Abstract

Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations to achieve a “parallel-in-real-time” implementation. These algorithms proceed as two-level integrators, with the fine integrator being given by the exponential Euler scheme in this work. Two choices for the coarse integrator are considered: the linear implicit Euler scheme and the exponential Euler scheme. It is proved that as the number of iterations increases, the order of convergence is limited by the regularity of the noise, whereas for the exponential Euler case, the order of convergence always increases. The influences on the performance of the parareal algorithms, of the choice of the coarse integrator, of the regularity of the noise, and of the number of parareal iterations are also illustrated by extensive numerical experiments.

Keywords

  1. parareal algorithm
  2. parabolic stochastic partial differential equations
  3. strong convergence
  4. implicit Euler scheme
  5. exponential Euler scheme

MSC codes

  1. 60H35
  2. 65M12
  3. 65W05

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Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 254 - 278
ISSN (online): 1095-7170

History

Submitted: 19 March 2019
Accepted: 4 November 2019
Published online: 9 January 2020

Keywords

  1. parareal algorithm
  2. parabolic stochastic partial differential equations
  3. strong convergence
  4. implicit Euler scheme
  5. exponential Euler scheme

MSC codes

  1. 60H35
  2. 65M12
  3. 65W05

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