Finite volume ADI scheme for hybrid dimension heat conduction problems set in a cross-shaped domain

Abstract

In this paper, we construct an alternating direction implicit (ADI) type finite volume numerical scheme to solve a nonclassical nonstationary heat conduction problem set in a 2D cross-shaped domain. We reduce the original model to a hybrid dimension model in a large part of the domain. We define special conjugation conditions between 2D and 1D parts. We apply the finite volume method to approximate spatial differential operators and use ADI splitting for time integration. The ADI scheme is unconditionally stable, and under a mix of Dirichlet and Neumann boundary conditions, the approximation error is of second order in space and time. The results of computational experiments confirm the theoretical error analysis. We compare visual representations and computational times for various sizes of reduced dimension zones.