The existing discrete variational derivative method is fully implicit and only second-order accurate for gradient flow. In this paper, we propose a framework to construct high-order implicit (original) energy stable schemes and second-order semi-implicit (modified) energy stable schemes. Combined with the Runge–Kutta process, we can build high-order and unconditionally (original) energy stable schemes based on the discrete variational derivative method. The new energy stable schemes are 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 solver. 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.

中文翻译：

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梯度流的高阶能量稳定离散变分导数格式


现有的离散变分导数方法是完全隐式的，并且对于梯度流只有二阶精度。在本文中，我们提出了一个构建高阶隐式（原始）能量稳定方案和二阶半隐式（修改）能量稳定方案的框架。结合龙格-库塔过程，我们可以基于离散变分导数方法构建高阶且无条件（原始）能量稳定的方案。新的能量稳定方案是隐式的，并导致每个时间步长的大型稀疏非线性代数系统，可以通过使用不精确的牛顿型求解器来有效地求解。为了避免求解非线性代数系统，我们提出了一种宽松的离散变分导数方法，该方法可以构造二阶、线性和无条件（修改）能量稳定方案。进行了多次数值模拟来研究新提出的方案的效率、稳定性和准确性。