Schwarz Waveform Relaxation Method Research Articles

Optimized Schwarz waveform relaxation method for the incompressible Stokes problem

We propose and analyse the optimized Schwarz waveform relaxation (OSWR) method for the unsteady incompressible Stokes equations. Well-posedness of the local subdomain problems with Robin boundary conditions is proved. Convergence of the velocity is shown through energy estimates; however, pressure converges only up to constant values in the subdomains, and an astute correction technique is proposed to recover these constants from the velocity. The convergence factor of the OSWR algorithm is obtained through a Fourier analysis, and allows to efficiently optimize the space-time Robin transmission conditions involved in the OSWR method. Then, numerical illustrations for the two-dimensional unsteady incompressible Stokes system are presented to illustrate the performance of the OSWR algorithm.

Quasi-optimal domain decomposition method for neural network-based computation of the time-dependent Schrödinger equation

In this paper, we derive and analyze the performance of optimal/quasi-optimal Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for the time-dependent Schrödinger equation when implement with neural network-based Partial Differential Equations (PDE) solvers. Optimal SWR methods, which are based on Dirichlet-to-Neumann operators, are known to have a higher convergence rate than classical or optimized SWR methods. However, they are usually considered prohibitive due to their computational costs with standard PDE solvers. Thanks to Physics Informed Neural Network acceleration within the Schwarz waveform relaxation process and an efficient computation of Dirichlet-to-Neumann transmission operators, we demonstrate that optimal and quasi-optimal SWR methods can be performed almost as efficiently as classical or optimized SWR methods while maintaining a faster convergence rate. We present a few numerical examples to illustrate the performance and convergence of the proposed method.

Convergence analysis of Schwarz waveform relaxation method to compute coupled advection–diffusion–reaction equations

We study the computation of coupled advection–diffusion–reaction equations with the same or different coefficients by the Schwarz waveform relaxation method. The study starts with linear equations and analyzes the computation’s convergence with a Dirichlet condition, a Robin condition, and a combination of them as the transmission conditions. Then, it presents an optimized algorithm for the Dirichlet condition, and this algorithm leads to a substantial speedup in the convergence. Furthermore, the optimized algorithm extends to the computation of nonlinear equations, including the viscous Burgers equation, and the algorithm largely remains effective for convergence speedup. Finally, the study compares waveform relaxation and conventional methods. Numerical experiments indicate that the former tends to be much more expensive than the latter regarding the number of times to solve the involved linear systems.

Convergence analysis and optimization of a Robin Schwarz waveform relaxation method for time-periodic parabolic optimal control problems

This paper is concerned with a novel convergence analysis of the optimized Schwarz waveform relaxation method (OSWRM) for the solution of optimal control problems governed by periodic parabolic partial differential equations (PDEs). The new analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition. This leads to a precise characterization of the convergence factor of the method at the semidiscrete level. Using this characterization, the optimal transmission condition parameter is obtained at the semidiscrete level and its asymptotic behavior as the time discretization converges to zero is analyzed in detail.

Fully discrete Schwarz waveform relaxation analysis for the heat equation on a finite spatial domain

Schwarz waveform relaxation methods provide space-time parallelism for the solution of time dependent partial differential equations. The algorithms are differentiated by the choice of the transmission conditions enforced at the introduced space-time boundaries. Early results considered the theoretical analysis of these algorithms in the continuous and semi-discrete (in space) settings for various families of linear partial differential equations. Later, fully discrete results were obtained under the simplifying assumption of an infinite spatial domain. In this paper, we provide a first analysis of a fully discrete classical Schwarz Waveform algorithm for the one-dimensional heat equation on an arbitrary but finite number of bounded subdomains. The θ-method is chosen as the time integrator. Convergence results are given in both the infinity norm and two norm, with an explicit contraction given in the case of a uniform partitioning. The results are compared to the numerics and to the earlier theoretical results.

Why Fourier mode analysis in time is different when studying Schwarz Waveform Relaxation

Schwarz waveform relaxation methods are Schwarz methods applied to evolution problems. Like for steady problems, they are based on an overlapping domain decomposition of the spatial domain, and an iteration which only requires subdomain solutions, now in space-time, to get better and better approximations of the global, monodomain solution. Fourier analysis has been used to study the convergence of both Schwarz and Schwarz waveform relaxation methods. We show here that their convergence is however quite different: for steady problems of diffusive type, Schwarz methods converge linearly, which is also well predicted by Fourier analysis. For a time dependent heat equation however, the Schwarz waveform relaxation algorithm first has a rapid convergence phase, followed by a slow down, and eventually convergence increases again to become superlinear, none of which is predicted by classical Fourier analysis. Introducing a new Fourier analysis combined with kernel estimates, we can explain this behavior for the heat equation. We then generalize our approach to the case of advection reaction diffusion problems. We illustrate all our results with numerical experiments.

