On the change of step size in multistep codes

Calvo, M.; Montijano, J. I.; Rández, L.

doi:10.1007/bf02144108

Cited by 11 publications

(4 citation statements)

References 5 publications

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“…BDF with variable time steps are widely used in computational codes (see e.g. [11,16]). A variable step size/variable order BDF implementation in combination with a (nonlinear) Galerkin method has been tested in [25,26] for the time integration of the Kuramoto-Sivashinsky and a reaction-diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator

Emmrich

2009

Bit Numer Math

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The initial-value problem for a first-order evolution equation is discretised in time by means of the two-step backward differentiation formula (BDF) on a variable time grid. The evolution equation is governed by a monotone and coercive potential operator. On a suitable sequence of time grids, the piecewise constant interpolation and a piecewise linear prolongation of the time discrete solution are shown to converge towards the weak solution if the ratios of adjacent step sizes are close to 1 and do not vary too much.

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Section: Introductionmentioning

confidence: 99%

Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator

Emmrich

2009

Bit Numer Math

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“…So, a reliable estimate of the local error is needed. For that purpose we consider the method of second order in (8) and the method of third order in (16). If we denote by y 1 n the approximation obtained with the method of second order and by y 2 n the approximation obtained with the third-order method, the difference ERR = |y 1 n − y 2 n | will be used as an estimate of the local error on each step.…”

Section: Formulation Using Variable Step-sizementioning

confidence: 99%

“…The strategy considered for changing the step size is that used on multistep codes, [8,20]: given a tolerance, TOL, for a selected norm, · , the classical step-size prediction derived from equating this tolerance to the norm of the local truncation error yields a new step-size h new given by…”

Section: Formulation Using Variable Step-sizementioning

confidence: 99%

Solving first-order initial-value problems by using an explicit non-standard A -stable one-step method in variable step-size formulation

Ramos

Singh

Kanwar

et al. 2015

Applied Mathematics and Computation

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“…All derivatives up to second order, which are used for the calculations of the Hessian needed for a robust performance of IPOPT, are calculated by automatic differentiation using CppAD, [9,8]. For the solution of the differential equations within the optimization problem, which are commonly stiff in chemical and biochemical applications, we have implemented a fully variable step, variable order (order 1 to 6), Backward differentiation formulae (BDF) method, based on Nordsiek array polynomial interpolation similar to the EPISODE BDF method by Byrne and Hindmarsh [16], but with the step size selection strategy of Calvo and Rández [17]. For the generation of sensitivities we have adopted the sophisticated principles of internal numerical differentiation developed by Albersmeyer and Bock [2,1] in forward and adjoint mode.…”

Section: Numerical Solution Of the Problem P θ Nmentioning

confidence: 99%

A robust optimization approach to experimental design for model discrimination of dynamical systems

Skanda

Lebiedz

2012

Math. Program.

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A high-ranking goal of interdisciplinary modeling approaches in science and engineering are quantitative prediction of system dynamics and model based optimization. Quantitative modeling has to be closely related to experimental investigations if the model is supposed to be used for mechanistic analysis and model predictions. Typically, before an appropriate model of an experimental system is found different hypothetical models might be reasonable and consistent with previous knowledge and available data. The parameters of the models up to an estimated confidence region are generally not known a priori. Therefore one has to incorporate possible parameter configurations of different models into a model discrimination algorithm which leads to the need for robustification. In this article we present a numerical algorithm which calculates a design of experiments allowing optimal discrimination of different hypothetic candidate models of a given dynamical system for the most inappropriate (worst case) parameter configurations within a parameter range. The design comprises initial values, system perturbations and the optimal placement of measurement time points, the number of measurements as well as the time points are subject to design. The statistical discrimination criterion is worked out rigorously for these settings, a derivation from the Kullback-Leibler divergence as optimization objective is presented for the case of discontinuous Heaviside-functions modeling the measurement decision which are replaced by continuous approximation during the optimization procedure. The resulting problem can be classified as a semiinfinite optimization problem which we solve in an outer approximations

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On the change of step size in multistep codes

Cited by 11 publications

References 5 publications

Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator

Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator

Solving first-order initial-value problems by using an explicit non-standard A -stable one-step method in variable step-size formulation

A robust optimization approach to experimental design for model discrimination of dynamical systems

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