  An explicit Milstein-type scheme for stochastic differential equation with
Markovian switching is derived and its strong convergence in
$\mathcal{L}^2$-sense is established without using It\^o-Taylor expansion
formula. Rate of strong convergence is shown to be equal to $1.0$ under the
assumptions that coefficients satisfy mild regularity conditions. More
precisely, coefficients are assumed to be only once differentiable which are
more relaxed conditions than those made in existing literature.
