In this work, we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker-Planck equation and the Keller-Segel equations. The two proposed schemes are first order accurate in time, explicitly solvable, and second order and fourth order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proven positivity-preserving and energy-dissipative: the second order scheme can achieve so unconditionally; the fourth order scheme only requires a mild time step and mesh size constraint. Furthermore, the fourth order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions.
翻译:在这项工作中,我们为具有梯度流结构的连续性方程式引入了半隐含或隐含的有限差异方案,此类方程式的例子包括线性Fokker-Planck方程式和Keller-Segel方程式。两种拟议方案在时间上第一顺序准确,明确可以溶解,在空间中第二顺序和第四顺序准确,通过对传统连续不变要素法的有限差异实施而获得。完全独立的方案证明是活性保留和能量分散的:第二顺序方案可以无条件实现;第四顺序方案只需要一个轻度的时间步骤和网状尺寸限制。此外,第四顺序方案是能够实现正值和能量衰减特性的第一种高度空间分解,这既适合长期模拟,又适合获得准确的稳定状态解决方案。