论文标题
能量减少隐式解释runge-梯度流的kutta方法
Energy diminishing implicit-explicit Runge--Kutta methods for gradient flows
论文作者
论文摘要
这项研究着重于对一组高阶隐式解释（IMEX）runge-kutta（RK）方法的开发和分析，这些方法适用于以Lipschitz的持续非线性的离散梯度流的离散梯度流。我们证明，由于稳定技术，这些IMEX-RK方法可以保留原始的能量耗散属性，而无需对时间步长进行任何限制。稳定常数仅取决于IMEX-RKS屠夫表所产生的最小特征值。此外，我们建立了一个简单的框架，该框架可以确定IMEX-RK方案是否能够保留原始能量耗散属性。我们还基于截断误差提供了启发式收敛分析。这是第一个证明线性高阶单步方案可以确保一般梯度流无条件的原始能量稳定性的研究。此外，我们提供了一些满足已建立框架的高级IMEX-RK计划。值得注意的是，我们发现了一种新的四阶段IMEX-RK方案，可降低能量。最后，我们提供了数值示例，以证明所提出方法的稳定性和准确性。
This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge--Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.
