Convergence Analysis of Schwarz Waveform Relaxation with Convolution Transmission Conditions

2017 ◽  
Vol 39 (3) ◽  
pp. A890-A921 ◽  
Author(s):  
Shu-Lin Wu ◽  
Yingxiang Xu
Keyword(s):  
Convergence Analysis ◽  
Transmission Conditions ◽  
Waveform Relaxation ◽  
Schwarz Waveform Relaxation

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

2020 ◽  
Vol 20 (3) ◽  
pp. 397-417
Author(s):  
Mohammad Al-Khaleel ◽  
Shu-Lin Wu
Keyword(s):  
Transmission Conditions ◽  
Parabolic Problems ◽  
Waveform Relaxation ◽  
Discrete Level ◽  
Convergence Factor ◽  
Schwarz Waveform Relaxation ◽  
Relaxation Algorithms ◽  
Time Dependent Problems ◽  
New Algorithms ◽  
Level Analysis

AbstractThe 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.

scholarly journals Convergence Analysis of Schwarz Waveform Relaxation for Nonlocal Diffusion Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-25
Author(s):  
Ke Li ◽  
Dali Guo ◽  
Yunxiang Zhao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Analysis ◽  
Good Choice ◽  
Nonlocal Diffusion ◽  
Waveform Relaxation ◽  
Partial Differential ◽  
Schwarz Waveform Relaxation ◽  
Singular Kernels ◽  
The One

Diffusion equations with Riemann–Liouville fractional derivatives are Volterra integro-partial differential equations with weakly singular kernels and present fundamental challenges for numerical computation. In this paper, we make a convergence analysis of the Schwarz waveform relaxation (SWR) algorithms with Robin transmission conditions (TCs) for these problems. We focus on deriving good choice of the parameter involved in the Robin TCs, at the continuous and fully discretized level. Particularly, at the space-time continuous level, we show that the derived Robin parameter is much better than the one predicted by the well-understood equioscillation principle. At the fully discretized level, the problem of determining a good Robin parameter is studied in the convolution quadrature framework, which permits us to precisely capture the effects of different temporal discretization methods on the convergence rate of the SWR algorithms. The results obtained in this paper will be preliminary preparations for our further study of the SWR algorithms for integro-partial differential equations.

scholarly journals Schwarz waveform relaxation algorithm for heat equations with distributed delay

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 659-667
Author(s):  
Shu-Lin Wu
Keyword(s):  
Numerical Computation ◽  
Distributed Delay ◽  
Computation Time ◽  
Analytic Solutions ◽  
Transmission Conditions ◽  
Waveform Relaxation ◽  
Heat Equations ◽  
Relaxation Algorithm ◽  
Schwarz Waveform Relaxation ◽  
The One

Heat equations with distributed delay are a class of mathematic models that has wide applications in many fields. Numerical computation plays an important role in the investigation of these equations, because the analytic solutions of partial differential equations with time delay are usually unavailable. On the other hand, duo to the delay property, numerical computation of these equations is time-consuming. To reduce the computation time, we analyze in this paper the Schwarz waveform relaxation algorithm with Robin transmission conditions. The Robin transmission conditions contain a free parameter, which has a significant effect on the convergence rate of the Schwarz waveform relaxation algorithm. Determining the Robin parameter is therefore one of the top-priority matters for the study of the Schwarz waveform relaxation algorithm. We provide new formula to fix the Robin parameter and we show numerically that the new Robin parameter is more efficient than the one proposed previously in the literature.

scholarly journals Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

2018 ◽  
Vol 52 (4) ◽  
pp. 1569-1596 ◽  
Author(s):  
Xavier Antoine ◽  
Fengji Hou ◽  
Emmanuel Lorin
Keyword(s):  
Domain Decomposition ◽  
Decomposition Methods ◽  
Nonlinear Schrödinger Equations ◽  
Waveform Relaxation ◽  
Schrödinger Equations ◽  
Two Dimensional ◽  
Domain Decomposition Methods ◽  
Schwarz Waveform Relaxation ◽  
Nonlinear Schrodinger Equations ◽  
Linear And Nonlinear

This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for linear equations with variable coefficients and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [X. Antoine and E. Lorin, Numer. Math. 137 (2017) 923–958] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

Sign in / Sign up

Export Citation Format

Share Document