Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case

The efficiency of numerically solving time-dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously. The theoretical properties of one such method, namely the discrete-time multigrid waveform relaxation method, are investigated for systems of ordinary differential equations obtained by spatial finite-element discretisation of linear parabolic initial-boundary value problems. The results are compared to the corresponding continuous-time results. The theory is illustrated for a one-dimensional and a two-dimensional model problem and checked against results obtained by numerical experiments

The Waveform Relaxation Method for Systems of Differential/Algebraic Equations

An extension of the waveform relaxation (WR) algorithm to systems of differential/algebraic equations (DAE) is presented. Although this type of application has been explored earlier in relation to VLSI circuits, the algorithm has not been generalized to include the vast array of DAE system structures. The solvability and convergence requirements of the WR algorithm for higher-index systems are established. Many systems in robotics and control applications are modeled with DAE systems having an index greater than two. Computer simulation of these systems has been hampered by numerical integration methods which perform poorly and must be explicitly tailored to the system. The WR algorithm presents a means by which these systems may be more efficiently simulated by breaking them into weakly coupled subsystems, many of which will no longer retain the limiting high-index properties

Application of the Waveform Relaxation Technique to the Co-Simulation of Power Converter Controller and Electrical Circuit Models

In this paper we present the co-simulation of a PID class power converter controller and an electrical circuit by means of the waveform relaxation technique. The simulation of the controller model is characterized by a fixed-time stepping scheme reflecting its digital implementation, whereas a circuit simulation usually employs an adaptive time stepping scheme in order to account for a wide range of time constants within the circuit model. In order to maintain the characteristic of both models as well as to facilitate model replacement, we treat them separately by means of input/output relations and propose an application of a waveform relaxation algorithm. Furthermore, the maximum and minimum number of iterations of the proposed algorithm are mathematically analyzed. The concept of controller/circuit coupling is illustrated by an example of the co-simulation of a PI power converter controller and a model of the main dipole circuit of the Large Hadron Collider

Preconditioned WR–LMF-based method for ODE systems

AbstractThe waveform relaxation (WR) method was developed as an iterative method for solving large systems of ordinary differential equations (ODEs). In each WR iteration, we are required to solve a system of ODEs. We then introduce the boundary value method (BVM) which is a relatively new method based on the linear multistep formulae to solve ODEs. In particular, we apply the generalized minimal residual method with the Strang-type block-circulant preconditioner for solving linear systems arising from the application of BVMs to each WR iteration. It is demonstrated that these techniques are very effective in speeding up the convergence rate of the resulting iterative processes. Numerical experiments are presented to illustrate the effectiveness of our methods

Multigrid waveform relaxation for the time-fractional heat equation

In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of $O(N M \log(M))$ operations, where $M$ is the number of time steps and $N$ is the number of spatial grid points. A semi-algebraic mode analysis is also developed to theoretically confirm the good results obtained. Several numerical experiments, including examples with non-smooth solutions and a nonlinear problem with applications in porous media, are presented

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

A note on windowing for the waveform relaxation

The technique of windowing has been often used in the implementation of the waveform relaxations for solving ODE's or time dependent PDE's. Its efficiency depends upon problem stiffness and operator splitting. Using model problems, the estimates for window length and convergence rate are derived. The electiveness of windowing is then investigated for non-stiff and stiff cases respectively. lt concludes that for the former, windowing is highly recommended when a large discrepancy exists between the convergence rate on a time interval and the ones on its subintervals. For the latter, windowing does not provide any computational advantage if machine features are disregarded. The discussion is supported by experimental results

Waveform Relaxation with asynchronous time-integration

We consider Waveform Relaxation (WR) methods for partitioned time-integration of surface-coupled multiphysics problems. WR allows independent time-discretizations on independent and adaptive time-grids, while maintaining high time-integration orders. Classical WR methods such as Jacobi or Gauss-Seidel WR are typically either parallel or converge quickly. We present a novel parallel WR method utilizing asynchronous communication techniques to get both properties. Classical WR methods exchange discrete functions after time-integration of a subproblem. We instead asynchronously exchange time-point solutions during time-integration and directly incorporate all new information in the interpolants. We show both continuous and time-discrete convergence in a framework that generalizes existing linear WR convergence theory. An algorithm for choosing optimal relaxation in our new WR method is presented. Convergence is demonstrated in two conjugate heat transfer examples. Our new method shows an improved performance over classical WR methods. In one example we show a partitioned coupling of the compressible Euler equations with a nonlinear heat equation, with subproblems implemented using the open source libraries DUNE and FEniCS