₁ = yₙ + h(k₁+2k₂+2k₃+k₄)/6) task step by step:

1
Task
Rework the example for “Classical Fourth-Order Runge-Kutta Method (RK4)” and explain why the check is valid.
Solution path
1. For y′=y, y(0)=1 and h=0.1: k₁=1, k₂=1.05, k₃=1.0525, k₄=1.10525; y₁=1+0.1(1+2·1.05+2·1.0525+1.10525)/6≈1.105170833.
2. Exact: e^0.1≈1.105170186, so the step error is about 6.47×10⁻⁷.

Frequently asked questions about Classical Fourth-Order Runge-Kutta Method (RK4)

What does the “Classical Fourth-Order Runge-Kutta Method (RK4)” relation describe?+

Classical RK4 approximates one step of an initial-value problem using four weighted slope evaluations. The notation y_n+1 = y_n + h(k₁+2k₂+2k₃+k₄)/6 shows which quantities are connected. Here, tn: Time at the start of the current step (s); yn: Approximate solution at the step start (unit of the state variable); h: Step size (s). t,h in time units; y in state units; k_i=f(t,y) in y per time unit; every stage argument must be dimensionally consistent. An application depends on whether the supplied quantities match the definitions and reference directions. Common uses include Numerical initial-value problems in mechanics, circuits, population models and dynamical systems.. Read the variables and their dimensions before substituting numbers; this often reveals whether the result should be a length, rate, probability, or dimensionless statistic.

How can I solve a “Classical Fourth-Order Runge-Kutta Method (RK4)” problem transparently?+

Start with a diagram or a list of known quantities, assign each one a symbol, and convert units to a consistent form. Then choose the appropriate equation and substitute values only afterward. Example: For y′=y, y(0)=1 and h=0.1: k₁=1, k₂=1.05, k₃=1.0525, k₄=1.10525; y₁=1+0.1(1+2·1.05+2·1.0525+1.10525)/6≈1.105170833. Exact: e^0.1≈1.105170186, so the step error is about 6.47×10⁻⁷. Verification is part of the calculation: For sufficiently smooth solutions, RK4 has global error O(h⁴) over a fixed interval and local one-step error O(h⁵) Also compare the magnitude and unit with the context. Round only at the end, since early rounding can affect multi-step calculations, especially when small differences or sums are involved.

What assumptions and limitations apply to “Classical Fourth-Order Runge-Kutta Method (RK4)”?+

For sufficiently smooth solutions, RK4 has global error O(h⁴) over a fixed interval and local one-step error O(h⁵). The k_i are slopes, not increments already multiplied by h. Classical RK4 supplies no error estimate and does not control stability; stiff problems often require implicit methods. Varying the step size or using an embedded method is necessary when error control is required. These are not minor footnotes: when assumptions fail, the formula can produce a precise but inappropriate result. Before calculating, identify which values are observed, estimated, or modeled, and whether special cases change the equation. Numerical initial-value problems in mechanics, circuits, population models and dynamical systems. For numerical methods, also check step size and stability; for statistics, sampling design and distribution; for calculus, differentiability, boundary, and orientation. Use another method when these conditions do not hold.

How is “Classical Fourth-Order Runge-Kutta Method (RK4)” related to Forward Euler Method?+

Both relations belong to the same topic, but answer different questions. “Classical Fourth-Order Runge-Kutta Method (RK4)”: Classical RK4 approximates one step of an initial-value problem using four weighted slope evaluations. The companion relation “Forward Euler Method” The forward Euler method approximates a solution of y′=f(t,y) by extending the tangent slope as constant over a step h. Choose based on the quantity sought and the model: Numerical initial-value problems in mechanics, circuits, population models and dynamical systems. The worked example For y′=y, y(0)=1 and h=0.1: k₁=1, k₂=1.05, k₃=1.0525, k₄=1.10525; y₁=1+0.1(1+2·1.05+2·1.0525+1.10525)/6≈1.105170833. Exact: e^0.1≈1.105170186, so the step error is about 6.47×10⁻⁷. shows which quantities actually enter this relation. Compare both formulas using the same data and identify the companion’s extra assumption. This clarifies which relation fits.

