Space-time Legendre-Gauss-Lobatto collocation method for two-dimensional generalized sine-Gordon equation

In this paper, we present a space-time Legendre-Gauss-Lobatto (LGL) collocation method for solving the generalized two-dimensional sine-Gordon equation with nonhomogeneous Dirichlet boundary conditions. The proposed method is based on the LGL collocation method to discretize in space, then use the LGL collocation method or block LGL collocation method to discretize in time. Our formulation has high-order accuracy in both space and time. We introduce a new H1 projection for the error analysis of collocation method for nonhomogeneous Dirichlet boundary conditions. This projection differs from the classical H1 projection. The new H1 projection is identically equal to Lagrange interpolation on boundary. The approximation properties of the new H1 projection are obtain. We derive error bounds in both discrete L2 and H1 norms for the (spatially) semidiscrete formulation, the analysis is based on new H1 projection results.