High-order multi-implicit spectral deferred correction methods for problems of reactive flow

Bourlioux, Anne; Layton, Anita T.; Minion, Michael L.

doi:10.1016/s0021-9991(03)00251-1

Cited by 128 publications

(167 citation statements)

References 27 publications

(63 reference statements)

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“…A detailed derivation of the SIPIDC methods for ODEs and for an advection-diffusion-reaction equation can be found in ] and [Bourlioux et al 2003]. The target ODE takes the form…”

Section: Sipidc Methodsmentioning

confidence: 99%

“…The quadrature ᏽ should have at least the same order of accuracy as the updated approximation u k+1 . As in [Bourlioux et al 2003;], the quadrature ᏽ m+1 m is computed as the integral of an interpolating polynomial over the subinterval [t m , t m+1 ] (see further discussion below).…”

Section: Sipidc Methodsmentioning

confidence: 99%

“…In [Bourlioux et al 2003;] the points t m are chosen to be Gauss-Lobatto quadrature nodes, which are more convenient in that the solution is directly computed at both endpoints of the time step interval. However, because Gauss-Lobatto nodes are not evenly spaced for orders of accuracy >2, predictors with higher than first order are less convenient to implement (nonetheless, it can be done; see ]).…”

Section: Sipidc Methodsmentioning

confidence: 99%

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Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods

Minion

2007

CAMCoS

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High-order semi-implicit Picard integral deferred correction (SIPIDC) methods have previously been proposed for the time-integration of partial differential equations with two or more disparate time scales. The SIPIDC methods studied to date compute a high-order approximation by first computing a provisional solution with a first-order semi-implicit method and then using a similar semi-implicit method to solve a series of correction equations, each of which raises the order of accuracy of the solution by one. This study assesses the efficiency of SIPIDC methods that instead use standard semi-implicit methods with orders two through four to compute the provisional solution. Numerical results indicate that using a method with more than first-order accuracy in the computation of the provisional solution increases the efficiency of SIPIDC methods in some cases. First-order PIDC corrections can improve the efficiency of semi-implicit integration methods based on backward difference formulae (BDF) or Runge-Kutta methods while maintaining desirable stability properties. Finally, the phenomenon of order reduction, which may be encountered in the integration of stiff problems, can be partially alleviated by the use of BDF methods in the computation of the provisional solution.

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“…A detailed derivation of the SIPIDC methods for ODEs and for an advection-diffusion-reaction equation can be found in ] and [Bourlioux et al 2003]. The target ODE takes the form…”

Section: Sipidc Methodsmentioning

confidence: 99%

Section: Sipidc Methodsmentioning

confidence: 99%

Section: Sipidc Methodsmentioning

confidence: 99%

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Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods

Minion

2007

CAMCoS

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“…Furthermore, SDC methods based on a semi-implicit or IMEX formulation have been constructed by applying an operator splitting approach to the correction equations [14,45,46,47]. Semi-implicit methods treat non-stiff terms in the equations explicitly and stiff terms implicitly, which can significantly reduce the overall cost for stiff problems compared to fully implicit methods.…”

Section: Introductionmentioning

confidence: 99%

“…Higher-order semi-implicit SDC schemes do not suffer from the stability limits of linear multistep schemes based on BDF [3,7,25], nor the difficulty of satisfying the coupling constraints in IMEX Runge-Kutta methods [8,12,13,19,37,49,57]. In fact, higher-order SDC methods with multiple implicit or explicit terms and varying time steps (multirate methods) have also been developed [14,15,41].…”

Section: Introductionmentioning

confidence: 99%

Semi-implicit Krylov deferred correction methods for differential algebraic equations

Huang

Minion

2012

Math. Comp.

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Abstract. In the recently developed Krylov deferred correction (KDC) methods for differential algebraic equation initial value problems (Huang, Jia, Minion, 2007), a Picard-type collocation formulation is preconditioned using low-order time integration schemes based on spectral deferred correction (SDC), and the resulting system is solved efficiently using Newton-Krylov methods. KDC methods have the advantage that methods with arbitrarily high order of accuracy can be easily constructed which have similar computational complexity as lower order methods. In this paper, we investigate semi-implicit KDC (SI-KDC) methods in which the stiff component of the preconditioner is treated implicitly and the non-stiff parts explicitly. For certain types of problems, such a semi-implicit treatment can significantly reduce the computational cost of the preconditioner compared to fully implicit KDC (FI-KDC) methods. Preliminary analysis and numerical experiments show that the convergence of Newton-Krylov iterations in the SI-KDC methods is similar to that in FI-KDC, and hence the SI-KDC methods offer a reduction in overall computational cost for such problems.

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High‐order TVB discretizations in time using Richardson's extrapolation

Amat

Busquier

Gómez

et al. 2009

Numer Methods Biomed Eng

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SUMMARYRichardson's extrapolation is applied to total variation diminishing (TVD) Runge-Kutta schemes in order to improve their accuracy without doubling their computational cost as classical high-order (more than three) TVD schemes. The obtained schemes preserve the total variation bounded (TVB) property. Since the extrapolation is performed with schemes that are really TVD, a stable behavior is expected. Using also TVB schemes in space, some numerical experiments showing good performances of the proposed schemes are presented. Up to our knowledge, it is the first time that extrapolation procedures are used in this way.

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High-order multi-implicit spectral deferred correction methods for problems of reactive flow

Cited by 128 publications

References 27 publications

Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods

Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods

Semi-implicit Krylov deferred correction methods for differential algebraic equations

High‐order TVB discretizations in time using Richardson's extrapolation

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