论文标题
基于DVD方法的梯度流的能量稳定方案
Energy stable schemes for gradient flows based on the DVD method
论文作者
论文摘要
现有的离散变分衍生方法仅是二阶准确且完全隐含的。在本文中，我们提出了一个框架，以构建任意的高阶隐式（原始）能量稳定方案和二阶半无限（修改）能量稳定方案。结合runge-kutta过程，我们可以基于离散的衍生化方法来构建任意的高阶和无条件（原始）能量稳定方案。新的能量稳定方案是隐式的，并在每个时间步骤中导致较大的稀疏非线性代数系统，可以使用不精确的牛顿型算法有效地求解。为了避免求解非线性代数系统，我们提出了一种松弛的离散变分衍生物方法，该方法可以构建二阶，线性和无条件（修改）能量稳定方案。进行了几项数值模拟，以研究新提出的方案的效率，稳定性和准确性。
The existing discrete variational derivative method is only second-order accurate and fully implicit. In this paper, we propose a framework to construct an arbitrary high-order implicit (original) energy stable scheme and a second-order semi-implicit (modified) energy stable scheme. Combined with the Runge--Kutta process, we can build an arbitrary high-order and unconditionally (original) energy stable scheme based on the discrete variational derivative method. The new energy stable scheme is implicit and leads to a large sparse nonlinear algebraic system at each time step, which can be efficiently solved by using an inexact Newton type algorithm. To avoid solving nonlinear algebraic systems, we then present a relaxed discrete variational derivative method, which can construct second-order, linear, and unconditionally (modified) energy stable schemes. Several numerical simulations are performed to investigate the efficiency, stability, and accuracy of the newly proposed schemes.
