We consider an autonomous initial value ODE
y
′
(
t
)
=
f
(
y
(
t
)
)
with
y
(
t
0
)
=
y
0
.
{\displaystyle y'(t)=f(y(t))\quad {\text{with}}\quad y(t_{0})=y_{0}\,.}
(ODE
Applying the Tradezoidal rule
gives the implicit Runge-Kutta method
y
n
+
1
=
y
n
+
h
2
(
f
(
y
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)
+
f
(
y
n
+
1
)
)
.
{\displaystyle y_{n+1}=y_{n}+{\frac {h}{2}}\left(f(y_{n})+f(y_{n+1})\right)\,.}
(method
We will show that (method
Expanding the true solution
y
(
t
n
+
h
)
{\displaystyle y(t_{n}+h)}
t
n
{\displaystyle t_{n}}
y
(
t
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+
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=
y
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+
h
y
′
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+
h
2
2
y
″
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t
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+
h
3
3
!
y
‴
(
t
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)
+
O
(
h
4
)
.
{\displaystyle y(t_{n}+h)=y(t_{n})+hy'(t_{n})+{\frac {h^{2}}{2}}y''(t_{n})+{\frac {h^{3}}{3!}}y'''(t_{n})+O(h^{4})\,.}
Since
y
(
t
)
{\displaystyle y(t)}
ODE
y
′
(
t
)
=
f
(
y
(
t
)
)
{\displaystyle y'(t)=f(y(t))}
y
(
t
n
+
h
)
=
y
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+
h
f
(
y
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)
+
h
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2
f
′
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)
f
(
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+
h
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3
!
[
f
″
(
y
(
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)
)
(
f
(
y
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2
+
(
f
′
(
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)
)
2
f
(
y
(
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)
)
]
+
O
(
h
4
)
.
{\displaystyle y(t_{n}+h)=y(t_{n})+hf(y(t_{n}))+{\frac {h^{2}}{2}}f'(y(t_{n}))f(y(t_{n}))+{\frac {h^{3}}{3!}}\left[f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right]+O(h^{4})\,.}
(true
In (method
y
n
=
y
(
t
n
)
{\displaystyle y_{n}=y(t_{n})}
method true
y
(
t
n
+
h
)
−
y
n
+
1
=
h
2
[
f
(
y
(
t
n
)
)
−
f
(
y
n
+
1
)
]
+
h
2
2
f
′
(
y
(
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)
)
f
(
y
(
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)
)
+
h
3
3
!
[
f
″
(
y
(
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n
)
)
(
f
(
y
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)
)
)
2
+
(
f
′
(
y
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)
)
2
f
(
y
(
t
n
)
)
]
+
O
(
h
4
)
.
{\displaystyle y(t_{n}+h)-y_{n+1}={\frac {h}{2}}\left[f(y(t_{n}))-f(y_{n+1})\right]+{\frac {h^{2}}{2}}f'(y(t_{n}))f(y(t_{n}))+{\frac {h^{3}}{3!}}\left[f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right]+O(h^{4})\,.}
(error1
In order to cancel more terms we need to expand
f
(
y
n
+
1
)
{\displaystyle f(y_{n+1})}
y
n
+
1
≠
y
(
t
n
+
h
)
{\displaystyle y_{n+1}\not =y(t_{n}+h)}
method
f
{\displaystyle f}
f
(
y
n
+
1
)
=
f
(
y
(
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n
)
+
h
2
(
f
(
y
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)
+
f
(
y
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+
1
)
)
)
=
f
(
y
(
t
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)
)
+
h
2
(
f
(
y
(
t
n
)
)
+
f
(
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+
1
)
)
f
′
(
y
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)
+
h
2
8
(
f
(
y
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)
+
f
(
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+
1
)
)
2
f
″
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)
+
O
(
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)
.
{\displaystyle {\begin{aligned}f(y_{n+1})&=f\left(y(t_{n})+{\frac {h}{2}}\left(f(y(t_{n}))+f(y_{n+1})\right)\right)\\&=f(y(t_{n}))+{\frac {h}{2}}\left(f(y(t_{n}))+f(y_{n+1})\right)f'(y(t_{n}))+{\frac {h^{2}}{8}}\left(f(y(t_{n}))+f(y_{n+1})\right)^{2}f''(y(t_{n}))+O(h^{3})\,.\end{aligned}}}
(implicit
Substituting (implicit error1
h
2
4
f
′
(
y
(
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)
)
[
f
(
y
(
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)
)
−
f
(
y
n
+
1
)
]
+
−
h
3
16
(
f
(
y
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)
)
+
f
(
y
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+
1
)
)
2
f
″
(
y
(
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)
)
+
h
3
3
!
