Defect Correction Algorithms for Stiff Ordinary Differential Equations

Frank, Reinhard; Hertling, Jörg; Lehner, H.

doi:10.1007/978-3-7091-7023-6_2

Cited by 3 publications

(7 citation statements)

References 7 publications

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“…This idea of deferred corrections is first used in [14] and the references therein, where it is applied to the differential equations (30) directly. In this work, we use the integral form of the differential equations as suggested in [4].…”

Section: Spectral Deferred Correctionsmentioning

confidence: 99%

Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. I: The Algorithm and Benchmarks

Zhou¹,

Forest²,

Wang³

2005

Multiscale Model. Simul.

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The Doi-Hess theory coupled with an anisotropic Marrucci-Greco distortional elasticity potential provides a kinetic, mean field description of the coupling between hydrodynamics, molecular orientation by excluded volume, and elastic distortions of flowing nematic liquid crystalline polymers (LCPs). In this paper we provide the first numerical algorithm and implementation of kinetic-scale models for structure formation in confined, planar Couette cells. The model and algorithm extend kinetic simulations 80-93] for homogeneous nematic polymers in imposed shear flows. The focus here is one-dimensional flow and LCP structures that form between oppositely moving, parallel plates, a classical model problem that has been studied in detail with continuum models [P. G. de Gennes and On weak plane Couette and Poiseuille flows of nematic polymers, SIAM J. Appl. Math., submitted]. The model consists of a Smoluchowski equation for the space-time evolution of the orientational probability distribution function, coupled with a momentum flow balance equation, a constitutive equation for the extra stress, and the continuity equation. The Smoluchowski equation is first reduced to a finite set of partial differential equations in time and space for spherical harmonic amplitudes. Then we discretize the spatial variables (by the method of lines) using high-order finite differences, which reduces the full system to a large set of ordinary differential equations. Adaptive grid generation techniques are implemented. To provide an accurate and stable time integration, we employ the newly developed spectral deferred correction algorithm. We close with an application of the kinetic theory and numerical code to explore steady state flow-molecular structures in slow planar Couette experiments (low Deborah number) and low Ericksen number, where distortional elasticity dominates short-range excluded volume effects. We confirm recent analytical results based on mesoscopic closure models derived from this kinetic model, both with equal [M. G. Forest et al., J. Rheol., 48 (2004), pp. 175-192] and distinct [Z. Cui, M. G. Forest, and Q. Wang, On weak plane Couette and Poiseuille flows of nematic polymers, SIAM J. Appl. Math., submitted] Frank elasticity constants, in the dual limit of low Deborah and low Ericksen numbers.

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Section: Spectral Deferred Correctionsmentioning

confidence: 99%

Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. I: The Algorithm and Benchmarks

Zhou¹,

Forest²,

Wang³

2005

Multiscale Model. Simul.

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“…(Here we are using " ij to denote the Kronecker ı to avoid confusion with the correction vector.) Combining (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) with (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) and solving in increasing k we conclude that…”

Section: Order Of Accuracymentioning

confidence: 93%

“…The most straightforward approach, used by Dutt et al [7] and Minion [21], is to apply (2-5) directly to (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) to obtain the correction formula:…”

Section: Spectral Deferred Correction With Splittingmentioning

confidence: 99%

“…Lastly we note that alternative correction formulas are also possible. The correction method we have employed in [12; 11; 24; 25] follows [9]:…”

Section: Spectral Deferred Correction With Splittingmentioning

confidence: 99%

“…However, the theoretical results in this paper only apply in general to corrections based on (2-12)- (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). To apply them to corrections based on (2-16) we must make the starting method and the correction method coincide, which is the case in [12; 11; 24; 25].…”

Section: Spectral Deferred Correction With Splittingmentioning

confidence: 99%

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

Hagstrom

Zhou

2006

CAMCoS

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Spectral deferred correction is a flexible technique for constructing high-order, stiffly-stable time integrators using a low order method as a base scheme. Here we examine their use in conjunction with splitting methods to solve initial-boundary value problems for partial differential equations. We exploit their close connection with implicit Runge-Kutta methods to prove that up to the full accuracy of the underlying quadrature rule is attainable. We also examine experimentally the stability properties of the methods for various splittings of advection-diffusion and reaction-diffusion equations.

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Survey of interval algorithms for ordinary differential equations

Corliss

1989

Proceedings of the Conference on Numerical Ordinary Differential Equations

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Defect Correction Algorithms for Stiff Ordinary Differential Equations

Cited by 3 publications

References 7 publications

Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. I: The Algorithm and Benchmarks

Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. I: The Algorithm and Benchmarks

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

Survey of interval algorithms for ordinary differential equations

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