In this paper, we extend fourth-order alternating direction implicit (ADI) methods to solve high-dimensional wave equations with nonseparable coefficients in each spatial direction. By combining a compact finite difference approximation with the linear \(\theta \) -method, we derive several ADI algorithms with fourth-order accuracy in space and second-order accuracy in time. For the intermediate solutions generated by the ADI splitting, the required boundary values are derived from the original boundary data and one-sided ghost-point extrapolations that preserve the overall accuracy, so that the compact solvers can be implemented without losing the spatial accuracy. Furthermore, the proposed algorithms are capable of handling nonseparable coefficients depending simultaneously on spatial and temporal variables. A stability analysis is also provided for a representative variable-coefficient problem, which provides theoretical insight into the stability of the proposed schemes. To match the spatial accuracy, Richardson extrapolation is employed to enhance the temporal accuracy. Three-dimensional algorithms are also constructed when coefficients are independent of time. Extensive numerical examples with different values of the parameter \(\theta \) are carried out to validate the accuracy.
