  This paper aims to investigate a full numerical approximation of
non-autonomous semilnear parabolic partial differential equations (PDEs) with
nonsmooth initial data. Our main interest is on such PDEs where the nonlinear
part is stronger than the linear part, also called reactive dominated transport
equations. For such equations, many classical numerical methods lose their
stability properties. We perform the space and time discretizations
respectively by the finite element method and an exponential integrator. We
obtain a novel explicit, stable and efficient scheme for such problems called
Magnus-Rosenbrock method. We prove the convergence of the fully discrete scheme
toward the exact solution. The result shows how the convergence orders in both
space and time depend on the regularity of the initial data. In particular,
when the initial data belongs to the domain of the family of the linear
operator, we achieve convergence orders $\mathcal{O}\left(h^{2}+\Delta
t^{2-\epsilon}\right)$, for an arbitrarily small $\epsilon>0$. Numerical
simulations to illustrate our theoretical result are provided.
