deriving an implicit Runge Kutta method from its Butcher tableau

Question

I would like to use the Gauss–Legendre method of order four (which is a particular Runge Kutta method) to solve numerically an ode but I find only it's Butcher Tableau and I fail to derive the method . You can find the Tableau here : https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods then search for Gauss-Legendre methods paragraph and it's the first tableau . using $$y_{n+1}=y_n + h \sum_{i=1}^s b_i k_i$$

with $$k_i=f(t_n+c_i h,y_n+ h \sum_{j=1}^s a_{ij}k_j)$$

I find for my tableau : $$y_{n+1}=y_n+\frac{h}{2} (k_1+k_2) $$ $$k_1=f(t_n+(\frac{1}{2}-\frac{\sqrt{3}}{6}) h,y_n+ \frac{h}{4} k_1 + (\frac{h}{4}+ \frac{h \sqrt{3}}{6}) k2)$$ $$k_2=f(t_n+(\frac{1}{2}+\frac{\sqrt{3}}{6}) h,y_n+ \frac{h}{4} k_2 + (\frac{h}{4}- \frac{h \sqrt{3}}{6}) k1)$$

using some substitution we can get : $$k_1=f(t_n+(\frac{1}{2}-\frac{\sqrt{3}}{6})h, \frac{1}{2}y_{n+1}+ \frac{1}{2} y_n + \frac{h \sqrt{3}}{6} k2)$$

there is this extra term $ \frac{h \sqrt{3}}{6} k2$ which I can't find any way to deal with. Any suggestion would be greatly appreciated .

This IS already the method. You have a system of two equations for $k_1$ and $k_2$ you need to solve. — maxmilgram, Apr 24 '19 at 13:26
ok thanks I was trying to find a simplification like for the trapezoidal rule but there is none right ? — nathan raynal, Apr 24 '19 at 13:31
Additional remark: For some, usually simple, RK schemes it is possible to express the scheme only in terms of $y_n$ and $y_{n+1}$. I dont think thats the case here though... — maxmilgram, Apr 24 '19 at 13:32
that's what I was trying to do but yes I can't find anyway to do it — nathan raynal, Apr 24 '19 at 13:34
As I said, I dont think it is doable due to the cross-dependency. — maxmilgram, Apr 24 '19 at 14:06

score 1 · Answer 1 · answered Apr 24 '19 at 18:09

Summary of my comments: :-)

This IS already the method. You have a system of two equations for $k_1$ and $k_2$ you need to solve.

For some, usually simple, RK schemes it is possible to express the scheme only in terms of $y_n$ and $y_{n+1}$. However due to the cross-dependency I am pretty confident that is not possible in this specific case...

Look here for an example of the derivation of the implict midpoint rule starting from the butcher tableau.

deriving an implicit Runge Kutta method from its Butcher tableau

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