Schwarz waveform relaxation-learning for advection-diffusion-reaction equations

This paper develops a physics-informed neural network (PINN) combined with a Schwarz waveform relaxation (SWR) method for solving local and nonlocal advection-diffusion-reaction equations. Specifically, we derive the algorithm by constructing subdomain-dependent local solutions by minimizing local loss functions, allowing the decomposition of the training process in different domains in an embarrassingly parallel procedure. Provided the convergence of PINN, the overall proposed algorithm is convergent. By constructing local solutions, one can, in particular, adapt the depth of the deep neural networks, depending on the solution's spectral space and time complexity in each subdomain. One of the main advantages of using NN compared to standard solvers, is that the PINN algorithm introduces some learning in the SWR algorithm allowing for an acceleration of the overall algorithm, especially close to SWR convergence. We present some numerical experiments based on classical and Robin-SWR algorithms to illustrate the performance and comment on the convergence of the proposed method.

A Schwarz waveform relaxation method for time-dependent space fractional Schrödinger/heat equations

This paper is dedicated to the derivation and analysis of a Schwarz waveform relaxation domain decomposition method for solving time-dependent linear/nonlinear space fractional Schrödinger and heat equations. Along with the details of the derivation of the method and some mathematical properties, we also propose some illustrating numerical experiments and conjectures on the rate of convergence of the method.

Coupling Parareal with Optimized Schwarz Waveform Relaxation for Parabolic Problems

We propose and analyse a parallel method, both in the time and space directions, that couples the Parareal algorithm with the Optimized Schwarz waveform relaxation (OSWR) method, with only few OSWR iterations in the fine propagator and with a simple coarse propagator deduced from the Backward Euler method. The analysis of this coupled method is presented for a one-dimensional advection-reaction-diffusion equation. We prove a general convergence result for this method via energy estimates. Numerical results for two-dimensional advection-diffusion problems and for a diffusion equation with strong heterogeneities are presented, to illustrate the performance of the coupled Parareal-OSWR algorithm.

Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients

In this paper, we investigate the effect of the space and time discretisation on the convergence properties of Schwarz Waveform Relaxation (SWR) algorithms. We consider a reaction-diffusion problem with discontinuous coefficients discretised on two non-overlapping domains with several numerical schemes (in space and time). A methodology to determine the rate of convergence of the classical SWR method with standard interface conditions (Dirichlet-Neumann or Robin-Robin) accounting for discretisation errors is presented. We discuss how such convergence rates differ from the ones derived at a continuous level (i.e. assuming an exact discrete representation of the continuous problem). In this work we consider a second-order finite difference scheme and a finite volume scheme based on quadratic spline reconstruction in space, combined with either a simple backward Euler scheme or a two-step “Padé” scheme (resembling a Diagonally Implicit Runge Kutta scheme) in time. We prove those combinations of space-time schemes to be unconditionally stable on bounded domains. We illustrate the relevance of our analysis with specifically designed numerical experiments.

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

<p style='text-indent:20px;'>This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.</p>

Wave-heat coupling in one-dimensional unbounded domains: artificial boundary conditions and an optimized Schwarz method

This paper deals with the coupling between one-dimensional heat and wave equations in unbounded subdomains, as a simplified prototype of fluid-structure interaction problems. First we apply appropriate artificial boundary conditions that yield an equivalent problem, but with bounded subdomains, and we carry out the stability analysis for this coupled problem in truncated domains. Then we devise an optimized Schwarz-in-time (or Schwarz Waveform Relaxation) method for the numerical solving of the coupled equations. Particular emphasis is made on the design of optimized transmission conditions. Notably, for this setting, the optimal transmission conditions can be expressed analytically in a very simple manner. This result is illustrated by some numerical experiments.

High-order Absorbing Boundary Condition, Domain Decomposition Method and Stratified Dispersive Wave Model

The high-order Absorbing Boundary Condition proposed by Hagstrom and Warburton was applied to various models as the wave equation, dispersive, convected with stratified materials. We propose an other way to expand the use of this condition with Domain Decomposition Method. We apply it here to a Stratified Dispersive Wave Model not just to limit the real unbounded domain but to use a non-overlapping domain decomposition method with the classical Schwarz Waveform Relaxation method using the Absorbing Boundary Condition taken as an approximation of the Transparent Operator for the interface condition. We must add terms to enrich the equations associated to the Absorbing Boundary Condition and compute them with a cumulative process. Numerical examples are used to show the performance of this Domain Decomposition Method.

Schwarz Waveform Relaxation Physics Informed Neural Networks for Solving Advection-Diffusion-Reaction Equations

This paper develops a physics-informed neural network (PINN) based on the Schwarz waveform relaxation (SWR) method for solving local and nonlocal advection-diffusion-reaction equations. Specifically, we derive the formulation by constructing subdomain-dependent local solutions by minimizing local loss functions, allowing the decomposition of the training process in different domains in an embarrassingly parallel procedure. Provided the convergence of PINN, the overall proposed algorithm is convergent. By constructing local solutions, one can, in particular, adapt the depth of the deep neural networks, depending on the solution's spectral space and time complexity in each subdomain. We present some numerical experiments based on classical and Robin-SWR to illustrate the performance and comment on the convergence of the proposed method.

