Unconditionally stable fully-discrete finite element numerical scheme for active fluid model

Abstract

In this paper, we propose a linear, decoupled, unconditionally stable fully-discrete finite element scheme for the active fluid model, which is derived from the gradient flow approach for an effective non-equilibrium free energy. The developed scheme is employed by an implicit-explicit treatment of the nonlinear terms and a second-order Gauge–Uzawa method for the decoupling of computations for the velocity and pressure. We rigorously prove the unique solvability and unconditional stability of the proposed scheme. Several numerical tests are presented to verify the accuracy, stability, and efficiency of the proposed scheme. We also simulate the self-organized motion under the various external body forces in 2D and 3D cases, including the motion direction of active fluid from disorder to order. Numerical results show that the scheme has a good performance in accurately capturing and handling the complex dynamics of active fluid motion.