0

I have the following exercise:

Consider an explicit two-stage Runge-Kutta method given by the Butcher table: $$ A = \begin{pmatrix} 0 & 0 \\ a & 0 \end{pmatrix}, \quad \vec{c} = \begin{pmatrix} 0 \\ c \end{pmatrix}, \quad \vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}. $$ Write down equations for the parameters such that the method is consistent of order two. Show that the order of consistency cannot be higher.

I don't know how to start this exercise. Can someone provide some idea?

NB: I want only an idea to start the exercise, not a complete step-by-step solution! Thank you.

1 Answers1

1

Consistency is given by

$$\sum_{j=1}^{i-1} a_{ij} = c_i$$

for $j=2,\ldots, s$, and $s$ is the number of stages of the method.

So this should be straightforward.

Also, for order $p$, you must choose the $b_i$ such that you have a nonzero $h^p$ term in the expansion, leaving only $\mathcal{O}(h^{p+1})$.

So, writing out

$$y_{m+1} = y_m + \sum_{i=1}^s b_ik_i$$

with $k_i = hf(t_i+c_ih,y_i+\sum_{j=1}^{i-1}a_{ij}k_j)$,

for each step in an order 2 method, you should get some very clear conditions for $b_i$, given what you know about $c$.

Emily
  • 35,688
  • 6
  • 93
  • 141