A second-order accurate and unconditionally energy stable numerical scheme for nonlinear sine-Gordon equation

In this work, a second-order finite difference method is proposed to solve a nonlinear sine-Gordon equation. The constructed implicit scheme is proved to be unconditionally energy stable. A linear iteration algorithm is used to solve this nonlinear numerical scheme, and we prove that this iteration algorithm is convergent with a negligible constraint for time step. By constructing a suitable high-precision numerical solution, and using the inverse inequality and refinement constraint Δt≤Ch, the error estimate in L∞(0,T;L∞) norm of the fully discrete scheme is obtained. Several numerical examples are given to confirm the sharpness of our theoretical analysis.