High Order Multiscale Methods for Advection-Diffusion Equation in Highly Oscillatory Regimes: Application to Surfactant Diffusion and Generalization to Arbitrary Domains (Communications in Computational Physics, 2025)
In this paper, we propose high order numerical methods to solve a 2D advection diffusion equation, in the highly oscillatory regime. We use an integratorstrategy that allows the construction of arbitrary high-order schemes leading to anaccurate approximation of the solution without any time step-size restriction. Thispaper focuses on the multiscale challenges in time of the problem, that come fromthe velocity, an $ε$-periodic function, whose expression is explicitly known. $ε$–uniformthird order in time numerical approximations are obtained. For the space discretization, this strategy is combined with high order finite difference schemes. Numericalexperiments 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 orderaccurate in space and time, with a particular attention to the dependency on a smallparameter 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 specialemphasis on the squared boundary, especially when a Dirichlet condition is assigned.In such case, we compute an ad hoc Taylor expansion of the solution to ensure thatthere is no degradation of the accuracy order at the boundary. On the other hand, thehigh 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 thepresence of two small parameters—$δ$ and $ε$—in space and time, respectively.
Keywords: High order discretization; time multi-scale; advection-diffusion equation; surfactant; oscillating trap; arbitrary domains.