Efficient Numerical Algorithms For Phase Field Model Of Two-phase Incompressible Fluids

Phase field model of two-phase incompressible fluids have been widely used in scientific and engineering applications,and although they have been a great deal of theoretical and experimental research,the following difficulties remain: stiffness problems due to small interface widths,handling of strongly coupled and strongly non-linear terms.These difficulties undoubtedly pose a great challenge for solving phase field model for two-phase incompressible fluids.Therefore,this paper proposes an efficient and easy-to-implement numerical algorithm for phase field model of the two-phase incompressible fluids,which still satisfies the energy dissipation law at the discrete level.The main research is as follows:Chapter 1 introduces the background and significance of phase field model of twophase incompressible fluids and the current state of research in China and abroad.Chapter 2 focuses on the numerical study of the modified phase field crystal(MPFC)model with long-range interactions,combining the Crank-Nicolson Leap-frog(CNLF)method and the Fourier spectrum method.The uniquely solvable,mass-conserving and energystable properties of this numerical scheme is also demonstrated.Finally,the efficiency,mass conservation and energy stability of the proposed scheme are verified by numerical examples in two and three dimensions.Chapter 3 provides the efficient numerical approximation of the Allen-Cahn equation by introducing an exponential integration factor,solving the nonlinear part by the explicit second order strong stability preserving Runge-Kutta(SSP-RK2)method and solving the linear part by the analytical method,constructing a second order efficient and easy-toimplement Strang splitting scheme.The presented scheme is rigorously analyzed for its maximum principle,energy stability and convergence.In the numerical example,not only the efficiency of the proposed scheme is verified,but also different nonlinear functions are applied to describe the variation of the phase field.Chapter 4 focuses on the CHNS model for two-phase incompressible fluids.In order to propose an easy-to-implement time stepping scheme,a linear,decoupled and unconditionally energy-stable numerical scheme based on the CNLF method and the artificially compressible method is implemented by introducing a non-local variable and designing an ordinary differential equation to help eliminate coupling terms and display treatment of non-linear terms.Finally,the accuracy,efficiency and stability of the algorithm are verified through extensive numerical experiments.Chapter 5 summarises the findings of this paper and looks at future research.