Space–time domain decomposition for two-phase flow between different rock types

In Ahmed et al. (2019) a space–time domain decomposition method was proposed for two-phase flow in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. For this nonlinear and degenerate parabolic problem, with nonlinear and discontinuous transmission conditions on the interface, the Optimized Schwarz waveform relaxation method (OSWR) with Robin or Ventcell transmission conditions was considered. A guaranteed and fully computable a posteriori error estimate was derived, which in particular took into account the domain decomposition error.In this paper we provide a mathematical and numerical analysis of this space–time domain decomposition method in the Robin case. Complete numerical approximation is achieved by a finite volume scheme in space and the lowest order discontinuous Galerkin method in time. We prove the existence of a weak solution of the two-phase flow subdomain problem with Robin boundary conditions by analyzing the convergence of the finite volume scheme. The domain decomposition algorithm is based on the solution of space–time nonlinear subdomain problems over the whole time interval, allowing for different time steps in different parts of the domain, adapted to the physical properties of each subdomain, and we show that such an algorithm is well-defined. Numerical experiments on three-dimensional problems with different rock types illustrate the performance of the domain decomposition method.

Dirichlet–Neumann waveform relaxation methods for parabolic and hyperbolic problems in multiple subdomains

In this paper, a new waveform relaxation variant of the Dirichlet–Neumann algorithm is introduced for general parabolic problems as well as for the second-order wave equation for decompositions with multiple subdomains. The method is based on a non-overlapping decomposition of the domain in space, and the iteration involves subdomain solves in space-time with transmission conditions of Dirichlet and Neumann type to exchange information between neighboring subdomains. Regarding the convergence of the algorithm, two main results are obtained when the time window is finite: for the heat equation, the method converges superlinearly, whereas for the wave equation, it converges after a finite number of iterations. The analysis is based on Fourier–Laplace transforms and detailed kernel estimates, which reveals the precise dependence of the convergence on the size of the subdomains and the time window length. Numerical experiments are presented to illustrate the performance of the algorithm and to compare its convergence behaviour with classical and optimized Schwarz Waveform Relaxation methods. Experiments involving heterogeneous coefficients and non-matching time grids, which are not covered by the theory, are also presented.

A discrete Schwarz waveform relaxation method of higher order for singularly perturbed parabolic reaction–diffusion problems

We consider singularly perturbed parabolic reaction–diffusion problems of scalar and vector types. We construct a discrete Schwarz waveform relaxation method of higher order for their numerical solution. The method is shown to be parameter-uniformly convergent having almost fourth order in space and first order in time. Further, interesting result proven is much faster convergence of iterative process for small perturbation parameter. Numerical results demonstrating the efficiency of the proposed method and validating the theoretically proven convergence results are given.

Quasi-overlapping Semi-discrete Schwarz Waveform Relaxation Algorithms: The Hyperbolic Problem

Abstract The Schwarz waveform relaxation (SWR) algorithms have many favorable properties and are extensively studied and investigated for solving time dependent problems mainly at a continuous level. In this paper, we consider a semi-discrete level analysis and we investigate the convergence behavior of what so-called semi-discrete SWR algorithms combined with discrete transmission conditions instead of the continuous ones. We shall target here the hyperbolic problems but not the parabolic problems that are usually considered by most of the researchers in general when investigating the properties of the SWR methods. We first present the classical overlapping semi-discrete SWR algorithms with different partitioning choices and show that they converge very slow. We then introduce optimal, optimized, and quasi optimized overlapping semi-discrete SWR algorithms using new transmission conditions also with different partitioning choices. We show that the new algorithms lead to a much better convergence through using discrete transmission conditions associated with the optimized SWR algorithms at the semi-discrete level. In the performed semi-discrete level analysis, we also demonstrate the fact that as the ratio between the overlap size and the spatial discretization size gets bigger, the convergence factor gets smaller which results in a better convergence. Numerical results and experiments are presented in order to confirm the theoretical aspects of the proposed algorithms and providing an evidence of their usefulness and their accuracy.

A robust domain decomposition method for singularly perturbed parabolic reaction-diffusion systems

In this paper we develop and analyse a Schwarz waveform relaxation (SWR) based domain decomposition method for solving a coupled system of singularly perturbed parabolic reaction-diffusion problems with distinct small positive parameters. The proposed discrete SWR method is based on decomposing the original computational domain into five overlapping subdomains and employing the central difference scheme in spatial direction and the backward Euler scheme in time direction to solve subdomain problems in the iterative steps. Further, we use appropriate interface conditions between space-time subdomain and present the convergence analysis of the method. In particular, it is proved that the proposed method converges uniformly of almost second order in spatial direction and first order in time direction. Finally, some numerical experiments are conducted in support of the theoretical results.

On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

This paper is dedicated to the analysis of the rate of convergence of the classical and quasi-optimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation with space-dependent potential. The strategy is based on i) the rewriting of the SWR algorithm as a fixed point algorithm in frequency space, and ii) the explicit construction of contraction factors thanks to pseudo-differential calculus. Some numerical experiments illustrating the analysis are also provided.