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.