An Order Bound for Runge–Kutta Methods

Let $u_0 ,u_1 , \cdots $ be defined by $u_0 = 5,u_{n + 1} = [{{(4u_n + 2n + 3)} / 3}]$, $(n = 0,1,2, \cdots )$ so that $u_1 = 7$, $u_2 = 11$, $u_3 = 17$, $u_4 = 25, \cdots $. The main result of this paper is that there does not exist an explicit Runge–Kutta method with s stages and order $p \geqq u_n $ unless $s > p + n$.