  The existence and uniqueness of the numerical invariant measure of the
backward Euler-Maruyama method for stochastic differential equations with
Markovian switching is yielded, and it is revealed that the numerical invariant
measure converges to the underlying invariant measure in the Wasserstein
metric. Under the polynomial growth condition of drift term the convergence
rate is estimated. The global Lipschitz condition on the drift coefficients
required by Bao et al., 2016 and Yuan et al., 2005 is released. Several
examples and numerical experiments are given to verify our theory.
