On the efficiency of the Peaceman-Rachford ADI-dG method forwave-type methods

The Peaceman{Rachford alternating direction implicit (ADI) method is considered for the time-integration of a class of wavetype equations for linear, isotropic materials on a tensorial domain, e.g., a cuboid in 3D or a rectangle in 2D. This method is known to be unconditionally stable and of classical order two. So far, it has been applied to specific problems and is mostly combined with finite differences in space, where it can be implemented at the cost of an explicit method. In this paper, we consider the ADI method for a discontinuous Galerkin (dG) space discretization.We characterize a large class of first-order differential equations for which we show that on tensorial meshes, the method can be implemented with optimal (linear) complexity.