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

Minion, Michael L.

doi:10.2140/camcos.2007.2.1

Cited by 35 publications

(35 citation statements)

References 34 publications

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“…[2]) in the prediction step of PIDC methods. This strategy was investigated in [25] as a possible remedy for order reduction in very stiff problems, and a paper which further investigates this potential is in preparation [22].…”

Section: Discussionmentioning

confidence: 99%

“…Although a qualitatively similar order reduction for PIDC methods is observed for both the van der Pol test problem and the simpler cosine test, it is not clear whether this behavior is completely generic, especially in regard to the differences in behavior between uniform and Gauss points. As just mentioned, the benefits of using semi-implicit BDF type predictors with PIDC methods to remedy order reduction is being more closely investigated [22].…”

Section: Discussionmentioning

confidence: 99%

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Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

Layton

Minion

2005

Bit Numer Math

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This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [tn, tn+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The main purpose of this paper is to present a detailed analysis of the implications of the location of quadrature nodes on the accuracy and stability of the overall method. Comparisons between Gauss-Legendre, Gauss-Lobatto, Gauss-Radau, and uniformly spaced points are presented. Also, for a given set of quadrature nodes, quadrature rules may be formulated that include or exclude function values computed at the left-hand endpoint tn. Quadrature rules that do not depend on the left-hand endpoint (which are referred to as right-hand quadrature rules) are shown to lead to L(α)-stable implicit methods with α ≈ π/2. The semiimplicit analog of this property is also discussed. Numerical results suggest that the use of uniform quadrature nodes, as opposed to nodes based on Gaussian quadratures, does not significantly affect the stability or accuracy of these methods for orders less than ten. In contrast, a study of the reduction of order for stiff equations shows that when uniform quadrature nodes are used in conjunction with a right-hand quadrature rule, the form and extent of order-reduction changes considerably. Specifically, a reduction of order to O( 2 ) is observed for uniform nodes as opposed to O( ∆t) for non-uniform nodes, where ∆t denotes the time step and a stiffness parameter such that → 0 corresponds to the problem becoming increasingly stiff. (2000): 65B05. AMS subject classification

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

Layton

Minion

2005

Bit Numer Math

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“…It is also possible to contruct IMEX methods using general linear multi-step schemes. Such methods however require multiple starting values, and their stabilities deteriorate at higher order [16,3,23].…”

Section: Introductionmentioning

confidence: 99%

Semi-implicit integral deferred correction constructed with additive Runge–Kutta methods

Christlieb

Morton

Ong

et al. 2011

Communications in Mathematical Sciences

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Abstract. In this paper, we construct high order semi-implicit integrators using integral deferred correction (IDC) to solve stiff initial value problems. The general framework for the construction of these semi-implicit methods uses uniformly distributed nodes and additive Runge-Kutta (ARK) integrators as base schemes inside an IDC framework, which we refer to as IDC-ARK methods. We establish under mild assumptions that, when an r th order ARK method is used to predict and correct the numerical solution, the order of accuracy of the IDC method increases by r for each IDC prediction and correction loop. Numerical experiments support the established theorems, and also indicate that higher order IDC-ARK methods present an efficiency advantage over existing implicit-explicit (IMEX) ARK schemes in some cases.

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“…The selection of quadrature nodes is discussed in [11], while [15] uses semi-implicit schemes to handle temporal multi-scale problems. The authors in [13,12] also study the choice of predictors and correctors to construct semi-implicit SDC methods. In [8,9], Krylov subspace methods are used to accelerate the convergence of SDC methods.…”

Section: Introductionmentioning

confidence: 99%

Integral deferred correction methods constructed with high order Runge–Kutta integrators

2009

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Abstract. Spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs) were introduced by Dutt, Greengard and Rokhlin (2000). It was shown in that paper that SDC methods can achieve arbitrary high order accuracy and possess nice stability properties. Their SDC methods are constructed with low order integrators, such as forward Euler or backward Euler, and are able to handle stiff and non-stiff terms in the ODEs. In this paper, we use high order Runge-Kutta (RK) integrators to construct a family of related methods, which we refer to as integral deferred correction (IDC) methods. The distribution of quadrature nodes is assumed to be uniform, and the corresponding local error analysis is given. The smoothness of the error vector associated with an IDC method, measured by the discrete Sobolev norm, is a crucial tool in our analysis. The expected order of accuracy is demonstrated through several numerical examples. Superior numerical stability and accuracy regions are observed when high order RK integrators are used to construct IDC methods.

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

Cited by 35 publications

References 34 publications

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

Semi-implicit integral deferred correction constructed with additive Runge–Kutta methods

Integral deferred correction methods constructed with high order Runge–Kutta integrators

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