Schwarz waveform relaxation (SWR) methods have been developed to solve a wide range of diffusion-dominated and reaction-dominated equations. The appeal of these methods stems primarily from their ability to use nonconforming space-time discretizations; SWR methods are consequently well-adapted for coupling models with highly varying spatial and time scales. The efficacy of SWR methods is questionable, however, since in each iteration, one propagates an error across the entire time interval. In this manuscript, we introduce an adaptive pipeline approach wherein one subdivides the computational domain into space-time blocks, and adaptively selects the waveform iterates which should be updated given a fixed number of computational workers. Our method is complementary to existing space and time parallel methods, and can be used to obtain additional speedup when the saturation point is reached for other types of parallelism. We analyze these waveform relaxation with adaptive pipelining (WRAP) methods to show convergence and the theoretical speedup that can be expected. Numerical experiments on solutions to the linear heat equation, the advection-diffusion equation, and a reaction-diffusion equation illustrate features and efficacy of WRAP methods for various transmission conditions.


  1. waveform relaxation
  2. domain decomposition
  3. adaptivity
  4. parallel computing

MSC codes

  1. 65Y05
  2. 65M20

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A339 - A364
ISSN (online): 1095-7197


Submitted: 20 October 2017
Accepted: 6 November 2018
Published online: 15 January 2019


  1. waveform relaxation
  2. domain decomposition
  3. adaptivity
  4. parallel computing

MSC codes

  1. 65Y05
  2. 65M20



Funding Information

National Natural Science Foundation of China https://doi.org/10.13039/501100001809

Funding Information

Research Grants Council, University Grants Committee https://doi.org/10.13039/501100002920

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