Second derivative methods applied to implicit first‐ and second‐order systems

Addison, Cliff; Gladwell, I.

doi:10.1002/nme.1620200704

Cited by 12 publications

(10 citation statements)

References 19 publications

(26 reference statements)

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“…The use of second derivatives was first suggested in [24] and later studied in detail in [25], [26] and [27], and the resulting formulas were shown to have good stability properties. A more recent study [28] reinforces the stability and potential efficiency gains for stiff systems through second-derivative methods.…”

Section: Methodsmentioning

confidence: 99%

“…A more recent study [28] reinforces the stability and potential efficiency gains for stiff systems through second-derivative methods. However, despite several published implementations [27,28], these methods have not yet found wide acceptance because, despite being A -stable, they are only stable at infinity if only the second derivative at t + h is used [25] (see Section S3 in Additional file 1 and Additional file 2 for details).…”

Section: Methodsmentioning

confidence: 99%

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A specialized ODE integrator for the efficient computation of parameter sensitivities

et al. 2012

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BackgroundDynamic mathematical models in the form of systems of ordinary differential equations (ODEs) play an important role in systems biology. For any sufficiently complex model, the speed and accuracy of solving the ODEs by numerical integration is critical. This applies especially to systems identification problems where the parameter sensitivities must be integrated alongside the system variables. Although several very good general purpose ODE solvers exist, few of them compute the parameter sensitivities automatically.ResultsWe present a novel integration algorithm that is based on second derivatives and contains other unique features such as improved error estimates. These features allow the integrator to take larger time steps than other methods. In practical applications, i.e. systems biology models of different sizes and behaviors, the method competes well with established integrators in solving the system equations, and it outperforms them significantly when local parameter sensitivities are evaluated. For ease-of-use, the solver is embedded in a framework that automatically generates the integrator input from an SBML description of the system of interest.ConclusionsFor future applications, comparatively ‘cheap’ parameter sensitivities will enable advances in solving large, otherwise computationally expensive parameter estimation and optimization problems. More generally, we argue that substantially better computational performance can be achieved by exploiting characteristics specific to the problem domain; elements of our methods such as the error estimation could find broader use in other, more general numerical algorithms.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

A specialized ODE integrator for the efficient computation of parameter sensitivities

et al. 2012

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“…Cash gives conditions for methods of the form (22) to be sixth order accurate when applied to the initial value problem (1). (Note that the value of y given by Cash is slightly in error.…”

Section: The Sixth Order Methodsmentioning

confidence: 99%

“…A similar use of complex arithmetic to obtain an efficient implementation of second derivative methods for solving systems of implicit first order ordinary differential equations was discussed by Enright, [5] and by Addison and Gladwell, [1], where further details may be found. )…”

Section: Im(r){xt~l-x+l Smentioning

confidence: 98%

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Phase properties of high order, almostP-stable formulae

Thomas¹

1984

BIT

202

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Abstract.Cash [3] and Chawla [4] derive families of two-step, symmetric, P-stable (hybrid) methods for solving periodic initial value problems numerically. Chawla demonstrates the existence of a family of fourth order methods while Cash derives both fourth order and sixth order methods. In this paper, we demonstrate that these methods, which are dependent on certain free parameters, have in phase particular solutions. We consider more general families of 2-step symmetric methods, including those derived by Cash and Chawla, and show that some members of these families have higher order phase lag and are almost P-stable. We also consider how the free parameters can be chosen so as to lead to an efficient implementation of the fourth order methods for large periodic systems.

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Some error bounds and numerical experiments in modal methods for dynamics of systems

Gladwell

Hanson

1984

Earthq Engng Struct Dyn

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SUMMARYWe present some error expressions for approximate modal solutions of viscously damped linear structural vibration equations; these approximations are calculated by a variety of modal techniques. Motivated by our numerical experience, we show the equivalence of some different modal solutions in the presence of classical damping and, in passing, we obtain a previously unremarked manifestation of this type of damping. We make some brief comments about obtaining error estirnafes and, finally, we show some numerical results for a standard test problem which illustrate properties of the modal solutions.

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Second derivative methods applied to implicit first‐ and second‐order systems

Cited by 12 publications

References 19 publications

A specialized ODE integrator for the efficient computation of parameter sensitivities

A specialized ODE integrator for the efficient computation of parameter sensitivities

Phase properties of high order, almostP-stable formulae

Some error bounds and numerical experiments in modal methods for dynamics of systems

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