Acceleration of implicit schemes for large systems of nonlinear differential-algebraic equations

Mouhamad Al Sayed Ali; Miloud Sadkane; Mouhamad Al Sayed Ali; Miloud Sadkane

doi:10.3934/math.2020040

AIMS Mathematics

2020, Volume 5, Issue 1: 603-618. doi: 10.3934/math.2020040

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Acceleration of implicit schemes for large systems of nonlinear differential-algebraic equations

Mouhamad Al Sayed Ali ¹,
Miloud Sadkane ^{2
,
,}

1.
Université Libanaise, Faculté des Sciences 1, Beyrouth, Liban
2.
Université de Brest, CNRS - UMR 6205, Mathéematiques, 6, Av. Le Gorgeu. 29238 Brest Cedex 3, France

Received: 03 October 2019 Accepted: 18 November 2019 Published: 16 December 2019
MSC : 65H10, 65L80, 65F10, 65K10

When solving large systems of nonlinear differential-algebraic equations by implicit schemes, each integration step requires the solution of a system of large nonlinear algebraic equations. The latter is solved by an inexact Newton method which, in its turn, leads to a set of large linear systems commonly solved by a Krylov subspace iterative method. The efficiency of the whole process depends on the initial guesses for the inexact Newton and the Krylov subspace methods. An inexpensive approach is proposed and justified that computes good initial guesses for these methods. It requires a subspace of small dimension and the use of line search and trust region for the inexact Newton method and Petrov-Galerkin for the Krylov subspace method. Numerical examples are included to illustrate the effectiveness of the proposed approach.
- nonlinear equations,
- nonlinear DAE systems,
- inexact Newton,
- GMRES,
- line search,
- trust region
Citation: Mouhamad Al Sayed Ali, Miloud Sadkane. Acceleration of implicit schemes for large systems of nonlinear differential-algebraic equations[J]. AIMS Mathematics, 2020, 5(1): 603-618. doi: 10.3934/math.2020040

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Abstract

When solving large systems of nonlinear differential-algebraic equations by implicit schemes, each integration step requires the solution of a system of large nonlinear algebraic equations. The latter is solved by an inexact Newton method which, in its turn, leads to a set of large linear systems commonly solved by a Krylov subspace iterative method. The efficiency of the whole process depends on the initial guesses for the inexact Newton and the Krylov subspace methods. An inexpensive approach is proposed and justified that computes good initial guesses for these methods. It requires a subspace of small dimension and the use of line search and trust region for the inexact Newton method and Petrov-Galerkin for the Krylov subspace method. Numerical examples are included to illustrate the effectiveness of the proposed approach.

References

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