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

Abstract

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-$δ$ and $\varepsilon$-in space and time, respectively.

Figures (14)

• Figure 1: (a): Experimental setup of the diffusion-trapping of surfactants in presence of a trap. The zoom-in on the right shows the composition of the anions when they are stuck at the surface of the air bubble: the hydrophobic tails are inside the air bubble, while the hydrophilic heads lay on the surface. (b) Scheme of the potential $V(x)$, defined in Eq. \ref{['eq_V_expr']}, where $\delta$ is the thickness of the attractive-repulsive layer.
• Figure 2: Shape of different domains: (a) ellipsoidal, (b) flower-shaped and (c) cardioid-shape domain.
• Figure 3: (a): Representation of the domain $\Omega$ in 2D, where $\Gamma_\mathcal{S}$ is the external wall, $\mathcal{B}$ is the bubble with boundary $\Gamma_{\mathcal{B}}$ and radius $R_\mathcal{B}$. The vectors $n$ and $\tau$ are the outer and tangential unit vectors to $\Gamma_B$, respectively. (b): Classification of the inside (blue circles), ghost (orange circles) and inactive (red hole circles) points.
• Figure 4: (a): 9-point stencil for the discrete operators $L_{{\rm i},h}^{\rm 4th}$ and $Q_{{\rm i},h}^{\rm 4th}$ for the internal points $P_{\rm i,j} = (x_i,y_j)$. (b): Representation of the upwind 16-points stencil associated with the ghost point $G$, boundary point $B$ and the relative outgoing normal vector $n$ to $\Gamma_\mathcal{B}$. The stencil is composed by 15 blue internal points, together with an orange ghost point G.
• Figure 5: Relative error in $L^1,L^2,L^\infty$--norms, with $t_{\rm fin } = 0.1,$ for a fixed $\Delta t_{\rm ref} = 10^{-5}$ for Eq. \ref{['eq_4th_space']} (a) and Eq. \ref{['eq_4th_space_variable']} (b). The domain is $\Omega = [-1,1]^2$, $N_{\rm ref} = 640$, the initial condition is defined in Eq. \ref{['eq_expr_IC_tests']} with $x_{m_1} = y_{m_1} = 0, \sigma = 0.1$ homogeneous Dirichlet boundary conditions (i.e., $f = 0$ in Eq. \ref{['eq_Dir_bc']}) and velocity in Eq. \ref{['eq_expr_velocity_space']} for (b).

Theorems & Definitions (4)

• Proposition 1
• Proposition 2
• Proposition 3
• proof