Sequential Dirichlet-to-Neumann coupling for the mixed-dimensional wave equation

Coupling of a two-dimensional (2D) sub-model and a one-dimensional (1D) sub-model, to form a hybrid mixed-dimensional model, is considered for the linear scalar wave equation. A Dirichlet -to-Neumann (DtN) method is used to perform this coupling, which is based on enforcing the continuity of the DtN map across the 2D-1D interface. The DtN map relates the primary variable to the flux, on each side of the interface. A previously proposed DtN-coupling formulation was monolithic, namely the simulation had to be simultaneously performed with the combined 2D and 1D sub-models. In contrast, here a new sequential formulation is proposed, in which the 2D problem is solved separately, after a fundamental 1D problem is solved. Thus, in the new method the 2D-1D coupling is one-way. This has a clear advantage over the monolithic formulation, as the 2D and 1D problems can be solved using different codes, and with space and time discretizations that do not have to match at the interface. The proposed method is applied here in conjunction with a Finite Element (FE) formulation. Numerical examples demonstrate the performance of the method.