Partitioned Time Stepping for a Parabolic Two Domain Problem

Abstract

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 Ω 1 , Ω 2 ⊂ R 2 adjoined by an interface I = Ω 1 ∩ Ω 2 ⊂ 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.

Multiphysics simulations: challenges and opportunities.

“…Since each subdomain requires data from the other subdomain, this data-passing approach is less flexible but is stable when κ is large compared to ν 1 and ν 2 . The data-passing approach also does not have a stability constraint on the timestep; see (Connors et al, 2009) for details and analysis. Partitioned time integration methods are often used in a sequential fashion as in Algorithm 2, first advancing a single physics component in time, followed by updating information on an interface, advancing a second physics component in time, and finally updating interface information from the second component.…”

Stability of algorithms for a two domain natural convection problem and observed model uncertainty

“…Hence, Algorithm 22 is referred to as the "two-way partitioned method with geometric averaging", or TWP-GA method. The approach used to decouple in the linear terms has been studied in [7] for the Navier-Stokes equations.…”

Partitioned time discretization for parallel solution of coupled ODE systems

“…Stability of some IMEX schemes for certain classes of PDEs has been studied (see [2]), but coupling terms vary in structure and complexity among applications. Examples include coupled heat equations, [8], and fluid-structure interaction [6,7].…”