### 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 language | English |
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Pages (from-to) | 229-246 |

Number of pages | 18 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 316 |

DOIs | |

Publication status | Published - 15 May 2017 |

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### Keywords

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

### Cite this

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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.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

AU - Kooij, G.L.

AU - Botchev, M.A.

AU - Geurts, B.J.

PY - 2017/5/15

Y1 - 2017/5/15

N2 - 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.

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.

KW - Block Krylov subspace

KW - Exponential integrators

KW - Parallel computing

KW - Parallel in time

KW - Partial differential equations

UR - http://www.scopus.com/inward/record.url?scp=85005790402&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2016.09.036

DO - 10.1016/j.cam.2016.09.036

M3 - Article

AN - SCOPUS:85005790402

VL - 316

SP - 229

EP - 246

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -