A Hybrid Semi-Lagrangian Cut Cell Method for Advection-Diffusion Problems with Robin Boundary Conditions in Moving Domains

TL;DR

This paper introduces a second-order accurate hybrid semi-Lagrangian cut cell method for advection-diffusion problems with Robin boundary conditions on moving domains, effectively handling complex geometries and boundary conversions.

Contribution

It combines cut cell finite volume discretization with a semi-Lagrangian scheme to address small cell issues and achieve high accuracy on dynamic, complex domains.

Findings

Achieves second order accuracy in multiple norms.

Successfully handles moving boundary concentration conversions.

Demonstrates effectiveness on analytic and numerical tests.

Abstract

We present a new discretization for advection-diffusion problems with Robin boundary conditions on complex time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme that uses a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the $L^1$, $L^2$, and $L^\infty$ norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme to handle conversion of one concentration field to another across a moving boundary.