HIGH ORDER NUMERICAL SIMULATIONS FOR THE BINARY FLUID-SURFACTANT SYSTEM USING THE DISCONTINUOUS GALERKIN AND SPECTRAL DEFERRED CORRECTION METHODS

摘要

In this paper, we propose a high order numerical scheme to simulate the binary fluid-surfactant system by combining the semi-implicit spectral deferred correction (SDC) method and the energy stable linear scheme, in the framework of discontinuous Galerkin (DG) methods. The linear scheme we develop in this paper is decoupled and unconditionally energy stable, which is based on the combination of the convex-concave splitting principle and the invariant energy quadratization approach. However, the scheme is only first order accurate with respect to time, and the SDC method can be employed to iteratively improve the temporal accuracy. Specially, the SDC scheme can be extremely accurate when coupled with an adaptive time stepping strategy. Our numerical scheme leads to a set of decoupled and linear algebraic equations; at each time step, we apply a multigrid solver to solve the equations efficiently. In particular, due to the local property of the DG methods, the resulting algebraic equations can be solved in an explicit way when coupled with the multigrid solver, which is an attractive advantage of the DG method. Various numerical experiments are performed to illustrate the high order accuracy, capability, and efficiency of the proposed methods when solving the binary fluid-surfactant system.