Deferred Correction Methods for Ordinary Differential Equations

Ong, Benjamin W.; Spiteri, Raymond J.

doi:10.1007/s10915-020-01235-8

Cited by 15 publications

(8 citation statements)

References 67 publications

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“…Suppose that the integrator used to construct the provisional solution has order p 0 , the integrator used to compute the correction approximation at level k has order p k , and m correction steps are taken. If we have p groups of N uniformly distributed nodes, then the SDC methods can attain order [26,27] O(h min(P m ,p+1) ),…”

Section: Order Of Convergencementioning

confidence: 99%

High-Order Schemes for Nonlinear Fractional Differential Equations

et al. 2022

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We propose high-order schemes for nonlinear fractional initial value problems. We split the fractional integral into a history term and a local term. We take advantage of the sum of exponentials (SOE) scheme in order to approximate the history term. We also use a low-order quadrature scheme to approximate the fractional integral appearing in the local term and then apply a spectral deferred correction (SDC) method for the approximation of the local term. The resulting one-step time-stepping methods have high orders of convergence, which make adaptive implementation and accuracy control relatively simple. We prove the convergence and stability of the proposed schemes. Finally, we provide numerical examples to demonstrate the high-order convergence and adaptive implementation.

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Section: Order Of Convergencementioning

confidence: 99%

High-Order Schemes for Nonlinear Fractional Differential Equations

et al. 2022

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“…Finally, we use the Arenstorf orbit problem [27,36] for further illustration of the proposed method's capabilities. The Arenstorf orbit problem describes a three body problem, where the movement of a light object is influenced by two heavy objects.…”

Section: Arenstorf Orbitmentioning

confidence: 99%

“…The solution is a closed orbit with a period of 17.065216560159. Similar to [36], we choose 10 5 equidistant timesteps to simulate one period of the problem. We choose the HBPC*(8, k max ) method with Alg.…”

Section: Arenstorf Orbitmentioning

confidence: 99%

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Parallel-in-time high-order multiderivative IMEX solvers

Schütz¹,

Seal²,

Zeifang³

2021

Preprint

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In this work, we present a novel class of parallelizable high-order time integration schemes for the approximate solution of additive ODEs. The methods achieve high order through a combination of a suitable quadrature formula involving multiple derivatives of the ODE's right-hand side and a predictor-corrector ansatz. The latter approach is designed in such a way that parallelism in time is made possible. We present thorough analysis as well as numerical results that showcase scaling opportunities of methods from this class of solvers.

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“…On other hand, the deferred correction method, proposed originally by Fox [32], is a well-established method for designing high-order approximations to the solution of the partial differential equations based on simple lower-order numerical methods by a process of iterated corrections (see [33] for a survey). Further, Skeel [34] has obtained computational estimates of the local discretization error by the deferred correction method without the necessity of demonstrating the existence of asymptotic expansions of the global error.…”

mentioning

confidence: 99%

Deferred defect‐correction finite element method for the Darcy‐Brinkman model

Zeng

Huang

2021

Z Angew Math Mech

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This paper proposes a deferred defect-correction method for the time-dependent nonlinear Darcy-Brinkman model based on the mixed finite element method. The presented method combines the deferred correction strategy with the defect-correction method and involves two steps. In the first step, a defect Darcy-Brinkman model is solved. Then in the second step, a correction Darcy-Brinkman model based on the deferred correction approach is solved. Moreover, unconditional stability and convergence of the presented method are deduced. Finally, several numerical examples are given to support the theoretical analysis.

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Deferred Correction Methods for Ordinary Differential Equations

Cited by 15 publications

References 67 publications

High-Order Schemes for Nonlinear Fractional Differential Equations

High-Order Schemes for Nonlinear Fractional Differential Equations

Parallel-in-time high-order multiderivative IMEX solvers

Deferred defect‐correction finite element method for the Darcy‐Brinkman model

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