  We propose a fourth-order unfitted characteristic finite element method to
solve the advection-diffusion equation on time-varying domains. Based on a
characteristic-Galerkin formulation, our method combines the cubic MARS method
for interface tracking, the fourth-order backward differentiation formula for
temporal integration, and an unfitted finite element method for spatial
discretization. Our convergence analysis includes errors of discretely
representing the moving boundary, tracing boundary markers, and the spatial
discretization and the temporal integration of the governing equation.
Numerical experiments are performed on a rotating domain and a severely
deformed domain to verify our theoretical results and to demonstrate the
optimal convergence of the proposed method.
