Operator splitting based structure-preserving numerical schemes for the mass-conserving convective Allen-Cahn equation

Abstract

The mass-conserving convective Allen-Cahn (MCAC) equation is an important component of phase field modeling for the multiphase fluid coupled with a flow field. It inherits the maximum bound principle (MBP) from the classic Allen-Cahn equation, in the sense that the time-dependent solution under appropriate initial and boundary conditions preserves a uniform point-wise bound in the absolute value for all time. In this paper, we develop two structure-preserving numerical schemes for the MCAC equation based on the operator splitting approach. In particular, the advancing of the MCAC equation at each time step is split into two (first-order splitting in time) or three (second-order splitting in time) stages, and each of the stages consists of either a mass-conserving AC equation or a transport equation. The mass-conserving AC part is then discretized by using the classic finite volume approximation in space and the stabilized exponential time differencings in time and the resulting system can be efficiently solved via fast Fourier transform based algorithms. The transport part is solved by explicit strong stability preserving Runge-Kutta substeppings combined with a maximum-principle-satisfying finite volume method. Optimal error estimates are derived for the proposed fully-discrete schemes, as well as preservation of the discrete MBP and conservation of the mass. Various numerical examples are also presented to verify the theoretical results and demonstrate the performance of the proposed schemes.