In this paper, we present a novel spectral renormalization exponential integrator method for solving gradient flow problems. Our method is specifically designed to simultaneously satisfy discrete analogues of the energy dissipation laws and achieve high-order accuracy in time. To accomplish this, our method first incorporates the energy dissipation law into the target gradient flow equation by introducing a time-dependent spectral renormalization (TDSR) factor. Then, the coupled equations are discretized using the spectral approximation in space and the exponential time differencing (ETD) in time. Finally, the resulting fully discrete nonlinear system is decoupled and solved using the Picard iteration at each time step. Furthermore, we introduce an extra enforcing term into the system for updating the TDSR factor, which greatly relaxes the time step size restriction of the proposed method and enhances its computational efficiency. Extensive numerical tests with various gradient flows are also presented to demonstrate the accuracy and effectiveness of our method as well as its high efficiency when combined with an adaptive time-stepping strategy for long-term simulations.

翻译：本文提出了一种新颖的谱重整指数积分方法，用于求解梯度流问题。该方法专为同时满足能量耗散律的离散类比并实现时间上的高阶精度而设计。为实现这一目标，我们首先通过引入时间依赖谱重整因子，将能量耗散律融入目标梯度流方程。随后，采用空间谱近似与时间指数时间差分对耦合方程进行离散。最后，对得到的全离散非线性系统进行解耦，并在每个时间步利用皮卡迭代求解。此外，我们在系统中引入一个额外的强制项以更新时间依赖谱重整因子，这极大放宽了所提方法的时间步长限制，并提升了其计算效率。通过针对多种梯度流的大量数值实验，我们验证了该方法在精度、有效性方面的表现，以及在结合自适应时间步进策略进行长期模拟时的高效性。