Analysis of a fourth‐order compact ADI method for a linear hyperbolic equation with three spatial variables

This paper is concerned with a three‐level alternating direction implicit (ADI) method for the numerical solution of a 3D hyperbolic equation. Stability criterion of this ADI method is given by using von Neumann method. Meanwhile, it is shown by a discrete energy method that it can achieve fourth‐order accuracy in both time and space with respect to H ¹‐ and L ²‐norms only if stable condition is satisfied. It only needs solution of a tri‐diagonal system at each time step, which can be solved by multiple applications of one‐dimensional tri‐diagonal algorithm. Numerical experiments confirming the high accuracy and efficiency of the new algorithm are provided.