Mixed-Dimensional Modeling of Time-Dependent Wave Problems Using the Panasenko Construction

Abstract

We consider the coupling of two-dimensional (2D) and one-dimensional (1D) models to form a single hybrid 2D–1D model for time-dependent linear wave problems. The 1D model is used to represent a 2D computational domain where the solution behaves approximately in a 1D way. This hybrid model, if designed properly, is a more efficient way to solve the full 2D model over the entire problem. Two important issues related to such hybrid 2D–1D models are (a) the design of the hybrid model and its validation (with respect to the original problem) and (b) the way the 2D–1D coupling is done, and the coupling error generated. This paper focuses on the second issue. The method used in this paper to couple the 1D and 2D models is the one proposed by Panasenko. This method has been used for mixed-dimensional coupling in many steady-state problems, and here it is being used for the first time for time-dependent problems. The hybrid formulation is derived, and the numerical accuracy and efficiency of the method are explored for a couple of basic problems.