Variable‐order variable‐step algorithms for second‐order systems. Part 2: The codes

Gladwell, I.; Thomas, R. M.

doi:10.1002/nme.1620260105

Cited by 19 publications

(8 citation statements)

References 4 publications

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“…The table also shows the number of iterations and the number of failed steps for each solution. The second-order convergence of the iterative Thomas-Gladwell and the non-iterative scheme is evident, confirming that condition (6) is weakened to (20) when the order of the governing equation is reduced from second to first. The empirical results are also consistent with the analytic stability assessment, which predicted no stability restrictions on the step size for both the iterative and non-iterative algorithms.…”

Section: Is the Volumetric Moisture Content [Dimensionless] Z Is Thesupporting

confidence: 54%

“…We seek to verify the order of accuracy and stability for fully non-linear DEs, as opposed to linear or quasi-linear forms. It is found that both constraints (6) and (7) can be weakened, indicating that Thomas-Gladwell methods are applicable to a wider range of problems than originally intended.…”

Section: M(u T) Du(t) Dt + K(u T)u(t) = F(u T)mentioning

confidence: 92%

“…Thomas-Gladwell methods have also been used in variable-order variable-step ODE codes [6] and are of practical significance since they include many common schemes as special cases [1].…”

Section: M(u T) Du(t) Dt + K(u T)u(t) = F(u T)mentioning

confidence: 99%

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Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

Kavetski

Binning

Sloan

2004

Numerical Meth Engineering

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SUMMARYThe consistency and stability of a Thomas-Gladwell family of multistage time-stepping schemes for the solution of first-order non-linear differential equations are examined. It is shown that the consistency and stability conditions are less stringent than those derived for second-order governing equations. Second-order accuracy is achieved by approximating the solution and its derivative at the same location within the time step. Useful flexibility is available in the evaluation of the non-linear coefficients and is exploited to develop a new non-iterative modification of the Thomas-Gladwell method that is secondorder accurate and unconditionally stable. A case study from applied hydrogeology using the non-linear Richards equation confirms the analytic convergence assessment and demonstrates the efficiency of the non-iterative formulation.

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Section: Is the Volumetric Moisture Content [Dimensionless] Z Is Thesupporting

confidence: 54%

Section: M(u T) Du(t) Dt + K(u T)u(t) = F(u T)mentioning

confidence: 92%

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Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

Kavetski

Binning

Sloan

2004

Numerical Meth Engineering

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“…The discussion has, so far, assumed that it is convenient to evaluate the external force rate, F "dF /dt, analytically in the overall forcing function de"ned by equation (20). For cases where this is not so, this derivative can be approximated using discrete values of the external force vector.…”

Section: Evaluation Of the Forcing Function Ratementioning

confidence: 99%

Biot consolidation analysis with automatic time stepping and error control Part 1: theory and implementation

Sloan

Abbo

1999

Int. J. Numer. Anal. Meth. Geomech.

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SUMMARYA new "nite element algorithm for solving elastic and elastoplastic coupled consolidation problems is described. The procedure treats the governing consolidation relations as a system of "rst-order di!erential equations and is based on the backward Euler and Thomas and Gladwell schemes with automatic subincrementation of a prescribed series of time increments. The prescribed time increments, which are called coarse time steps, serve to start the procedure and are chosen by the user. The automatic consolidation algorithm attempts to select the time subincrements such that, for a given mesh, the time-stepping (or temporal discretisation) error in the displacements lies close to a speci"ed tolerance.Unlike existing solution techniques, the new algorithm computes not only the displacements and pore pressures, but also their derivatives with respect to time. These extra variables permit a family of unconditionally stable integration algorithms to be constructed which automatically provide an estimate of the local truncation error for each time step. This error estimate is inexpensive to compute and may be used to develop a simple and e$cient automatic time stepping mechanism. For the elastic case, the displacements and pore pressures at the end of each subincrement may be solved directly without the need for iteration. For elastoplastic behaviour, however, the governing relationships are non-linear and a system of non-linear equations must be solved to compute the updates.

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“…Here, we consider the special second-order system Mx = f(t, x), x(O), x(0) given (3) such as arises commonly in nonlinear oscillation problems. We derive an efficient formula pair which is particularly suitable for such problems.…”

Section: Introduction Thomas and Gladwell' Developed A Class Of Mulmentioning

confidence: 99%

A single‐step algorithm for oscillatory problems

Thomas¹,

Evans²

1989

Commun. appl. numer. methods

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SUMMARYWe consider a class of multistage single-step methods introduced by Zienkicwicz ef al. ' and developed by Thomas and Gladwell.' From this class, we derive methods which are particularly suitable for the numerical solution of oscillatory problems. We propose a particular pair of methods which permit local error estimation while retaining a 'one-step' mode of operation. We discuss the implementation of this pair for both linear and nonlinear problems, and illustrate the discussion with numerical results.

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Variable‐order variable‐step algorithms for second‐order systems. Part 2: The codes

Cited by 19 publications

References 4 publications

Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

Biot consolidation analysis with automatic time stepping and error control Part 1: theory and implementation

A single‐step algorithm for oscillatory problems

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