Runge-Kutta methods and Butcher tableau What does the Butcher tableau of a Runge-Kutta method tell me about the method, besides the coefficients in its formulation? In particular, what requirements about it guarantee consistency and therefore convergence? I have been told something necessary is the row-sum condition, i.e.:
$$c_i=\sum\limits_{j=1}^na_{ij}.$$
What does this guarantee or what is this necessary for? And could you give me proofs of any results you mention in your answers? Or links to them anyway. Thanks.
 A: The Butcher Tableau determines the stability function $R(z)$ of the corresponding method.
In particular, for the Linear Test equation due to Dahlquist
$$u'(t) = \lambda u(t) \Rightarrow u(t) = u_0 e^{\lambda (t - t_0)}$$
the stability function determines how the approximation $u_{n+1}$ follows from the previous iterate $u_n$:
$$ u_{n+1} = R(z) u_n, \quad z = \lambda \Delta t_{n+1}$$
This stability function can actually be computed as (see for instance [1] or [2])
$$ R(z) = \frac{\text{det}\left(I-zA + z \boldsymbol 1 \boldsymbol b  \right) }{\text{det}\left(I-zA\right)}$$
which simplifies for explicit methods with strictly lower triangular matrix $A$ to
$$ R(z) =\text{det}\left(I-zA + z \boldsymbol 1 \boldsymbol b  \right). $$
This stability function determines (as the name suggests) the region of absolute stability:
$$ z \in \mathbb C, \text{Re}(z): \vert R(z) \vert \leq 1. $$
Reason I mention this is that convergence is not guaranteed for convergent methods - the method has also to be a stable for the employed finite timesteps $\Delta t$.

For explicit methods, $R(z)$ is actually a polynomial
$$ R(z) = \sum_{j=0}^S \alpha_j z^j$$
and one can directly check the order of consistency by checking to what power the coefficients $\alpha_j$ agree with the terms of the exponential, i.e.,
$$ \alpha_j = \frac{1}{j!}, j = 0, \dots , p.$$
For implicit methods, however, the order of consistency cannot be that easily read-off from $R(z)$.
