Why is it assumed that $c_i = \sum_{j=1}^sa_{i,j}$ in the butcher tableau of a RK-method?

Question

In my textbook it is stated that we make a "simplifying assumption" $$c_i = \sum_{j=1}^sa_{i,j}, $$ where $c_i, a_{i,j}$ are the constants in the butcher tableau.

What's the relevancy of this assumption? Is it only for explicit RK-methods?

Answer 1

You want that the method produces on the autonomous version of a non-autonomous system the same values in both versions. Remember that you get the autonomous version by enlarging the state vector $y=(y_1,...,y_n)$ to $$\bar y=(y_0,y_1,...,y_n)$$ with the added equation $\dot y_0=\bar {f_0}(\bar y)=1$, $y_0(t_0)=t_0$, that is, $$\frac{d\bar y}{dt}=\bar f(\bar y)=(1, f_1(y_0,y),...,f_1(y_0,y)),\\ \bar y(t_0)=(t_0, y_{1,0},...,y_{n,0}).$$

Now insert into the Runge-Kutta mechanism to find that in the first component of the intermediate direction vectors $$ \bar k_i=f(\bar y+h\sum_j a_{ij}\bar k_j) $$ for the autonomous system you have $\bar k_{i,0}=1$, so that the first component of the $i$-th intermediate point in the argument of above formula is $$y_0+h\sum_ja_{ij}k_{j,0}=y_0+h\sum_ja_{ij}.$$ This compares in the non-autonomous implementation to the evaluation of $$ k_i=f(t+hc_i,y+\sum_j a_{ij}k_j) $$ to the first argument $t+c_ih$, so that for both to be equal one needs $$ c_i=\sum_ja_{ij}. $$