The Problem
I have been struggling to determine the coefficients $b_i$ and $a_{ij}$ for a second order exponential Runge-Kutta scheme, for which I am provided with the generic formulation as well as Butcher's tableaux. For more context, this scheme is explained here (pg. 221, 226).
The differential equation is of the form: \begin{equation} u'(t) + Au(t) = g(u(t)), \quad u(0) = u_0 \end{equation}
The exponential RK scheme generic formulas are given as:
\begin{align} u_{n+1} & = u_n + h_n\sum^s_{i=1}b_i(-h_nA)(G_{ni}-Au_n) \\ U_{ni} & = u_n + h_n\sum^s_{j=1}a_{ij}(-h_nA)(G_{nj}-Au_n) \\ G_{nj} & = g(U_{nj}) \end{align}
The Butcher's tableaux for the second order method I want to use is given as:
\begin{equation} \begin{array}{c | c c} 0 & & \\ c_2 & c_2\phi_{1,2} & \\ \hline & \phi_1 - \frac{1}{c_2}\phi_2 & \frac{1}{c_2}\phi_2 & \end{array} \end{equation}
where $\phi_{1,2},\,\phi_1$ and $\phi_2$ are known functions of $h_nA$.
Attempt
From the above, by observation I can deduce the following:
\begin{align} u_{n+1} & = u_n + h_n\left[ b_1(-h_nA)(G_{n1}-Au_n) + b_2(-h_nA)(G_{n2}-Au_n) \right]\\ U_{n1} & = u_n,\quad\text{hence}\quad G_{n1} = g(u_n) \\ U_{n2} & = u_n + h_n\left[a_{2,1}(-h_nA)(G_{n1}-Au_n) \right], \quad\text{hence}\quad G_{n2} = g(U_{n2}) \\ \end{align}
where
\begin{align} a_{2,1}(-h_nA) & = c_2\phi_{1,2} \\ b_1(-h_nA) & = \phi_1 - \frac{1}{c_2}\phi_2 \\ b_2(-h_nA) & = \frac{1}{c_2}\phi_2 \end{align}
However, when using the above formulation I attempt to solve a very simple differential equation, for which the exponential Euler scheme was able to solve, the exponential RK is extremely unstable and eventually diverges. Moreover, if I increase $c_2$ to some absurdly large number (e.g. 100), the exponential RK as I formulated above, approximates the exponential Euler scheme which does not seem to make sense. So my only conclusion is that I am formulating the RK scheme in a wrong way.