Modified Defect Correction Algorithms for ODEs. Part I: General Theory

Auzinger, Winfried; Hofstätter, Harald; Kreuzer, W.; Weinmüller, Ewa

doi:10.1023/b:numa.0000033129.73715.7f

Cited by 22 publications

(29 citation statements)

References 14 publications

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“…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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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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Section: Introductionmentioning

confidence: 99%

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

2009

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“…They also extend to other defect correction methods (e.g., [1]). The authors are presently exploring strong stability preserving [20,15] and other nonlinear stability properties of these RIDC schemes.…”

Section: 3mentioning

confidence: 97%

“…In this paper, we construct and analyze a class of novel time integrators for initial value problems (IVP), known as Revisionist Integral Deferred Correction methods (RIDC), which can be efficiently implemented with multi-core architectures. We adopt the "revisionist" terminology to highlight that (1) this is a revision of the standard IDC formulation, and (2) successive corrections, running in parallel but lagging in time, revise and improve the approximation to the solution. Although the RIDC schemes are presented in the integral deferred correction framework, this new method can equivalently be viewed as a sequence of time steps with an occasional "reset".…”

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confidence: 99%

“…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%

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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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“…Many numerical techniques have been developed for the accurate and efficient solution of ordinary and partial differential equation initial value problems with algebraic constraints, including linear multi-step methods, Runge-Kutta methods, and operator splitting techniques [6,10,16,20,30,46,54,59,61,62]. Their applications include (among others) numerical simulations in fluid and solid mechanics, circuits design, electrical power systems, and diffusion-reaction processes in biological and chemical systems.…”

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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Modified Defect Correction Algorithms for ODEs. Part I: General Theory

Cited by 22 publications

References 14 publications

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

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

Parallel High-Order Integrators

Semi-implicit Krylov deferred correction methods for differential algebraic equations

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