On the spectral deferred correction of splitting methods for initial value problems

Hagstrom, Thomas; Zhou, Ruhai

doi:10.2140/camcos.2006.1.169

Cited by 45 publications

(50 citation statements)

References 22 publications

(92 reference statements)

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“…We will show in Lemma 4.2 that the theorem holds for k = 0, and in Lemma 4.3 that an inductive argument is satisfied. Note that this theorem is also implied by discussions in [5]. …”

Section: M) and Forward Euler Integrators For The Prediction And Kmentioning

confidence: 62%

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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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“…We will show in Lemma 4.2 that the theorem holds for k = 0, and in Lemma 4.3 that an inductive argument is satisfied. Note that this theorem is also implied by discussions in [5]. …”

Section: M) and Forward Euler Integrators For The Prediction And Kmentioning

confidence: 62%

“…Their analysis and concept of smoothness of the error vector motivated this paper. Theoretical convergence results for SDC methods constructed using low order integrators are discussed in various papers [5,1,18]. The application of SDC methods to PDEs, through the method of lines approach, can be found in [16,10,2,18,14].…”

Section: Introductionmentioning

confidence: 99%

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

2009

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“…The general notion of SDC, introduced in [14], leaves room for many variations to be explored; see, e.g., [15,16] and the references therein. For simplicity, we forgo the generality of the presentation and restrict ourselves to a single method for standard ODEs and the two methods proposed above for FDEs.…”

Section: Integral Deferred Correctionmentioning

confidence: 99%

“…Different variations of the IDC approach have been studied for standard ODEs; see e.g. [15,16] and the references therein. To reduce the number of correction iterations required to obtain a prescribed order, RK integrators have been proposed as inner schemes of methods on equidistant quadrature nodes [16].…”

Section: Introductionmentioning

confidence: 99%

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

Baffet

Hesthaven

2017

J Sci Comput

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High-order adaptive methods for fractional differential equations are proposed. The methods rely on a kernel compression scheme for the approximation and localization of the history term. To avoid complications typical to multistep methods, we focus our study on 1-step methods and approximate the local part of the fractional integral by integral deferred correction to enable high order accuracy. We study the local truncation error of integral deferred correction schemes for Volterra equations and present numerical results obtained with both implicit and the explicit methods applied to different problems.

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“…Integral deferred correction (IDC) methods, also known in some instances as spectral deferred correction (SDC), were first introduced by Dutt, Greengard and Rokhlin [9], and were further developed in [1,22,21,18,19,7,6,29,23,17]. An iterative correction procedure is used to correct an estimate of the solution by solving an integral formulation of the error equation.…”

mentioning

confidence: 99%

Parallel High-Order Integrators

Christlieb¹,

Macdonald²,

Ong³

2010

SIAM J. Sci. Comput.

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Abstract. In this work we discuss a class of defect correction methods which is easily adapted to create parallel time integrators for multi-core architectures and is ideally suited for developing methods which can be order adaptive in time. The method is based on Integral Deferred Correction (IDC), which was itself motivated by Spectral Deferred Correction by Dutt, Greengard and Rokhlin (BIT-2000).The method presented here is a revised formulation of explicit IDC, dubbed Revisionist IDC, which can achieve p th -order accuracy in "wall-clock time" comparable to a single forward Euler simulation on problems where the time to evaluate the right-hand side of a system of differential equations is greater than latency costs of inter-processor communication, such as in the case of the N -body problem. The key idea is to re-write the defect correction framework so that, after initial startup costs, each correction loop can be lagged behind the previous correction loop in a manner that facilitates running the predictor and M = p − 1 correctors in parallel on an interval which has K steps, where K p. We prove that given an r th -order Runge-Kutta method in both the prediction and M correction loops of RIDC, then the method is order r × (M + 1).The parallelization in Revisionist IDC uses a small number of cores (the number of processors used is limited by the order one wants to achieve). Using a four-core CPU, it is natural to think about fourth-order RIDC built with forward Euler, or eighth-order RIDC constructed with secondorder Runge-Kutta. Numerical tests on an N -body simulation show that RIDC methods can be significantly faster than popular Runge-Kutta methods such as the classical fourth-order RungeKutta scheme.In a PDE setting, one can imagine coupling RIDC time integrators with parallel spatial evaluators, thereby increasing the parallelization. The ideas behind RIDC extend to implicit and semiimplicit IDC and have high potential in this area.

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On the spectral deferred correction of splitting methods for initial value problems

Cited by 45 publications

References 22 publications

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

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

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

Parallel High-Order Integrators

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