具有退化迁移率相场模型的高阶自适应半隐式时间离散方法

Phase field models with degenerate mobility are highly nonlinear PDEs with high order space derivatives, which requires severe time step restriction for explicit time marching methods to maintain stability. Traditional semi-implicit methods are not efficient for phase field models with degenerate mobility, in which case the stiff and non-stiff components can not be well separated. Thus, it is of paramount challenge to develop efficient numerical methods with high order accuracy. To fix the problem, the project pays special attention to developing high order adaptive semi-implicit time marching methods for phase field models with degenerate mobility. We start with a convex splitting scheme, analyze the error estimate and prove the unconditional energy stability. However, the convex splitting scheme is only first order accurate in time. To improve the temporal accuracy, a high order semi-implicit Runge-Kutta and a semi-implicit spectral deferred correction method are employed here. The numerical simulations of phase field problems need long time to reach steady state, and then adaptive time step is effective and of paramount importance, which is determined based on the time derivative of the corresponding energy. Numerical results are given to illustrate that the combination of local discontinuous Galerkin spatial discretization, the high order temporal scheme with adaptive time-stepping strategy and the multigrid solver is a robust, accurate and efficient simulation tool when solving phase field problems with degenerate mobility.

对退化迁移率相场模型方程进行高效、精确数值求解的挑战在于：其高阶非线性性使显式时间离散对时间步长的要求较为严苛，而其刚性和非刚性部分不易分离的特点使得传统半隐式时间离散方法不再适用。为了解决这一难题，本项目研究针对此类方程的高阶自适应半隐式时间离散方法。首先，构造一种线性半隐式凸分解格式，分析其误差特性，并给出其能量稳定性的理论证明。在此基础上，引入高阶半隐式 Runge-Kutta 时间离散和半隐式谱延迟校正方法，实现时间上的高阶精度。其次，在保证数值模拟精确性的前提下，结合相场模型能量变化实现时间步长的自适应化，提高数值模拟效率。最后，将所提出的高阶自适应半隐式时间离散方法与局部间断有限元空间离散及线性多重网格方法结合，形成一套针对退化迁移率相场模型方程的精确、高效、鲁棒的数值求解算法，促进相场模型在诸多物理现象模拟中的应用。

本项目针对不同的相场模型问题（比如 modified phase field crystal 方程、Cahn-Hilliard-Navier-Stokes 模型、 binary fluid surfactant 模型），构造间断有限元空间离散和一阶或二阶时间离散格式，并证明了相应格式的能量稳定性。为了提高时间精度，我们提出高阶半隐式谱延迟校正方法，该方法具有一般性，可被用于求解一系列非线性偏微分方程中。为了进一步提高计算效率，我们还引入了时间步长的自适应思想，避免了时间步长过小造成的计算浪费和时间步长过大造成的精度损失。对于时间和空间离散之后产生的线性或者非线性代数方程组，采用多重网格方法进行快速求解，并数值上表明其具有最优的收敛速度。本项目发表标注 SCI 论文6篇。 这些期刊包括 SIAM Journal on Scientific Computing, Journal of Scientific Computing, Journal of Computational Physics, Communications in Computational Physics，均为专业顶级杂志。