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In this paper, we propose high order numerical methods to solve a 2Dadvection diffusion equation, in the highly oscillatory regime. We use anintegrator strategy that allows the construction of arbitrary high-orderschemes {leading} to an accurate approximation of the solution without any timestep-size restriction. This paper focuses on the multiscale challenges {intime} of the problem, that come from the velocity, an $\varepsilon-$periodicfunction, whose expression is explicitly known. $\varepsilon$-uniform thirdorder in time numerical approximations are obtained. For the spacediscretization, this strategy is combined with high order finite differenceschemes. Numerical experiments show that the proposed methods {achieve} theexpected order of accuracy, and it is validated by several tests across diversedomains and boundary conditions. The novelty of the paper consists ofintroducing a numerical scheme that is high order accurate in space and time,with a particular attention to the dependency on a small parameter in the timescale. The high order in space is obtained enlarging the interpolation stencilalready established in [44], and further refined in [46], with a specialemphasis on the squared boundary, especially when a Dirichlet condition isassigned. In such case, we compute an \textit{ad hoc} Taylor expansion of thesolution to ensure that there is no degradation of the accuracy order at theboundary. On the other hand, the high accuracy in time is obtained extendingthe work proposed in [19]. The combination of high-order accuracy in both spaceand time is particularly significant due to the presence of two smallparameters-$\delta$ and $\varepsilon$-in space and time, respectively.

High order multiscale methods for advection-diffusion equation in highly  oscillatory regimes: application to surfactant diffusion and generalization  to arbitrary domains