In this paper, we develop a novel staggered mesh (SM) approach for general nonlinear dissipative systems with arbitrary energy distributions (including cases with known or unknown energy lower bounds). Based on this framework, we propose several second-order semi-discrete schemes that maintain linearity, computational decoupling, and unconditional energy stability. Firstly, for dissipative systems with known energy lower bounds, we introduce a positive auxiliary variable $V(t)$ to substitute the total energy functional, subsequently discretizing it on staggered temporal meshes to ensure that the energy remains non-increasing regardless of the size of time step. The newly developed schemes achieve full computational decoupling, maintaining essentially the same computational expense as conventional implicit-explicit methods while demonstrating significantly improved accuracy. Furthermore, we rigorously establish the positivity preservation of the discrete variable $V^{n+1/2}$ which is a crucial property ensuring numerical stability and accuracy. Theoretical analysis confirms second-order temporal convergence for the proposed SM schemes. Secondly, for dissipative systems lacking well-defined energy lower bounds, we devise an alternative auxiliary variable formulation and extend the SM framework to maintain unconditional energy stability while preserving numerical effectiveness and accuracy. Finally, comprehensive numerical experiments, including benchmark problem simulations, validate the proposed schemes' efficacy and demonstrate their superior performance characteristics.

翻译：本文针对具有任意能量分布（包括已知或未知能量下界的情形）的一般非线性耗散系统，提出了一种新颖的交错网格方法。基于此框架，我们提出了几种保持线性、计算解耦及无条件能量稳定性的二阶半离散格式。首先，对于能量下界已知的耗散系统，我们引入一个正辅助变量 $V(t)$ 来替代总能量泛函，随后在交错时间网格上对其进行离散，以确保无论时间步长如何，能量始终保持非增。新开发的格式实现了完全的计算解耦，其计算开销与传统隐式-显式方法基本相当，同时展现出显著提升的精度。此外，我们严格证明了离散变量 $V^{n+1/2}$ 的保正性，这是确保数值稳定性和精度的关键性质。理论分析证实了所提交错网格格式具有二阶时间收敛性。其次，对于缺乏明确定义能量下界的耗散系统，我们设计了另一种辅助变量形式，并扩展了交错网格框架，以在保持数值有效性和精度的同时，维持无条件能量稳定性。最后，包括基准问题模拟在内的综合数值实验验证了所提格式的有效性，并展示了其优越的性能特征。