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We are interested in numerically solving a transitional model derived fromthe Bloch model. The Bloch equation describes the time evolution of the densitymatrix of a quantum system forced by an electromagnetic wave. In a highfrequency and low amplitude regime, it asymptotically reduces to a non-stiffrate equation. As a middle ground, the transitional model governs the diagonalpart of the density matrix. It fits in a general setting of linear problemswith a high-frequency quasi-periodic forcing and an exponentially decayingforcing. The numerical resolution of such problems is challenging. Adaptinghigh-order averaging techniques to this setting, we separate the slow (rate)dynamics from the fast (oscillatory and decay) dynamics to derive a newmicro-macro problem. We derive estimates for the size of the micro part of thedecomposition, and of its time derivatives, showing that this new problem isnon-stiff. As such, we may solve this micro-macro problem with uniform accuracyusing standard numerical schemes. To validate this approach, we presentnumerical results first on a toy problem and then on the transitional Blochmodel.

Multi-frequency averaging and uniform accuracy towards numerical approximations for a Bloch model