  In this paper, we develop a novel, linearly implicit and local
energy-preserving scheme for the sine-Gordon equation. The basic idea is from
the invariant energy quadratization approach to construct energy stable schemes
for gradient systems, which are energy dispassion. We here take the sine-Gordon
equation as an example to show that the invariant energy quadratization
approach is also an efficient way to construct linearly implicit and local
energy-conserving schemes for energy-conserving systems. Utilizing the
invariant energy quadratization approach, the sine-Gordon equation is first
reformulated into an equivalent system, which inherits a modified local energy
conservation law. The new system are then discretized by the conventional
finite difference method and a semi-discretized system is obtained, which can
conserve the semi-discretized local energy conservation law. Subsequently, the
linearly implicit structure-preserving method is applied for the resulting
semi-discrete system to arrive at a fully discretized scheme. We prove that the
resulting scheme can exactly preserve the discrete local energy conservation
law. Moveover, with the aid of the classical energy method, an unconditional
and optimal error estimate for the scheme is established in discrete
$H_h^1$-norm. Finally, various numerical examples are addressed to confirm our
theoretical analysis and demonstrate the advantage of the new scheme over some
existing local structure-preserving schemes.
