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The purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF) algorithms with variable time step size, and the adaptive time-stepping in BDF algorithms is demonstrated for efficient time-dependent simulations in fluid flow and heat transfer.

The Lagrange interpolation polynomial is used to predict the time derivative, and then the accurate primary result is obtained by the Gauss integral, which is applied to evaluate the local error. Not only the generalized formula of the proposed error estimator is presented but also the specific expression for the widely applied BDF1/2/3 is illustrated. Two essential executable MATLAB functions to implement the proposed error estimator are appended for practical applications. Then, the adaptive time-stepping is demonstrated based on the newly proposed error estimator for BDF algorithms.

The validation tests show that the newly proposed error estimator is accurate such that the effectivity index is always close to unity for both linear and nonlinear problems, and it avoids under/overestimation of the exact local error. The applications for fluid dynamics and coupled fluid flow and heat transfer problems depict the advantage of adaptive time-stepping based on the proposed error estimator for time-dependent simulations.

In contrast to existing error estimators for BDF algorithms, the present work is more accurate for the local error estimation, and it can be readily extended to practical applications in engineering with a few changes to existing codes, contributing to efficient time-dependent simulations in fluid flow and heat transfer.

This paper presents an approach to the time discretization of electron and hole continuity equations in semiconductors. We propose a nth order backward differentiation formula with variable stepsize determined by a local truncation error evaluation. Up to the fifth order is examined. As an example, the effect of a heavy ion (63Cu , 70 MeV) on a silicon diode is simulated with a 3D axi‐symmetrical model, using various time stepping schemes. The results obtained are compared. For this particular problem and the mesh considered, the third order is the most efficient one to reduce the calculation time.

Developing an efficient second‐order integration method of transient analysis of nonlinear dynamic circuits which overcomes the main drawback of the trapezoidal rule.

Dynamic circuits including transistors and operational amplifiers are considered. A new family of two‐step, second‐order numerical integration algorithms has been developed using a polynomial approximation.

The algorithms have been worked out which are implicit, A‐stable and they depend on a parameter which is allowed to be changed during the computation process according to a proposed strategy. Also the variable step‐size formula has been derived enabling us to eliminate a restarting procedure. The method has been implemented and tested using several representative circuits. It has been compared, both theoretically and via numerical examples, with the alternative well known algorithms: the trapezoidal rule and the backward differentiation formula of order two.

The algorithms developed in the paper are two‐step and second‐order, consequently the step size cannot be too large and the algorithms are not L‐stable.

A new family of two‐step implicit integration algorithms is developed. It can be useful for the analysis and design of electronic circuits.

This paper aims to focus on characterization of interactions between hp-adaptive time-integrators based on backward differentiation formulas (BDF) and adaptive meshing based on Zhu and Zienkiewicz error estimation approach. If mesh adaptation only occurs at user-supplied times and results in a completely new mesh, it is necessary to stop the time-integration at these same times. In these conditions, one challenge is to find an efficient and reliable way to restart the time-integration. The authors investigate what impact grid-to-grid interpolation errors have on the relaunch of the computation.

Two restart strategies of the time-integrator were used: one based on resetting the time-step size h and time-integrator order p to default values (used in the initial startup phase), and another designed to restart with the time-step size h and order p used by the solver prior to remeshing. The authors also investigate the benefits of quadratically interpolate the solution on the new mesh. Both restart strategies were used to solve laminar incompressible Navier–Stokes and the Unsteady Reynolds Averaged Naviers-Stokes (URANS) equations.

The adaptive features of our time-integrators are excellent tools to quantify errors arising from the data transfer between two grids. The second restart strategy proved to be advantageous only if a quadratic grid-to-grid interpolation is used. Results for turbulent flows also proved that some precautions must be taken to ensure grid convergence at any time of the simulation. Mesh adaptation, if poorly performed, can indeed lead to losing grid convergence in critical regions of the flow.

This study exhibits the benefits and difficulty of assessing both spatial error estimates and local error estimates to enhance the efficiency of unsteady computations.

The purpose of this study is to investigate the interaction between magnetotactic bacteria and Fe₃O₄–water nanofluid (NF) in a wavy enclosure in the presence of 2D natural convection flow.

Uniform magnetic field (MF), Brownian and thermophoresis effects are also contemplated. The dimensionless, time-dependent equations are governed by stream function, vorticity, energy, nanoparticle concentration and number of bacteria. Radial basis function-based finite difference method for the space derivatives and the second-order backward differentiation formula for the time derivatives are performed. Numerical outputs in view of isolines as well as average Nusselt number, average Sherwood number and flux density of microorganisms are presented.

Convective mass transfer rises if any of Lewis number, Peclet number, Rayleigh number, bioconvection Rayleigh number and Brownian motion parameter increases, and the flux density of microorganisms is an increasing function of Rayleigh number, bioconvection Rayleigh number, Peclet number, Brownian and thermophoresis parameters. The rise in buoyancy ratio parameter between 0.1 and 1 and the rise in Hartmann number between 0 and 50 reduce all outputs average Nusselt, average Sherwood numbers and flux density of microorganisms.

This study implies the importance of the presence of magnetotactic bacteria and magnetite nanoparticles inside a host fluid in view of heat transfer and fluid flow. The limitation is to check the efficiency on numerical aspect. Experimental observations would be more effective.

