PARTITIONED TIME STEPPING FOR A PARABOLIC TWO DOMAIN PROBLEM

• Published in: SIAM journal on numerical analysis 2009-01, Vol.47 (5), p.3526-3549
• Main Authors: CONNORS, JEFFREY M., HOWELL, JASON S., LAYTON, WILLIAM J.
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Boundary conditions; Computational fluid dynamics; Decomposition methods; Energy; Error rates; Fluid flow; Fluid-structure interaction; Fluids; Heat equation; Heat equations; Hilbert spaces; Inner products; Interfaces; Iterative methods; Lagrange multiplier; Lagrange multipliers; Mathematical models; Methods; Numerical analysis; Perceptron convergence procedure; Stability
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/080740891

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 $\Omega _{1},\Omega _{2}\subset {\Bbb R}^{2}$ adjoined by an interface $I=\Omega _{1},\Omega _{2}\subset {\Bbb R}$ . 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.