A fourth-order cut-cell method for solving the two-dimensional advection-diffusion equation with moving boundaries

TL;DR

This paper develops a fourth-order cut-cell method for the two-dimensional advection-diffusion equation with moving boundaries on Cartesian grids, embodied by $\partial_t \rho + \mathbf{u}\cdot\nabla\rho = \frac{1}{\mathrm{Pe}}\Delta \rho$ in $\Omega(t)$. It combines Yin-set geometry, ARMS interface tracking, a cell-merging strategy to address topology changes and small cells, PLG-based fourth-order spatial discretization, and a fourth-order IMEX Runge-Kutta time integrator. Theoretical error analysis shows $O(h^4)$ spatial and $O(k^4)$ temporal accuracy, and numerical tests on moving disks and vortex flows confirm the method’s high-order convergence and robustness for moving-boundary problems. This approach enables high-accuracy simulations of moving-boundary phenomena on fixed grids without re-meshing, with potential extensions to more complex fluid-structure interactions and Navier–Stokes/Boussinesq systems.

Abstract

We propose a fourth-order cut-cell method for solving the two-dimensional advection-diffusion equation with moving boundaries on a Cartesian grid. We employ the ARMS technique to give an explicit and accurate representation of moving boundaries, and introduce a cell-merging technique to overcome discontinuities caused by topological changes in cut cells and the small cell problem. We use a polynomial interpolation technique base on poised lattice generation to achieve fourth-order spatial discretization, and use a fourth-order implicit-explicit Runge-Kutta scheme for time integration. Numerical tests are performed on various moving regions, with advection velocity both matching and differing from boundary velocity, which demonstrate the fourth-order accuracy of the proposed method.

Figures (10)

• Figure 1: A sketch of a cut cell in the staggered grid. $\rho$ is stored as cell-averaged values, and different components of velocity ${\mathbf{u}}$ are stored as face-averaged values in different directions.
• Figure 1: Numerical concentration fields at different times. The moving boundary is depicted in a black curve. The simulation is run to a final time of $T=1$ with a grid resolution of $h=1/80$, the CFL numbers are $C_\mathrm{CFL}\approx0.17$ and $0.5$ , respectively.
• Figure 2: Example of ${\@fontswitch{}{\mathcal{}} S}_{{\mathbf{i}}}(t)$ sweeping a corner of $\mathbf{C}_{{\mathbf{i}}}$, where ${\@fontswitch{}{\mathcal{}} S}_{{\mathbf{i}}}(t)$ are the thick lines and ${\@fontswitch{}{\mathcal{}} C}_{{\mathbf{i}}}(t)$ are the regions colored by light gray.
• Figure 2: Numerical concentration fields at different times. The moving boundary is depicted in a black curve. The simulation is run to a final time of $T=1$ with a grid resolution of $h=1/80$ and the CFL number of $C_\mathrm{CFL}\approx 0.05$.
• Figure 3: Example of the boundary entering a cell $\mathbf{C}_{{\mathbf{i}}}$ in a time step without sweeping across any corner nodes.

Theorems & Definitions (14)

• Definition 2.1
• Definition 2.2: Yin space Zhang2020YinSet
• Theorem 2.3: Global topology of Yin sets Zhang2020YinSet
• Theorem 2.4
• Definition 2.5: General IT
• Definition 2.6: MARS method Zhang2016MARS
• Definition 2.7
• Theorem 2.8
• Proof 1
• Definition 2.9: The ARMS strategy Hu2024ARMS