In this paper, we propose high order numerical methods to solve a 2D advection diffusion equation, in the highly oscillatory regime. We use an integrator strategy that allows the construction of arbitrary high-order schemes {leading} to an accurate approximation of the solution without any time step-size restriction. This paper focuses on the multiscale challenges {in time} of the problem, that come from the velocity, an $\varepsilon-$periodic function, whose expression is explicitly known. $\varepsilon$-uniform third order in time numerical approximations are obtained. For the space discretization, this strategy is combined with high order finite difference schemes. Numerical experiments show that the proposed methods {achieve} the expected order of accuracy, and it is validated by several tests across diverse domains and boundary conditions. The novelty of the paper consists of introducing 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 time scale. The high order in space is obtained enlarging the interpolation stencil already established in [44], and further refined in [46], with a special emphasis on the squared boundary, especially when a Dirichlet condition is assigned. In such case, we compute an \textit{ad hoc} Taylor expansion of the solution to ensure that there is no degradation of the accuracy order at the boundary. On the other hand, the high accuracy in time is obtained extending the work proposed in [19]. The combination of high-order accuracy in both space and time is particularly significant due to the presence of two small parameters-$\delta$ and $\varepsilon$-in space and time, respectively.

翻译：本文提出高阶数值方法求解二维对流扩散方程在高度振荡区域的问题。我们采用积分器策略，可构造任意高阶格式，从而在无任何时间步长限制条件下获得解的高精度近似。本文重点研究该问题在时间维度上的多尺度挑战，该挑战源于速度场——一个表达式已知的 $\varepsilon-$周期函数。我们获得了 $\varepsilon$ 一致的三阶时间数值近似。在空间离散化方面，该策略与高阶有限差分格式相结合。数值实验表明，所提方法达到了预期的精度阶数，并通过多种不同区域和边界条件的测试验证了其有效性。本文的创新点在于提出了一种在空间和时间上均具有高阶精度的数值格式，并特别关注了时间尺度上对小参数的依赖性。空间高阶精度通过扩展[44]中已建立并[46]中进一步完善的插值模板获得，尤其针对方形边界区域，特别是当给定狄利克雷边界条件时。在此情况下，我们计算解的\textit{ad hoc}泰勒展开，以确保边界处精度阶数不降低。另一方面，时间高精度通过扩展[19]中提出的工作实现。空间和时间双高阶精度的结合尤为重要，因为空间和时间分别存在两个小参数——$\delta$ 和 $\varepsilon$。