In practical point of view, in a heat transfer and fluid flow system involving magnetite nanoparticles, the inclusion of magnetotactic bacteria and MF effect provide control over fluid flow and heat transfer.

This is a scientific study. However, this idea may be extended to sustainable energy or biofuel studies, too. This means that a better world may create better social environment between people.

The presence of magnetotactic bacteria inside a Fe₃O₄–water NF under the effect of a MF is a good controller on fluid flow and heat transfer. Since the magnetotactic bacteria is fed by nanoparticles Fe₃O₄ which has strong magnetic property, varying nanoparticle concentration and Brownian and thermophoresis effects are first considered.

This article aims to develop a fast numerical method for solving the one-dimensional heat and mass transfer problem within a desiccant rotor.

The collocation method is used for discretizing the axial dimension and reducing the number of dependent variables. The resulting system of equation is then solved through backward differentiation formulas.

The numerical results obtained here focus on verifying the accuracy and the computation time of the proposed method with respect to the finite difference method. The proposed numerical solution method resulted faster than, and as much accurate as, the finite difference method, over a large range of operating conditions that are of interest in desiccant cooling applications.

For heat and mass transfer analysis, constant average transfer coefficients are used. The results are calculated for NTU between 2 and 15 and for Le number between 0.5 and 2.

The results can be used in designing desiccant heat exchangers and desiccant cooling systems including complex rotor arrangements.

Different from other simplified solution techniques, the proposed method relies on few parameters that retain physical meaning and applies also to complex rotor configurations.

Using appropriate solution techniques for transformer inrush current transient study is of great prominence owing to the inevitable inclusion of differential equations leading to complicated analysis procedures. This study aims to propose an analytical-numerical method to accurately analyze the three-phase three-limb core-type transformer inrush current in different cases considering the nonlinear behavior of the iron core.

The proposed method focuses on acquiring equations for inrush current and also the magnetic core flux by the application of a simulation-based iterative approach. In this regard, multiple integral equations are solved taking the time intervals into account. Then several derivations and integrations of matrix terms are substituted into the obtained results so as to simplify the solution process.

The method provides notable enhancements in computation time and also excellent qualities of accuracy compared with conventional numerical methods.

The proposed method is simulated for two three-phase transformers via MATLAB software. The obtained simulation results have been also compared with experimental tests.

Actually, the analytical-numerical method is capable of computing higher number of iterations in a shorter time efficiently, while making use of the conventional numerical procedures may not result in expected convergences. The simulation results of the proposed analytical-numerical technique illustrate a close agreement with the experimental test, and hence, verify the method preciousness.

This paper aims to present an unconditionally energy-stable scheme for approximating the convective heat transfer equation.

The scheme stems from the generalized positive auxiliary variable (gPAV) idea and exploits a special treatment for the convection term. The original convection term is replaced by its linear approximation plus a correction term, which is under the control of an auxiliary variable. The scheme entails the computation of two temperature fields within each time step, and the linear algebraic system resulting from the discretization involves a coefficient matrix that is updated periodically. This auxiliary variable is given by a well-defined explicit formula that guarantees the positivity of its computed value.

Compared with the semi-implicit scheme and the gPAV-based scheme without the treatment on the convection term, the current scheme can provide an expanded accuracy range and achieve more accurate simulations at large (or fairly large) time step sizes. Extensive numerical experiments have been presented to demonstrate the accuracy and stability performance of the scheme developed herein.

This study shows the unconditional discrete energy stability property of the current scheme, irrespective of the time step sizes.

This paper aims to develop a meshfree algorithm based on local radial basis functions (RBFs) combined with the differential quadrature (DQ) method to provide numerical approximations of the solutions of time-dependent, nonlinear and spatially one-dimensional reaction-diffusion systems and to capture their evolving patterns. The combination of local RBFs and the DQ method is applied to discretize the system in space; implicit multistep methods are subsequently used to discretize in time.

In a method of lines setting, a meshless method for their discretization in space is proposed. This discretization is based on a DQ approach, and RBFs are used as test functions. A local approach is followed where only selected RBFs feature in the computation of a particular DQ weight.

The proposed method is applied on four reaction-diffusion models: Huxley’s equation, a linear reaction-diffusion system, the Gray–Scott model and the two-dimensional Brusselator model. The method captured the various patterns of the models similar to available in literature. The method shows second order of convergence in space variables and works reliably and efficiently for the problems.

The originality lies in the following facts: A meshless method is proposed for reaction-diffusion models based on local RBFs; the proposed scheme is able to capture patterns of the models for big time T; the scheme has second order of convergence in both time and space variables and Nuemann boundary conditions are easy to implement in this scheme.

This paper discusses algorithms for large displacement analysis of interconnected flexible and rigid multibody systems. Hydrostatic and hydrodynamic loads for systems being submerged in water are also considered. The systems may consist of cables and beams and may combine very flexible parts with rigid parts. Various ways of introducing structural joints are discussed. A special implementation of the Hilber‐Hughes‐Taylor time integration scheme for constrained non‐linear systems is outlined. The formulation is general and allows for displacements and rotational motion of unlimited size. Aspects concerning efficient solution of constrained dynamic problems are discussed. These capabilities have been implemented in a general purpose non‐linear finite element program. Applications involving static and dynamic analysis of a bi‐articulated tower and a floating tripod platform kept in place by three anchor lines are discussed.