A class of one-parameter alternating direction implicit methods for two-dimensional wave equations with discrete and distributed time-variable delays

For solving the initial-boundary value problem of two-dimensional wave equations with discrete and distributed time-variable delays, in the present paper, we first construct a class of basic one-parameter methods. In order to raise the computational efficiency of this class methods, we remold the methods as one-parameter alternating direction implicit (ADI) methods. Under the suitable conditions, the remolded methods are proved to be stable and convergent of second order in both of time and space. With several numerical experiments, the computational effectiveness and theoretical exactness of the methods are confirmed. Moreover, it is illustrated that the proposed one-parameter ADI method has the better advantage in computational efficiency than the basic one-parameter methods.