Embedded methods List of Runge–Kutta methods

1 embedded methods

1.1 heun–euler
1.2 fehlberg rk1(2)
1.3 bogacki–shampine
1.4 fehlberg
1.5 cash-karp
1.6 dormand–prince

embedded methods

the embedded methods designed produce estimate of local truncation error of single runge-kutta step, , result, allow control error adaptive stepsize. done having 2 methods in tableau, 1 order p , 1 order p-1.

the lower-order step given by

y

n
+
1


∗


=

y

n


+
h

∑

i
=
1


s



b

i


∗



k

i


,


{\displaystyle y_{n+1}^{*}=y_{n}+h\sum _{i=1}^{s}b_{i}^{*}k_{i},}

where




k

i




{\displaystyle k_{i}}

same higher order method. error is

e

n
+
1


=

y

n
+
1


−

y

n
+
1


∗


=
h

∑

i
=
1


s


(

b

i


−

b

i


∗


)

k

i


,


{\displaystyle e_{n+1}=y_{n+1}-y_{n+1}^{*}=h\sum _{i=1}^{s}(b_{i}-b_{i}^{*})k_{i},}

which



o
(

h

p


)


{\displaystyle o(h^{p})}

. butcher tableau kind of method extended give values of




b

i


∗




{\displaystyle b_{i}^{*}}

c

1





a

11





a

12




…



a

1
s







c

2





a

21





a

22




…



a

2
s






⋮


⋮


⋮


⋱


⋮





c

s





a

s
1





a

s
2




…



a

s
s








b

1





b

2




…



b

s








b

1


∗





b

2


∗




…



b

s


∗








{\displaystyle {\begin{array}{c|cccc}c_{1}&a_{11}&a_{12}&\dots &a_{1s}\\c_{2}&a_{21}&a_{22}&\dots &a_{2s}\\\vdots &\vdots &\vdots &\ddots &\vdots \\c_{s}&a_{s1}&a_{s2}&\dots &a_{ss}\\\hline &b_{1}&b_{2}&\dots &b_{s}\\&b_{1}^{*}&b_{2}^{*}&\dots &b_{s}^{*}\\\end{array}}}



heun–euler

the simplest adaptive runge–kutta method involves combining heun s method, order 2, euler method, order 1. extended butcher tableau is:

0





1


1





1

/

2


1

/

2





1


0






{\displaystyle {\begin{array}{c|cc}0&\\1&1\\\hline &1/2&1/2\\&1&0\end{array}}}

the error estimate used control stepsize.

fehlberg rk1(2)

the fehlberg method has 2 methods of orders 1 , 2. extended butcher tableau is:

the first row of b coefficients gives first-order accurate solution, , second row has order two.

bogacki–shampine

the bogacki–shampine method has 2 methods of orders 3 , 2. extended butcher tableau is:

the first row of b coefficients gives third-order accurate solution, , second row has order two.

fehlberg

the runge–kutta–fehlberg method has 2 methods of orders 5 , 4. extended butcher tableau is:

the first row of b coefficients gives fifth-order accurate solution, , second row has order four.

cash-karp

cash , karp have modified fehlberg s original idea. extended tableau cash–karp method is

the first row of b coefficients gives fifth-order accurate solution, , second row has order four.

dormand–prince

the extended tableau dormand–prince method is

the first row of b coefficients gives fifth-order accurate solution , second row gives fourth-order accurate solution.