Space-time spectral method for the two-dimensional generalized sine-Gordon equation

In this paper, we propose a space-time spectral method for solving the generalized two-dimensional sine-Gordon equation with nonhomogeneous Dirichlet boundary conditions and initial conditions. The proposed method is based on the Legendre-Galerkin spectral method in space and the spectral collocation method (or block spectral collocation method) in time. We derive a priori error estimates in both L² and H¹ norms for the semidiscrete formulation. Our formulation has spectral accuracy in both space and time. Our numerical results confirm the exponential convergence of the proposed method in both space and time.