Abstract
By introducing appropriate intermediate variables and simplifying inhomogeneous terms of the traditional ADI schemes, an improved ADI scheme is proposed for linear hyperbolic equations and this ADI scheme is extended to nonlinear hyperbolic equations and compact ADI schemes in the present work. Meanwhile, the boundary and initial conditions are carefully analyzed to match the accuracy of these improved ADI schemes. Although the common (without compactification) and compact improved ADI schemes have the same accuracy with the corresponding traditional ADI schemes, respectively, it is not just a simple variant of ADI schemes for hyperbolic equations. Both theoretical analysis and numerical experiments show that compared with the traditional ADI scheme, the improved ADI scheme is more efficient for linear hyperbolic equations and more stable for nonlinear hyperbolic equations without loss of accuracy.