A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

G.L. Kooij, M.A. Botchev, B.J. Geurts

Research output: Contribution to journalArticleAcademicpeer-review

7 Citations (Scopus)

Abstract

A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as source terms, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection–diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection–diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.

Original languageEnglish
Pages (from-to)229-246
Number of pages18
JournalJournal of Computational and Applied Mathematics
Volume316
DOIs
Publication statusPublished - 15 May 2017

Fingerprint

Krylov Subspace
Parallel Methods
Time Integration
Burgers Equation
Nonlinear Partial Differential Equations
Partial differential equations
Scaling
Waveform Relaxation
Krylov Methods
Iteration
Advection-diffusion Equation
Krylov Subspace Methods
Block Method
Linear partial differential equation
Source Terms
Superposition
Initial Value Problem
Nonlinear Equations
Initial value problems
Numerical Experiment

Keywords

  • Block Krylov subspace
  • Exponential integrators
  • Parallel computing
  • Parallel in time
  • Partial differential equations

Cite this

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abstract = "A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as source terms, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection–diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection–diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.",
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A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. / Kooij, G.L.; Botchev, M.A.; Geurts, B.J.

In: Journal of Computational and Applied Mathematics, Vol. 316, 15.05.2017, p. 229-246.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - Botchev, M.A.

AU - Geurts, B.J.

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AB - A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as source terms, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection–diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection–diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.

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