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Efficient and accurate numerical simulation of seismic wave propagation isimportant in various Geophysical applications such as seismic full waveforminversion (FWI) problem. However, due to the large size of the physical domainand requirement on low numerical dispersion, many existing numerical methodsare inefficient for numerical modelling of seismic wave propagation in aheterogeneous media. Despite the great efforts that have been devoted duringthe past decades, it still remains a challenging task in the development ofefficient and accurate finite difference method for the multi-dimensionalacoustic wave equation with variable velocity. In this paper, we proposed aPad\'{e} approximation based finite difference scheme for solving the acousticwave equation in three-dimensional heterogeneous media. The new method isobtained by combining the Pad\'{e} approximation and a novel algebraicmanipulation. The efficiency of the new algorithm is further improved throughthe Alternative Directional Implicit (ADI) method. The stability of the newalgorithm has been theoretically proved by the energy method. The new method isconditionally stable with a better Courant - Friedrichs - Lewy condition (CFL)condition, which has been verified numerically. Extensive numerical exampleshave been solved, which demonstrated that the new method is accurate, efficientand stable.

A compact high order Alternating Direction Implicit method for three-dimensional acoustic wave equation with variable coefficient