Explicit methods List of Runge–Kutta methods
1 explicit methods
1.1 forward euler
1.2 explicit midpoint method
1.3 heun s method
1.4 ralston s method
1.5 generic second-order method
1.6 kutta s third-order method
1.7 classic fourth-order method
1.8 3/8-rule fourth-order method
explicit methods
the explicit methods matrix
[
a
i
j
]
{\displaystyle [a_{ij}]}
lower triangular.
forward euler
the euler method first order. lack of stability , accuracy limits popularity use simple introductory example of numeric solution method.
0
0
1
{\displaystyle {\begin{array}{c|c}0&0\\\hline &1\\\end{array}}}
explicit midpoint method
the (explicit) midpoint method second-order method 2 stages (see implicit midpoint method below):
0
0
0
1
/
2
1
/
2
0
0
1
{\displaystyle {\begin{array}{c|cc}0&0&0\\1/2&1/2&0\\\hline &0&1\\\end{array}}}
heun s method
heun s method second-order method 2 stages (also known explicit trapezoid rule):
0
0
0
1
1
0
1
/
2
1
/
2
{\displaystyle {\begin{array}{c|cc}0&0&0\\1&1&0\\\hline &1/2&1/2\\\end{array}}}
ralston s method
ralston s method second-order method 2 stages , minimum local error bound:
0
0
0
2
/
3
2
/
3
0
1
/
4
3
/
4
{\displaystyle {\begin{array}{c|cc}0&0&0\\2/3&2/3&0\\\hline &1/4&3/4\\\end{array}}}
generic second-order method
0
0
0
x
x
0
1
−
1
2
x
1
2
x
{\displaystyle {\begin{array}{c|ccc}0&0&0\\x&x&0\\\hline &1-{\frac {1}{2x}}&{\frac {1}{2x}}\\\end{array}}}
kutta s third-order method
0
0
0
0
1
/
2
1
/
2
0
0
1
−
1
2
0
1
/
6
2
/
3
1
/
6
{\displaystyle {\begin{array}{c|ccc}0&0&0&0\\1/2&1/2&0&0\\1&-1&2&0\\\hline &1/6&2/3&1/6\\\end{array}}}
classic fourth-order method
the original runge–kutta method.
0
0
0
0
0
1
/
2
1
/
2
0
0
0
1
/
2
0
1
/
2
0
0
1
0
0
1
0
1
/
6
1
/
3
1
/
3
1
/
6
{\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\1/2&1/2&0&0&0\\1/2&0&1/2&0&0\\1&0&0&1&0\\\hline &1/6&1/3&1/3&1/6\\\end{array}}}
3/8-rule fourth-order method
this method doesn t have notoriety classical method, classical because proposed in same paper (kutta, 1901).
0
0
0
0
0
1
/
3
1
/
3
0
0
0
2
/
3
−
1
/
3
1
0
0
1
1
−
1
1
0
1
/
8
3
/
8
3
/
8
1
/
8
{\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\1/3&1/3&0&0&0\\2/3&-1/3&1&0&0\\1&1&-1&1&0\\\hline &1/8&3/8&3/8&1/8\\\end{array}}}
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