Which common reasoning error can undermine a “Classical Fourth-Order Runge-Kutta Method (RK4)” calculation?+

For an ODE, initial values are part of the problem; for a numerical step, also check step size and stability. A formally correct iteration does not guarantee an accurate approximation. The symbols must also match their definitions exactly: tn: Time at the start of the current step (s); yn: Approximate solution at the step start (unit of the state variable); h: Step size (s). Do not calculate mechanically; write the assumption beside the setup, then check dimensions and sign. Also note: For sufficiently smooth solutions, RK4 has global error O(h⁴) over a fixed interval and local one-step error O(h⁵) The example For y′=y, y(0)=1 and h=0.1: k₁=1, k₂=1.05, k₃=1.0525, k₄=1.10525; y₁=1+0.1(1+2·1.05+2·1.0525+1.10525)/6≈1.105170833. Exact: e^0.1≈1.105170186, so the step error is about 6.47×10⁻⁷. lets you follow these checks directly. If the assumptions do not fit, choose another relation from the chapter instead of forcing values into an unsuitable formula.

Quanta exam set

Curated exam set for "Classical Fourth-Order Runge-Kutta Method (RK4)":

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Which formula describes Classical Fourth-Order Runge-Kutta Method (RK4)?

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How do you rearrange Classical Fourth-Order Runge-Kutta Method (RK4) for A complete RK4 step?

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Which common mistake happens with Classical Fourth-Order Runge-Kutta Method (RK4)?

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]MIT OpenCourseWare, RES.18-009: Classical Runge-Kutta, ODE4

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Symbol

Meaning

Unit

tn

Time at the start of the current step

yn

Approximate solution at the step start

unit of the state variable

h

Step size

f(t,y)

Right-hand side of the ODE y′=f(t,y)

unit of y / time

k1…k4

Slopes at the four Runge-Kutta stages

unit of y / time

Derivation & background

Exam blueprint

Validity range

Derivation steps

Classical RK4 approximates one step of an initial-value problem using four weighted slope evaluations.

1Identify the quantities in the equation and check its assumptions: t,h in time units; y in state units; k_i=f(t,y) in y per time unit; every stage argument must be dimensionally consistent.
2Set up the model for “Classical Fourth-Order Runge-Kutta Method (RK4)” and assign each value to a symbol.
3Work through the example and verify it with an independent relation or limiting-case check: For y′=y, y(0)=1 and h=0.1: k₁=1, k₂=1.05, k₃=1.0525, k₄=1.10525; y₁=1+0.1(1+2·1.05+2·1.0525+1.10525)/6≈1.105170833. Exact: e^0.1≈1.105170186, so the step error is about 6.47×10⁻⁷.

How do you calculate with Classical Fourth-Order Runge-Kutta Method (RK4)?

Here is how to work through a typical Classical Fourth-Order Runge-Kutta Method (RK4) (yₙ₊₁ = yₙ + h(k₁+2k₂+2k₃+k₄)/6) task step by step:

Task

Rework the example for “Classical Fourth-Order Runge-Kutta Method (RK4)” and explain why the check is valid.

Solution path

For y′=y, y(0)=1 and h=0.1: k₁=1, k₂=1.05, k₃=1.0525, k₄=1.10525; y₁=1+0.1(1+2·1.05+2·1.0525+1.10525)/6≈1.105170833.
Exact: e^0.1≈1.105170186, so the step error is about 6.47×10⁻⁷.

Frequently asked questions about Classical Fourth-Order Runge-Kutta Method (RK4)

What does the “Classical Fourth-Order Runge-Kutta Method (RK4)” relation describe?+

How can I solve a “Classical Fourth-Order Runge-Kutta Method (RK4)” problem transparently?+

What assumptions and limitations apply to “Classical Fourth-Order Runge-Kutta Method (RK4)”?+

How is “Classical Fourth-Order Runge-Kutta Method (RK4)” related to Forward Euler Method?+

Which common reasoning error can undermine a “Classical Fourth-Order Runge-Kutta Method (RK4)” calculation?+

Retain Classical Fourth-Order Runge-Kutta Method (RK4) for exams

Create a curated FSRS exam set for yₙ₊₁ = yₙ + h(k₁+2k₂+2k₃+k₄)/6: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential