The Legacy of ADI and LOD methods and an operator splitting algorithm for solving highly oscillatory wave problems

Abstract

Different splitting methods have been playing an important role in computations of numerical solutions of partial differential equations. Modern numerical strategies including mesh adaptations, linear and nonlinear transformations are also utilized together with splitting algorithms in applications. This survey concerns two cornerstones of the splitting methods, that is, the Alternating Direction Implicit (ADI) and Local One-Dimensional (LOD) methods, as well as their applications together with an eikonal mapping for solving highly oscillatory paraxial Helmholtz equations in slowly varying envelope approximations of active laser beams. The resulted finite difference scheme is not only oscillation-free, but also asymptotically stable. This ensures the high efficiency and applicability in optical wave applications.