Runge–Kutta methods

In numerical analysis, the Runge–Kutta methods (English: RUUNG-ə-KUUT-tah) are a family of explicit and implicit iterative methods, which include the Euler method, used in discretization for the approximate solutions of differential equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta.

The fourth-order Runge–Kutta method

The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method".

Let an initial value problem be specified as follows:

d

y

d

t

=

f

(

t

,

y

)

,

y

(

t

0

)

=

y

0

.

{\displaystyle {\frac {dy}{dt}}=f(t,y),\quad y(t_{0})=y_{0}.}

Here

y

{\displaystyle y}

is an unknown function (scalar or vector) of time

t

{\displaystyle t}

, which we would like to approximate; we are told that

d

y

d

t

{\displaystyle {\frac {dy}{dt}}}

, the rate at which

y

{\displaystyle y}

changes, is a function of

t

{\displaystyle t}

and of

y

{\displaystyle y}

itself. At the initial time

t

0

{\displaystyle t_{0}}

the corresponding

y

{\displaystyle y}

value is

y

0

{\displaystyle y_{0}}

. The function

f

{\displaystyle f}

and the initial conditions

t

0

{\displaystyle t_{0}}

,

y

0

{\displaystyle y_{0}}

are given.

Now we pick a step-size h > 0 and define:

y

n

+

1

=

y

n

+

h

6

(

k

1

+

2

k

2

+

2

k

3

+

k

4

)

,

t

n

+

1

=

t

n

+

h

{\displaystyle {\begin{aligned}y_{n+1}&=y_{n}+{\frac {h}{6}}\left(k_{1}+2k_{2}+2k_{3}+k_{4}\right),\\t_{n+1}&=t_{n}+h\\\end{aligned}}}

for n = 0, 1, 2, 3, ..., using

k

1

=

f

(

t

n

,

y

n

)

,

k

2

=

f

(

t

n

+

h

2

,

y

n

+

k

1

h

2

)

,

k

3

=

f

(

t

n

+

h

2

,

y

n

+

k

2

h

2

)

,

k

4

=

f

(

t

n

+

h

,

y

n

+

h

k

3

)

.

{\displaystyle {\begin{aligned}k_{1}&=\ f(t_{n},y_{n}),\\k_{2}&=\ f\!\left(t_{n}+{\frac {h}{2}},y_{n}+k_{1}{\frac {h}{2}}\right),\\k_{3}&=\ f\!\left(t_{n}+{\frac {h}{2}},y_{n}+k_{2}{\frac {h}{2}}\right),\\k_{4}&=\ f\!\left(t_{n}+h,y_{n}+hk_{3}\right).\end{aligned}}}

(Note: the above equations have different but equivalent definitions in different texts.)

Here

y

n

+

1

{\displaystyle y_{n+1}}

is the RK4 approximation of

y

(

t

n

+

1

)

{\displaystyle y(t_{n+1})}

, and the next value (

y

n

+

1

{\displaystyle y_{n+1}}

) is determined by the present value (

y

n

{\displaystyle y_{n}}

) plus the weighted average of four increments, where each increment is the product of the size of the interval, h, and an estimated slope specified by function f on the right-hand side of the differential equation.

k

1

{\displaystyle k_{1}}

is the slope at the beginning of the interval, using

y

{\displaystyle y}

(Euler's method);

k

2

{\displaystyle k_{2}}

is the slope at the midpoint of the interval, using

y

{\displaystyle y}

and

k

1

{\displaystyle k_{1}}

;

k

3

{\displaystyle k_{3}}

is again the slope at the midpoint, but now using

y

{\displaystyle y}

and

k

2

{\displaystyle k_{2}}

;

k

4

{\displaystyle k_{4}}

is the slope at the end of the interval, using

y

{\displaystyle y}

and

k

3

{\displaystyle k_{3}}

.