A linearly energy-preserving Fourier pseudospectral method based on energy quadratization for the sine-Gordon equation

In this paper, we develop a linear-implicit energy-preserving Fourier pseudospectral method for the sine-Gordon equation. Based on (invariant) energy quadratization technique, we first reformulate the sine-Gordon equation to an equivalent form with a quadratic energy functional. Then the reformulated system is discretized by the Fourier pseudospectral method in space and the linear-implicit Crank-Nicolson method in time. The new fully discrete scheme is proved to preserve the modified energy conservation law in the full-discrete level, and the linear system resulting from the scheme is shown to be uniquely solvable. Numerical examples are conducted to demonstrate the conservation of energy and the numerical performance of the scheme.