A Fast θ Scheme Combined with the Legendre Spectral Method for Solving a Fractional Klein–Gordon Equation

In the current work, a fast θ scheme combined with the Legendre spectral method was developed for solving a fractional Klein–Gordon equation (FKGE). The numerical scheme was provided by the Legendre spectral method in the spatial direction, and for the temporal direction, a θ scheme of order O(τ2) with a fast algorithm was taken into account. The fast algorithm could decrease the computational cost from O(M2) to O(MlogM), where M denotes the number of time levels. In addition, correction terms could be employed to improve the convergence rate when the solutions have weak regularity. We proved theoretically that the scheme is unconditionally stable and obtained an error estimate. The numerical experiments demonstrated that our numerical scheme is accurate and efficient.