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This paper aims to investigate a full numerical approximation ofnon-autonomous semilnear parabolic partial differential equations (PDEs) withnonsmooth initial data. Our main interest is on such PDEs where the nonlinearpart is stronger than the linear part, also called reactive dominated transportequations. For such equations, many classical numerical methods lose theirstability properties. We perform the space and time discretizationsrespectively by the finite element method and an exponential integrator. Weobtain a novel explicit, stable and efficient scheme for such problems calledMagnus-Rosenbrock method. We prove the convergence of the fully discrete schemetoward the exact solution. The result shows how the convergence orders in bothspace 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 linearoperator, we achieve convergence orders $\mathcal{O}\left(h^{2}+\Deltat^{2-\epsilon}\right)$, for an arbitrarily small $\epsilon>0$. Numericalsimulations to illustrate our theoretical result are provided.

Convergence analysis of the Magnus-Rosenbrock type method for the finite  element discretization of semilinear non-autonomous parabolic PDEs with  nonsmooth initial data