In this paper, a family of linear structure-preserving (energy conservation) schemes with second-order accuracy in the time direction is developed to numerically solve the undamped sine–Gordon equation. To be specific, first, transformation of the undamped sine–Gordon system into an equivalent new system is made by introducing an improved scalar auxiliary variable (SAV), and generalization of the conservative Crank–Nicolson scheme is made by applying a non-negative family parameter $\vartheta$ to discretize time-dependent variables at the time step $(n+\vartheta )$ instead of just $(n+1∕2)$, thereby to establish a family of second-order conservative semi-discrete schemes. Further, based on the advantages of the improved SAV method, not only does the newly introduced scalar auxiliary variable be uncoupled with the original variables at the discrete level, it also requires the family of approximations, at each time step, no more efforts than the solution of a second-order linear differential equation of elliptic type with constant coefficients, making the computational cost of this method only half of the original one, which thus is particularly effective. Finally, several numerical experiments are presented to demonstrate the efficiency, stability, accuracy and energy conservation of the family of schemes developed herein.

在本文中，开发了一系列在时间方向上具有二阶精度的线性结构保留（能量守恒）方案，以数值求解未阻尼的正弦-Gordon方程。具体来说，首先，通过引入改进的标量辅助变量（SAV）将无阻尼的正弦-戈登系统转换为等效的新系统，并通过应用非负族来推广保守的Crank-Nicolson方案。范围$\vartheta$ 在时间步离散时间相关变量 $（ñ+\vartheta ）$ 而不只是 $（ñ+\mathrm{1个}∕\mathrm{2个}）$，从而建立了一个二阶保守半离散方案族。此外，基于改进的SAV方法的优点，不仅新引入的标量辅助变量在离散级别上与原始变量解耦，而且在每个时间步上都需要近似值族，而所做的工作不超过求解具有常数系数的椭圆型二阶线性微分方程，使得该方法的计算成本仅为原始方法的一半，因此特别有效。最后，提出了几个数值实验，以证明本文开发的一系列方案的效率，稳定性，准确性和节能性。