[
f
″
(
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(
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)
(
f
(
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2
+
(
f
′
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2
f
(
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)
]
+
O
(
h
4
)
.
{\displaystyle {\frac {h^{2}}{4}}f'(y(t_{n}))\left[f(y(t_{n}))-f(y_{n+1})\right]+{\frac {-h^{3}}{16}}\left(f(y(t_{n}))+f(y_{n+1})\right)^{2}f''(y(t_{n}))+{\frac {h^{3}}{3!}}\left[f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right]+O(h^{4})\,.}
(error2
This substitution was productive since the
h
{\displaystyle h}
implicit
O
(
h
2
)
{\displaystyle O(h^{2})}
h
2
{\displaystyle h^{2}}
O
(
h
4
)
{\displaystyle O(h^{4})}
implicit
f
(
y
n
+
1
)
{\displaystyle f(y_{n+1})}
error2
h
3
2
{
f
″
(
y
(
t
n
)
)
[
1
3
(
f
(
y
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n
)
)
)
2
+
−
1
8
(
f
(
y
(
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n
)
)
+
f
(
y
n
+
1
)
)
2
]
+
(
f
′
(
y
(
t
n
)
)
)
2
[
1
3
f
(
y
(
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n
)
)
+
−
1
4
(
f
(
y
(
t
n
)
)
+
f
(
y
n
+
1
)
)
]
}
+
O
(
h
4
)
.
{\displaystyle {\frac {h^{3}}{2}}\left\{f''(y(t_{n}))\left[{\frac {1}{3}}(f(y(t_{n})))^{2}+{\frac {-1}{8}}\left(f(y(t_{n}))+f(y_{n+1})\right)^{2}\right]+(f'(y(t_{n})))^{2}\left[{\frac {1}{3}}f(y(t_{n}))+{\frac {-1}{4}}\left(f(y(t_{n}))+f(y_{n+1})\right)\right]\right\}+O(h^{4})\,.}
(error3
This substitution was productive since the
h
2
{\displaystyle h^{2}}
implicit
O
(
h
)
{\displaystyle O(h)}
implicit error3
−
h
3
12
{
f
″
(
y
(
t
n
)
)
(
f
(
y
(
t
n
)
)
)
2
+
(
f
′
(
y
(
t
n
)
)
)
2
f
(
y
(
t
n
)
)
}
+
O
(
h
4
)
.
{\displaystyle {\frac {-h^{3}}{12}}\left\{f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right\}+O(h^{4})\,.}
Since the
h
3
{\displaystyle h^{3}}
O
(
h
3
)
{\displaystyle O(h^{3})}
![{\displaystyle y(t_{n}+h)=y(t_{n})+hf(y(t_{n}))+{\frac {h^{2}}{2}}f'(y(t_{n}))f(y(t_{n}))+{\frac {h^{3}}{3!}}\left[f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right]+O(h^{4})\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33519fe1dd55869aa93e134791ff9cbc7b5c37fd)
![{\displaystyle y(t_{n}+h)-y_{n+1}={\frac {h}{2}}\left[f(y(t_{n}))-f(y_{n+1})\right]+{\frac {h^{2}}{2}}f'(y(t_{n}))f(y(t_{n}))+{\frac {h^{3}}{3!}}\left[f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right]+O(h^{4})\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e9c493fc6a16fcb1fe8e02df82466ca183d6ed)
![{\displaystyle {\frac {h^{2}}{4}}f'(y(t_{n}))\left[f(y(t_{n}))-f(y_{n+1})\right]+{\frac {-h^{3}}{16}}\left(f(y(t_{n}))+f(y_{n+1})\right)^{2}f''(y(t_{n}))+{\frac {h^{3}}{3!}}\left[f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right]+O(h^{4})\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/735b116cfb64fb948c90d28ae837a58fdcf83b3e)
![{\displaystyle {\frac {h^{3}}{2}}\left\{f''(y(t_{n}))\left[{\frac {1}{3}}(f(y(t_{n})))^{2}+{\frac {-1}{8}}\left(f(y(t_{n}))+f(y_{n+1})\right)^{2}\right]+(f'(y(t_{n})))^{2}\left[{\frac {1}{3}}f(y(t_{n}))+{\frac {-1}{4}}\left(f(y(t_{n}))+f(y_{n+1})\right)\right]\right\}+O(h^{4})\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf72614bbaded1ef6cec2fa33bf48f6b7e068bc)
