# Does an implicit Runge Kutta scheme applied on a nonlinear ODE give a nonlinear set of equations to solve in each step?

We want to approximately solve an ODE $$\frac{dy}{dt} = f(y,t)$$ with the Runge Kutta method $$y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i$$ $$k_i = f\left(y_n + h \sum_{j=1}^s a_{ij} k_j,\,t_n + c_i h\right)$$

For explicit methods the matrix $a$ is strictly lower triangular and thus you can determine $k_i$ explicitly without solving any set of equations. For implicit methods on the other hand, $a$ is not lower triangular and calculating $k_i$ becomes more complicated since you have to solve a set of equations.

I was of the impression that for implicit RK methods you would have to solve a linear set of equations, but looking at the definition of $k_i$ it looks to me as if $f(y,t)$ is nonlinear the system of equations determining $k_i$ will also be nonlinear. Is this correct?

If so, does this render implicit RK methods quite useless for nonlinear $f(y,t)$s because of the heavy computational power required to solve the nonlinear equationset?

For more background on RK methods take a look at: List of RK methods, RK methods in general.