The Peaceman–Rachford ADI-dG method for linear wave-type problems

In this thesis the discretization in space and time of a broad class of linear first order wave-type problems is investigated. The full discretization is achieved by using a method of lines approach utilizing a central-fluxes discontinuous Galerkin (dG) method in space and the Peaceman–Rachford scheme in time. Rigorous error bounds are derived, which show full order of convergence in space and time if the exact solution is sufficiently regular.

Additionally, despite the fact that the Peaceman–Rachford scheme is an implicit method exhibiting unconditional stability, it is shown that by combining it with an alternating direction implicit (ADI) approach, it can be applied to certain problems at roughly the cost of an explicit time integration scheme. Problems for which this is possible are characterized precisely, and the implementation used to achieve this efficiency is described in detail. This class of problems comprises, e.g., the 2D advection and wave equations and the 3D Maxwell's equations, for which the ADI method was previously proposed in literature.