Partitioned Time Stepping for a Parabolic Two Domain Problem

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 Ω1,Ω2 ⊂ R2 adjoined by an interface I = Ω1 ∩Ω2 ⊂ 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 (IMEX) 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.