变密度不可压缩Navier-Stokes方程具有保结构形式的若干高效分裂算法研究

The incompressible Navier-Stokes equations with variable density are widely used in practical problems, such as the multiphase flows, the ground water flows, etc., and attract the extensive attention of scientific researchers. Numerical simulation of the incompressible Navier-Stokes equations with variable density becomes very difficult because this problem is a nonlinear coupled system with hyperbolic, parabolic and elliptic features and has important physical conservation. In view of a basic understanding of splitting algorithms and finite element algorithms for incompressible Navier-Stokes equations with constant density and other nonlinear problems, in this project, the decoupling of density-velocity-pressure of the variable-density incompressible Navier-Stokes equations is taken as the basic entry point and we will study: (1) we will study the projection algorithm and the fractional step algorithm with structure preserving and weakly coupling form to realize the decoupling of density-velocity-pressure; (2) based upon the discontinuous finite element method for the continuity equation and the conforming finite element method for momentum equation, we will study the stabilities, convergences and the approximation accuracies for the finite element fully discrete systems of these splitting algorithms; (3) for the interfacial dynamics about immiscible and incompressible two-phase flows, we will study the unconditionally stable mixed finite element algorithms for Navier-Stokes-Allen-Cahn equations and Navier-Stokes-Cahn-Hilliard equations with the help of the method of phase field models, and study the convergences of these algorithms under weak regularities. The research of this project has important theoretical and practical value in promoting the research of numerical methods in this field and other related fields.

变密度不可压缩NS方程在多相流、地下水流等实际问题中有广泛的应用，吸引着科研工作者的广泛关注。由于该方程是具有双曲、抛物和椭圆的非线性耦合系统，并具有重要的物理守恒性，使得数值模拟该问题变得十分困难。基于我们对常密度不可压缩NS方程及其他非线性问题分裂算法和有限元算法的初步认识，本项目以变密度不可压缩NS方程的密度-速度-压力的解耦为基本切入点，拟研究（1）具有保结构性和弱耦合形式的投影算法、分数步长算法，使之实现密度-速度-压力的解耦；（2）基于连续性方程的间断有限元和动量方程的协调有限元，研究这些分裂算法有限元全离散系统的稳定性、收敛性及逼近精度；（3）对两种不相混合且不可压缩流体的界面演变，借助相场模型方法，研究NS相场方程的无条件稳定混合有限元算法，并研究这些算法在弱正则性条件下的收敛性。本项目的研究对推动本领域及其他关联问题数值方法的研究都具有重要的理论和应用价值。

变密度不可压缩Navier-Stokes方程在多相流、地下水流等实际问题中有广泛的应用，吸引着科研工作者的广泛关注。由于该方程是具有双曲、抛物和椭圆的非线性耦合系统，并具有重要的物理守恒性，使得数值模拟该问题变得十分困难。基于我们对常密度不可压缩Navier-stokes方程及其他非线性问题分裂算法和有限元算法的初步认识，本项目以设计和构造求解变密度不可压缩流体方程和其他非线性抛物方程高效、稳定的数值算法为目的，主要研究了求解变密度不可压缩Navier-Stokes方程、变密度不可压缩MHD方程组和La ndau-Lifshitz方程的高效数值算法，具体包括：(1) 求解变密度不可压缩Navier-Stokes方程的分数步长算法、迭代加罚方法、算子分裂方法；(2).求解变密度不可压缩MHD方程的一阶Euler时间离散算法；（3）求解带有质量扩散的变密度不可压缩Navier-Stokes方程的一阶Euler有限元算法；(4)求解非线性Landau-Lifshitz方程的一阶Euler和二阶Crank-Nicolson有限差分和有限元算法；(5) 求解常密度Navier-Stokes方程的二阶分数步长算法；(6) 求解一类MHD方程组的一阶Euler和二阶BDF有限元算法、投影算法、分数步长算法。对这些算法我们从理论上分析了算法的稳定性和收敛性，给出了算法的收敛精度，并通过数值实验验证了算法的稳定性和可靠性。本项目的研究对推动本领域及其他关联问题数值方法的研究都具有重要的理论和应用价值。