Philipp Birken
Lund University
We discuss iterative methods for the partitioned time-integration of multiphysics problems, which commonly exhibit a multiscale behavior, requiring independent time-grids. Specifically, we consider unsteady thermal fluid structure interaction (FSI), modelled by heterogeneous coupled linear heat equations The ideal method for solving these problems allows independent and adaptive time-grids, higher order time-discretizations, is fast and robust, and allows the coupling of existing subsolvers, executed in parallel.
Waveform relaxation (WR) methods can potentially have all of these properties. These iterate on continuous-in-time interface functions, obtained via suitable interpolation. The difficulty is to find suitable convergence acceleration, which is required for the iteration to converge quickly. We present a fast and highly robust, second order in time, adaptive WR method. Basis is a Dirichlet-Neumann coupling at the interface. The method uses an analytically optimal relaxation parameter derived for the fully-discrete scheme in 1D. This parameter depends on discretization and problem, and in practice leads to very fast linear convergence. Numerical results demonstrate the robustness of the approach.
We further present a novel, parallel WR method, using asynchronous communication techniques during time integration to accelerate convergence. Instead of exchanging interpolated time-dependent functions at the end of each time-window or iteration, we exchange time-point data immediately after each timestep, by making use of one sided communication. The analytical description and convergence results of this method generalize existing WR theory.