The sine–Gordon and Allen–Cahn equations are two typical models in the fields of conservative Hamiltonian systems and dissipative gradient flows, respectively. As the demand for numerical methods that respect intrinsic energy conservation/dissipation laws turns into a fundamental principle, the subsequent computational efficiency is getting more and more desired. Linearly implicit methods, which only require solutions of linear algebraic systems at each time step, have become a popular way to design efficient schemes. However, the overall computational cost of these methods depends heavily on the speed at which the linear system can be solved. In this paper, we propose an alternative approach for developing highly efficient energy-conservative or dissipative schemes for the sine–Gordon and Allen–Cahn equations. Our approach is based on spatial finite difference approximations and is fully implicit at first glance, but actually it is completely decoupled point by point, allowing for the implementation of a scalar nonlinear equation successively with a lower complexity that is comparable only to the degrees of freedom. We further establish the connections between our approach and the classic Itoh–Abe discrete gradient and the partitioned averaged vector field methods, and then propose the generalized Itoh–Abe discrete gradient method, which offers great flexibility in the ordering of updating each unknown point. One immediate benefit is that we can specifically choose an ordering to achieve a naturally parallel computation of the resulting scheme, significantly improving the computational efficiency. Various numerical experiments are presented to illustrate the performance of the proposed schemes.

中文翻译：

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正弦-戈登和艾伦-卡恩方程的并行和能量保守/耗散格式

正弦-戈登方程和艾伦-卡恩方程分别是保守哈密顿系统和耗散梯度流领域的两个典型模型。随着对尊重内在能量守恒定律/耗散定律的数值方法的需求成为基本原理，后续的计算效率变得越来越理想。线性隐式方法仅需要在每个时间步求解线性代数系统，已成为设计高效方案的流行方法。然而，这些方法的总体计算成本在很大程度上取决于线性系统的求解速度。在本文中，我们提出了一种为正弦-戈登和艾伦-卡恩方程开发高效节能或耗散方案的替代方法。我们的方法基于空间有限差分近似，乍一看是完全隐式的，但实际上它是完全逐点解耦的，允许以较低的复杂度连续实现标量非线性方程，仅与自由度相当。我们进一步建立了我们的方法与经典 Itoh-Abe 离散梯度和分区平均向量场方法之间的联系，然后提出了广义 Itoh-Abe 离散梯度方法，该方法在更新每个未知点的顺序方面提供了很大的灵活性。一个直接的好处是我们可以专门选择一种排序来实现结果方案的自然并行计算，从而显着提高计算效率。提出了各种数值实验来说明所提出方案的性能。