Implementation of an Adaptive BDF2 Formula and Comparison with the MATLAB Ode15s

Alberdi, Elisabete; Aguirrezabala, Juan José Anza; Chatzipantelidis, Panagiotis

doi:10.1016/j.procs.2014.05.091

Cited by 31 publications

(24 citation statements)

References 10 publications

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“…Within each time step a nonlinear system of equations is likely and must therefore been solved. The first order predictor of the implemented second order backward differential formulation (bdf) is a given in [2] as (3) The bdf with a linear damping matrix and a nonlinear stiffness matrix is given as (4) According to [2], the parameter is weighting the current and previous time step by .…”

Section: Methodsmentioning

confidence: 99%

“…The multiplier is generated based on the number of currently available and potentially contributing software agents. If only one agent is available is chosen according to [2]. In case of more agents, larger time steps speed up the calculation.…”

Section: Methodsmentioning

confidence: 99%

“…The number of degrees of freedom (dofs) is given as , and represent desired tolerances, is the approximation of the solution and is the solver's estimation of the (local) absolute error [5]. The optimal step size is (2) with as order of the method and as previous step. The optimal steps size is multiplied by a safety factor and used for calculating the next time step [2].…”

Section: Introductionmentioning

confidence: 99%

“…The optimal step size is (2) with as order of the method and as previous step. The optimal steps size is multiplied by a safety factor and used for calculating the next time step [2]. If is satisfied the solution is used and a next time step is calculated.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Distributed Method for Transient Simulations That Dynamically Considers Supplementary Results From Autonomous Software Agents

Jüttner

Grabmaier

Rohloff

et al. 2018

IAPGOŚ

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Based on autonomous software agents capable of calculating individual numerical field problems, a distributed method for solving transient field problems is presented. The software agents are running on distributed resources connected via a network and represent a dynamic calculation environment. Communication and data exchange between multiple agents enables their collaboration and allows decisions based on distributed overall knowledge. As unique characteristics, no central unit influences the solution process at any time. The presented simulation example and its evaluated calculation process proves the method to benefit from redundant resources.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Distributed Method for Transient Simulations That Dynamically Considers Supplementary Results From Autonomous Software Agents

Jüttner

Grabmaier

Rohloff

et al. 2018

IAPGOŚ

View full text Add to dashboard Cite

show abstract

“…A dynamically adaptive time-step algorithm can automatically adjust the time-step according to the degree of stiffness, resulting in a more optimal use of computational resources. For example, Celaya et al [17] proposed a method to select or reject a proposed time-step in adaptive algorithms by evaluating the local truncation error at each time-step to maintain the error below a given threshold.…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff Ordinary Differential Equations: Application to Solve Activated Sludge Models

Alikhani¹,

Shoghli²,

Bhowmik³

et al. 2016

AJCM

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A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system's stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.

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A time averaged steady state method for the Navier–Stokes equations

Zhao

Zorrilla

Rossi

et al. 2021

Numerical Methods in Fluids

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This work derives an incompressible variational multiscales time-averagedNavier-Stokes (NS) formulation that aims at obtaining accurate steady state solutions. Rather than using the standard time instantaneous velocity and pressure, the new formulation devises a time averaging procedure based on rewriting and solving the NS equations in terms of the newly defined time-averaged velocity and pressure. Hence, the method could be understood as a convenient change of variable so that the problem is rewritten directly in terms of the steady state quantities. The important advantage of such a point of view is that it can in principle be applied to any other formulation. Such time averaging procedure is complemented by two time step modification strategies in order to accelerate the convergence to the steady state. The guidelines of an integrated framework are presented in the article, starting with the description of the proposed numerical technique applied to general incompressible flows. The explanation is enhanced with a one-dimensional (1D) nonlinear oscillator example. Several results are presented concerning analytical benchmarks, simulation of flows in laminar, transitional and turbulent regimes with and without an inherently steady solution. K E Y W O R D Scomputational fluid dynamics, finite element method, Navier-Stokes, steady state, time-averaged, variational multiscale method This project was conducted in cooperation with International Centre for Numerical Methods in Engineering (CIMNE).This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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Implementation of an Adaptive BDF2 Formula and Comparison with the MATLAB Ode15s

Cited by 31 publications

References 10 publications

A Distributed Method for Transient Simulations That Dynamically Considers Supplementary Results From Autonomous Software Agents

A Distributed Method for Transient Simulations That Dynamically Considers Supplementary Results From Autonomous Software Agents

An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff Ordinary Differential Equations: Application to Solve Activated Sludge Models

A time averaged steady state method for the Navier–Stokes equations

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