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We consider the second-order in time Strang-splitting approximation forvector-valued and matrix-valued Allen-Cahn equations. Both the linearpropagator and the nonlinear propagator are computed explicitly. For thevector-valued case, we prove the maximum principle and unconditional energydissipation for a judiciously modified energy functional. The modified energyfunctional is close to the classical energy up to $\mathcal O(\tau)$ where$\tau$ is the splitting step. For the matrix-valued case, we prove a sharpmaximum principle in the matrix Frobenius norm. We show modified energydissipation under very mild splitting step constraints. We exhibit severalnumerical examples to show the efficiency of the method as well as thesharpness of the results.

Stability and convergence of Strang splitting. Part II: tensorial Allen-Cahn equations