A family of effective structure-preserving schemes with second-order accuracy for the undamped sine–Gordon equation

Abstract

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 ϑ to discretize time-dependent variables at the time step (n+ϑ) 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.