93 research outputs found
Pipeline Implementations of Neumann-Neumann and Dirichlet-Neumann Waveform Relaxation Methods
This paper is concerned with the reformulation of Neumann-Neumann Waveform
Relaxation (NNWR) methods and Dirichlet-Neumann Waveform Relaxation (DNWR)
methods, a family of parallel space-time approaches to solving time-dependent
PDEs. By changing the order of the operations, pipeline-parallel computation of
the waveform iterates are possible without changing the final solution. The
parallel efficiency and the increased communication cost of the pipeline
implementation is presented, along with weak scaling studies to show the
effectiveness of the pipeline NNWR and DNWR algorithms.Comment: 20 pages, 8 figure
A linear domain decomposition method for partially saturated flow in porous media
The Richards equation is a nonlinear parabolic equation that is commonly used
for modelling saturated/unsaturated flow in porous media. We assume that the
medium occupies a bounded Lipschitz domain partitioned into two disjoint
subdomains separated by a fixed interface . This leads to two problems
defined on the subdomains which are coupled through conditions expressing flux
and pressure continuity at . After an Euler implicit discretisation of
the resulting nonlinear subproblems a linear iterative (-type) domain
decomposition scheme is proposed. The convergence of the scheme is proved
rigorously. In the last part we present numerical results that are in line with
the theoretical finding, in particular the unconditional convergence of the
scheme. We further compare the scheme to other approaches not making use of a
domain decomposition. Namely, we compare to a Newton and a Picard scheme. We
show that the proposed scheme is more stable than the Newton scheme while
remaining comparable in computational time, even if no parallelisation is being
adopted. Finally we present a parametric study that can be used to optimize the
proposed scheme.Comment: 34 pages, 13 figures, 7 table
Parallel-In-Time Simulation of Eddy Current Problems Using Parareal
In this contribution the usage of the Parareal method is proposed for the
time-parallel solution of the eddy current problem. The method is adapted to
the particular challenges of the problem that are related to the differential
algebraic character due to non-conducting regions. It is shown how the
necessary modification can be automatically incorporated by using a suitable
time stepping method. The paper closes with a first demonstration of a
simulation of a realistic four-pole induction machine model using Parareal
A multirate Neumann-Neumann waveform relaxation method for heterogeneous coupled heat equations
An important challenge when coupling two different time dependent problems is
to increase parallelization in time. We suggest a multirate Neumann-Neumann
waveform relaxation algorithm to solve two heterogeneous coupled heat
equations. In order to fix the mismatch produced by the multirate feature at
the space-time interface a linear interpolation is constructed. The heat
equations are discretized using a finite element method in space, whereas two
alternative time integration methods are used: implicit Euler and SDIRK2. We
perform a one-dimensional convergence analysis for the nonmultirate fully
discretized heat equations using implicit Euler to find the optimal relaxation
parameter in terms of the material coefficients, the stepsize and the mesh
resolution. This gives a very efficient method which needs only two iterations.
Numerical results confirm the analysis and show that the 1D nonmultirate
optimal relaxation parameter is a very good estimator for the multirate 1D case
and even for multirate and nonmultirate 2D examples using both implicit Euler
and SDIRK2.Comment: 32 pages, 12 figure
Towards hybrid two-phase modelling using linear domain decomposition
The viscous flow of two immiscible fluids in a porous medium on the Darcy scale is governed by a system of nonlinear parabolic equations. If infinite mobility of one phase can be assumed (e.g., in soil layers in contact with the atmosphere) the system can be substituted by the scalar Richards model. Thus, the porous medium domain may be partitioned into disjoint subdomains where either the full two-phase or the simplified Richards model dynamics are valid. Extending the previously considered one-model situations we suggest coupling conditions for this hybrid model approach. Based on an Euler implicit discretization, a linear iterative (L-type) domain decomposition scheme is proposed, and proved to be convergent. The theoretical findings are verified by a comparative numerical study that in particular confirms the efficiency of the hybrid ansatz as compared to full two-phase model computations.publishedVersio
A New Parareal Algorithm for Problems with Discontinuous Sources
The Parareal algorithm allows to solve evolution problems exploiting
parallelization in time. Its convergence and stability have been proved under
the assumption of regular (smooth) inputs. We present and analyze here a new
Parareal algorithm for ordinary differential equations which involve
discontinuous right-hand sides. Such situations occur in various applications,
e.g., when an electric device is supplied with a pulse-width-modulated signal.
Our new Parareal algorithm uses a smooth input for the coarse problem with
reduced dynamics. We derive error estimates that show how the input reduction
influences the overall convergence rate of the algorithm. We support our
theoretical results by numerical experiments, and also test our new Parareal
algorithm in an eddy current simulation of an induction machine
Adaptive time-integration for goal-oriented and coupled problems
We consider efficient methods for the partitioned time-integration of multiphysics problems, which commonly exhibit a multiscale behavior, requiring independent time-grids. Examples are fluid structure interaction in e.g., the simulation of blood-flow or cooling of rocket engines, or ocean-atmosphere-vegetation interaction. 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. 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
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