Nonlinear parallel-in-time Schur complement solvers for ordinary differential equations

Santiago Badia, Marc Olm

Research output: Contribution to journalArticleResearchpeer-review


In this work, we propose a parallel-in-time solver for linear and nonlinear ordinary differential equations. The approach is based on an efficient multilevel solver of the Schur complement related to a multilevel time partition. For linear problems, the scheme leads to a fast direct method. Next, two different strategies for solving nonlinear ODEs are proposed. First, we consider a Newton method over the global nonlinear ODE, using the multilevel Schur complement solver at every nonlinear iteration. Second, we state the global nonlinear problem in terms of the nonlinear Schur complement (at an arbitrary level), and perform nonlinear iterations over it. Numerical experiments show that the proposed schemes are weakly scalable, i.e., we can efficiently exploit increasing computational resources to solve for more time steps the same problem.

Original languageEnglish
Pages (from-to)794-806
Number of pages13
JournalJournal of Computational and Applied Mathematics
Publication statusPublished - 15 Dec 2018
Externally publishedYes


  • Domain decomposition
  • Nonlinear solver
  • Ordinary differential equations
  • Scalability
  • Time parallelism

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