Partitioned Time Stepping for a Parabolic Two Domain Problem

Abstract

There have been many numerical simulations but few analytical results of stability and accuracy of algorithms for computational modeling of fluid-fluid and fluid-structure interaction problems, where two domains corresponding to different fluids (ocean-atmosphere) or a fluid and deformable solid (blood flow) are separated by an interface. As a simplified model of the first examples, this report considers two heat equations in $\Omega_1,\Omega_2\subset\mathbb{R}^2$ adjoined by an interface $I=\Omega_1\cap\Omega_2\subset\mathbb{R}$. The heat equations are coupled by a condition that allows energy to pass back and forth across the interface I while preserving the total global energy of the monolithic, coupled problem. To compute approximate solutions to the above problem only using subdomain solvers, two first-order in time, fully discrete methods are presented. The methods consist of an implicit-explicit approach, in which the action across I is lagged, and a partitioned method based on passing interface values back and forth across I. Stability and convergence results are derived for both schemes. Numerical experiments that support the theoretical results are presented.