Adaptive time-integration for goal-oriented and coupled problems
Summary, in English
We consider Waveform relaxation (WR) methods, which can have all of these properties. WR methods 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 converge quickly.
We develop a fast and highly robust, second order in time, adaptive WR method for unsteady thermal fluid structure interaction (FSI), modelled by heterogeneous coupled linear heat equations. We use a Dirichlet-Neumann coupling at the interface and an analytical optimal relaxation parameter derived for the fully-discrete scheme. While this method is sequential, it is notably faster and more robust than similar parallel methods.
We further develop 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. The analytical description and convergence results of this method generalize existing WR theory.
Since WR methods allow coupling of problems in a relative black-box manner, we developed adapters to PDE-subsolvers implemented using DUNE and FEniCS. We demonstrate this coupling in a thermal FSI test case.
Lastly, we consider adaptive time-integration for goal-oriented problems, where one is interested in a quantity of interest (QoI), which is a functional of the solution. The state-of-the-art method is the dual-weighted residual (DWR) method, which is extremely costly in both computation and implementation. We develop a goal oriented adaptive method based on local error estimates, which is considerably cheaper in computation. We prove convergence of the error in the QoI for tolerance to zero under a controllability assumption. By analyzing global error propagation with respect to the QoI, we can identify possible issues and make performance predictions. Numerical results verify these results and show our method to be more efficient than the DWR method.
- Computational Mathematics
- Mathematical Analysis
- goal-oriented problems
- error estimation
- coupled problems
- waveform relaxation
- asynchronous methods
- conjugate heat transfer
- eSSENCE@LU 4:4 - 3M: Multiphysics, multicore, multirate solution of coupled dynamic problems
- Numerical Analysis
- ISBN: 978-91-7895-930-3
- ISBN: 978-91-7895-929-7
2 September 2021
MH:H, Lund. Join via zoom: https://lu-se.zoom.us/j/65642304248
- Andreas Dedner (Associate professor (Reader))