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In this paper, we present a Crank-Nicolson leapfrog (CNLF) time stepping decoupled scheme for the fluid-fluid interaction problems. The scheme is based on the modular grad-div stabilized (MGDS) method which is very efficient for the fluid-fluid interaction problems. The algorithm is very simple to implement and retain the benefits of grad-div stabilization. The unconditional stability of the proposed scheme is proven rigorously. Numerical tests are given to verify the efficiency and robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

We present the modular grad-div stabilization and defect-deferred correction (MGDDC) method for the incompressible Navier-Stokes equations. The new algorithm does not suffer from either solver breakdown or debilitating slow down for large values of grad-div parameters. Furthermore, it is improving two aspects of the numerical simulations for the problem. In the first step, a deferred correction method lifts the numerical order of accuracy from first order to second order in terms of the time interval. In the second step, a further defect correction is included as well, which is often included to control numerical noise. Stability analysis and optimal-order convergence of the method are proven. Numerical tests are presented to highlight the accuracy and effectiveness of the algorithms. [ABSTRACT FROM AUTHOR]

In this work, we present two linearly and unconditionally energy stable numerical schemes for a vesicle membrane model that exactly satisfies the conservation of volume constraint and penalizes the surface area constraint. First of all, we introduce some auxiliary variables to transform the original model into an equivalent system using the energy quadratization(EQ) technique. Moreover, with the transformed free energy a quadratic functional with respect to the new variables and the modified energy dissipative law is satisfied. Then an implicit-explicit(IMEX) time discrete technique for the semi-discrete scheme is used to construct two linearly and unconditionally energy stable fully discrete schemes for the model. Finally, numerical experiments are presented to demonstrate the accuracy and unconditionally energy-stability of the proposed schemes. [ABSTRACT FROM AUTHOR]