IMEX HERMITE-GALERKIN SPECTRAL SCHEMES WITH ADAPTIVE TIME STEPPING FOR THE COUPLED NONLOCAL GORDON-TYPE SYSTEMS IN MULTIPLE DIMENSIONS

This study is devoted to constructing some implicit-explicit (IMEX) Hermite-Galerkin spectral schemes for simulating the dynamics of coupled nonlocal Gordon-type systems with fractional Laplacian, where the systems are defined in the multidimensional unbounded domains R-d (d=1, 2, and 3). For this purpose, we apply the Hermite-Galerkin spectral method with scaling factor for the spatial approximation. The linearized IMEX schemes, including the IMEX Crank-Nicolson scheme with adaptive time stepping and the IMEX second-order backward differentiation formula scheme with adaptive time stepping, are employed for the temporal discretization. The existence and uniqueness of the numerical solutions are strictly proved for these two kinds of schemes. The main advantages of our algorithms are that (i) the errors and singularities introduced by the domain truncation can be avoided because the original problems are directly solved in the unbounded domains; (ii) we don't need to solve a nonlinear algebra system at each time step because of the employment of linearized IMEX schemes in time. Numerical examples are conducted to validate the stability and accuracy of the schemes, which show the resulting IMEX Hermite-Galerkin spectral schemes are highly accurate, efficient, and robust when used in conjunction with the adaptive time stepping strategies. The proposed schemes are showcased by solving several nonlinear coupled nonlocal Gordon-type models with fractional Laplacian, including the sine-Gordon, Klein-Gordon, and Klein-Gordon-Zakharov systems, together with the numerical simulations of the interactions of vector solitons.