Abstract
The application of partitioned schemes to fluid–structure interaction (FSI) allows the use of already developed solvers specifically designed for the efficient solution of the corresponding subproblems. In this work, we propose and describe a loosely coupled partitioned scheme based on the recently introduced generalized-structure additively partitioned Runge-Kutta (GARK) framework. The resulting scheme combines implicit-explicit (IMEX) and multirate approaches while coupling of the subproblems is realized both on the level of the discrete time steps and at the level of interior Runge-Kutta stages. Specifically, we allow for varying micro step sizes for the fluid subproblem and therefore extend the multirate GARK framework based on constant micro steps. Furthermore, we derive the order conditions for this extension allowing for coupled time integration schemes of up to third order and discuss specific choices of the Runge-Kutta coefficients complying with the geometric conservation law. Finally, numerical experiments are carried out for uniform flow on a moving grid as well as the classical FSI test case of a moving piston.
Highlights
Coupled systems as in the context of fluid–structure interaction (FSI) often consist of subsystems with significantly different time scales
It is reasonable to allow each subsystem of the coupled problem to advance with its preassigned time integration scheme which is adapted to its stiffness and time scales
Concerning the concrete choice of an multirate generalized-structure additively partitioned Runge-Kutta (MGARK) scheme in practice, we notice that the computational effort needs to be low enough in order to not exceed the cost of solving the coupled system with constant micro step sizes per macro step
Summary
Coupled systems as in the context of fluid–structure interaction (FSI) often consist of subsystems with significantly different time scales. We combine the IMEX approach with the concept of multirate schemes which allows for different time step sizes of the subsolvers, corresponding to the subcycling strategy in [3,4]. Sandu and Günther [25] constructed a generalized-structure approach to additively partitioned Runge-Kutta (GARK) methods. Günther and Sandu made use of the GARK framework to derive a multirate scheme solely based on implicit and explicit Runge-Kutta schemes [26]. The GARK formalism includes a variety of well-known schemes for partitioned problems, like the classical implicit-explicit (IMEX) Runge-Kutta schemes [20,23], as well as former multirate methods as in [19,21]. Compliance with the geometric conservation law is demonstrated for uniform flow on an artificially moving grid, the viability of adaptive micro steps is shown by an investigation of the step size statistics and we investigate the efficiency of higher order time integration for small tolerances by a comparison of first, second and third order schemes in terms of error vs. CPU time
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