In this paper we propose a numerical method to solve a 2D advection-diffusion equation, in the highly oscillatory regime. We use an efficient and robust integrator which leads to an accurate approximation of the solution without any time step-size restriction. Uniform first and second order numerical approximations in time are obtained with errors, and at a cost, that are independent of the oscillation frequency. {This work is part of a long time project, and the final goal is the resolution of a Stokes-advection-diffusion system, in which the expression for the velocity in the advection term, is the solution of the Stokes equations.} This paper focuses on the time multiscale challenge, coming from the velocity that is an $\varepsilon-$periodic function, whose expression is explicitly known. We also introduce a two--scale formulation, as a first step to the numerical resolution of the complete oscillatory Stokes-advection-diffusion system, that is currently under investigation. This two--scale formulation is also useful to understand the asymptotic behaviour of the solution.

翻译：本文提出了一种数值方法，用于求解高度振荡机制下的二维平流-扩散方程。我们采用高效且稳健的积分器，在无需时间步长限制的情况下，即可获得解的精确近似。在时间方向实现了与振荡频率无关的一阶和二阶均匀数值近似，其误差和计算成本均不依赖于振荡频率。本研究属于长期项目的一部分，最终目标是求解斯托克斯-平流-扩散系统，其中平流项中的速度表达式为斯托克斯方程的解。本文聚焦于时间多尺度挑战，该挑战源于速度显式已知的$\varepsilon$周期函数。此外，我们引入了双尺度公式，作为数值求解完整振荡斯托克斯-平流-扩散系统的初步步骤（该系统目前正在研究中）。该双尺度公式还有助于理解解的渐近行为。