Semi-implicit Krylov deferred correction methods for differential algebraic equations

Bu, Sunyoung; Huang, Jingfang; Minion, Michael L.

doi:10.1090/s0025-5718-2012-02564-6

Cited by 12 publications

(8 citation statements)

References 54 publications

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“…Appendix: Pseudo-code of the parareal/SDC method The following is the pseudocode for a semi-implicit parareal/SDC implementation using the first-order time-stepping method in (12) and (15). FOR k = 1 .…”

Section: Discussionmentioning

confidence: 99%

A hybrid parareal spectral deferred corrections method

Minion¹

2010

CAMCoS

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The parareal algorithm introduced in 2001 by Lions, Maday, and Turinici is an iterative method for the parallelization of the numerical solution of ordinary differential equations or partial differential equations discretized in the temporal direction. The temporal interval of interest is partitioned into successive domains which are assigned to separate processor units. Each iteration of the parareal algorithm consists of a high accuracy solution procedure performed in parallel on each domain using approximate initial conditions and a serial step which propagates a correction to the initial conditions through the entire time interval. The original method is designed to use classical single-step numerical methods for both of these steps. This paper investigates a variant of the parareal algorithm first outlined by Minion and Williams in 2008 that utilizes a deferred correction strategy within the parareal iterations. Here, the connections between parareal, parallel deferred corrections, and a hybrid parareal-spectral deferred correction method are further explored. The parallel speedup and efficiency of the hybrid methods are analyzed, and numerical results for ODEs and discretized PDEs are presented to demonstrate the performance of the hybrid approach. MSC2000: 65L99.

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Section: Discussionmentioning

confidence: 99%

A hybrid parareal spectral deferred corrections method

Minion¹

2010

CAMCoS

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“…In this section, we present our numerical scheme for solving (7), (8), and (9). Evaluating v involves layer potentials and Fourier differentiation, and for concentrated suspensions, this can be costly.…”

Section: Numerical Schemementioning

confidence: 99%

“…To reduce the cost of the linear solves, we used a block-diagonal preconditioner in [31] which is formed and factorized in matrix form at each time step. Using this preconditioner, the number of preconditioned GM-RES iterations depends only on the magnitude of the inter-vesicle interactions, which in turn is a function 1 If e n x j converges to 0, then r n+1 j = r n j , and by (8), we have solved (7) up to quadrature error.…”

Section: Preconditioningmentioning

confidence: 99%

“…SDC was first introduced by Dutt et al [13]. SDC has been applied to partial differential equations [21,39], differential algebraic equations [7,20], and integro-differential equations [19], but not vesicle flows. In [26], Minion extended SDC to implicit-explicit (IMEX) methods by studying the problem ẋ(t) = F E (x, t) + F I (x, t) where F E is non-stiff and F I is stiff.…”

Section: Related Workmentioning

confidence: 99%

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Adaptive time stepping for vesicle suspensions

Quaife

Biros

2016

Journal of Computational Physics

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We present an adaptive arbitrary-order accurate time-stepping numerical scheme for the flow of vesicles suspended in Stokesian fluids. Our scheme can be summarized as an approximate implicit spectral deferred correction (SDC) method. Applying a textbook fully implicit SDC scheme to vesicle flows is prohibitively expensive. For this reason we introduce several approximations. Our scheme is based on a semi-implicit linearized low-order time stepping method. (Our discretization is spectrally accurate in space.) We also use invariant properties of vesicle flows, constant area and boundary length in two dimensions, to reduce the computational cost of error estimation for adaptive time stepping. We present results in two dimensions for single-vesicle flows, constricted geometry flows, converging flows, and flows in a Couette apparatus. We experimentally demonstrate that the proposed scheme enables automatic selection of the step size and high-order accuracy.

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“…There are many research topics [2,4,5,8,10,11,12,15,16,17,18] in developing numerical methods for solving initial value problems (IVPs) described by…”

Section: Introductionmentioning

confidence: 99%

An Error Embedded Runge-Kutta Method for Initial Value Problems

Bu¹,

Jung²,

Kim³

2016

Kyungpook mathematical journal

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Abstract. In this paper, we propose an error embedded Runge-Kutta method to improve the traditional embedded Runge-Kutta method. The proposed scheme can be applied into most explicit embedded Runge-Kutta methods. At each integration step, the proposed method is comprised of two equations for the solution and the error, respectively. These solution and error are obtained by solving an initial value problem whose solution has the information of the error at each integration step. The constructed algorithm controls both the error and the time step size simultaneously and possesses a good performance in the computational cost compared to the original method. For the assessment of the effectiveness, the van der Pol equation and another one having a difficulty for the global * Corresponding Author.

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Semi-implicit Krylov deferred correction methods for differential algebraic equations

Cited by 12 publications

References 54 publications

A hybrid parareal spectral deferred corrections method

A hybrid parareal spectral deferred corrections method

Adaptive time stepping for vesicle suspensions

An Error Embedded Runge-Kutta Method for Initial Value Problems

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