Studying time-dependent behavior in lasers is analytically difficult due to the saturating nonlinearity inherent in the Maxwell-Bloch equations and numerically demanding because of the computational resources needed to discretize both time and space in conventional finite-difference time-domain approaches. We describe here an efficient spectral method to overcome these shortcomings in complex lasers of arbitrary shape, gain medium distribution, and pumping profile. We apply this approach to a quasidegenerate two-mode laser in different dynamical regimes and compare the results in the long-time limit to the steady-state ab initio laser theory (SALT), which is also built on a spectral method but makes a more specific ansatz about the long-time dynamical evolution of the semiclassical laser equations. Analyzing a parameter regime outside the known domain of validity of the stationary inversion approximation, we find that for only a narrow regime of pump powers the inversion is not stationary, and that this, as pump power is further increased, triggers a synchronization transition upon which the inversion becomes stationary again. We provide a detailed analysis of mode synchronization (also known as cooperative frequency locking), revealing interesting dynamical features of such a laser system in the vicinity of the synchronization threshold.