The Parareal algorithm allows one 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.


  1. evolution problems
  2. parallel-in-time solution
  3. Parareal
  4. ODEs with discontinuous inputs
  5. convergence analysis

MSC codes

  1. 34A34
  2. 34A36
  3. 34A37
  4. 65L20
  5. 78M10

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: B375 - B395
ISSN (online): 1095-7197


Submitted: 15 March 2018
Accepted: 14 February 2019
Published online: 2 April 2019


  1. evolution problems
  2. parallel-in-time solution
  3. Parareal
  4. ODEs with discontinuous inputs
  5. convergence analysis

MSC codes

  1. 34A34
  2. 34A36
  3. 34A37
  4. 65L20
  5. 78M10



Funding Information

Bundesministerium für Bildung und Forschung https://doi.org/10.13039/501100002347 : 05M2018RDA (PASIROM)

Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : SCHO1562/1-2

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