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The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization of the transient heat conduction equation subject to several boundary conditions, and for cases with volumetric heat generation, using both fully implicit and Crank-Nicolson time discretizations. The goal is to determine which combination of numerical boundary condition implementation and time discretization produces the most accurate solutions with the least computational effort.

The paper studies several benchmark cases including constant temperature, convective heating, constant heat flux, time-varying heat flux, and volumetric heating, and compares the convergence rates and local to analytical or semi-analytical solutions.

The Crank-Nicolson method coupled with second-order expression for the boundary derivatives produces the most accurate solutions on the coarsest meshes with the least computation times. The Crank-Nicolson method allows up to 16X larger time step for similar accuracy, with nearly negligible additional computational effort compared with the implicit method.

The findings can be used by researchers writing similar codes for quantitative guidance concerning the effect of various numerical boundary condition approximations for a large class of boundary condition types for two common time discretization methods.

The paper provides a comprehensive study of accuracy and convergence of the finite volume discretization for a wide range of benchmark cases and common time discretization methods.

The purpose of this paper is to obtain the nonlinear Schrodinger equation (NLSE) numerical solutions in the presence of the first-order chromatic dispersion using a second-order, unconditionally stable, implicit finite difference method. In addition, stability and accuracy are proved for the resulting scheme.

The conserved quantities such as mass, momentum and energy are calculated for the system governed by the NLSE. Moreover, the robustness of the scheme is confirmed by conducting various numerical tests using the Crank-Nicolson method on different cases of solitons to discuss the effects of the factor considered on solitons properties and on conserved quantities.

The Crank-Nicolson scheme has been derived to solve the NLSE for optical fibers in the presence of the wave packet drift effects. It has been founded that the numerical scheme is second-order in time and space and unconditionally stable by using von-Neumann stability analysis. The effect of the parameters considered in the study is displayed in the case of one, two and three solitons. It was noted that the reliance of NLSE numeric solutions properties on coefficients of wave packets drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws. Accordingly, by comparing our numerical results in this study with the previous work, it was recognized that the obtained results are the generalized formularization of these work. Also, it was distinguished that our new data are regarding to the new communications modes that depend on the dispersion, wave packets drift and nonlinearity coefficients.

The present study uses the first-order chromatic. Also, it highlights the relationship between the parameters of dispersion, nonlinearity and optical wave properties. The study further reports the effect of wave packet drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws.

The purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational framework, applied for the fluid dynamics with illustrations to incompressible Navier–Stokes equations.

The proposed error estimator is universal and versatile that it works for the entire subsets of the GS4-1 framework, encompassing the nondissipative Crank–Nicolson method, the most dissipative backward differential formula and anything in between. It is new and novel that the cumbersome design work of error estimation for specific time integration algorithms can be avoided. Regarding the numerical implementation, the local error estimation has a compact representation that it is determined by the time derivative variables at four successive time levels and only involves vector operations, which is simple for numerical implementation. Additionally, the adaptive time-stepping is further illustrated by the proposed error estimator and is used to solve the benchmark problems of lid-driven cavity and flow past a cylinder.

The proposed computational procedure is capable of eliminating the nonphysical oscillations in GS4-1(1,1)/Crank–Nicolson method; being CPU-efficient in both dissipative and nondissipative schemes with better solution accuracy; and detecting the complex physics and hence selecting a suitable time step according to the user-defined error threshold.

To the best of the authors’ knowledge, for the first time, this study applies the general purpose GS4-1 family of time integration algorithms for transient simulations of incompressible Navier–Stokes equations in fluid dynamics with constant and adaptive time steps via a novel and universal error estimator. The proposed computational framework is simple for numerical implementation and the time step selection based on the proposed error estimation is efficient, benefiting to the computational expense for transient simulations.

The purpose of this paper is to obtain numerical solutions of a two‐dimensional mixed space‐time PDE modelling the flow of a second‐grade.

The paper derives conditionally stable Crank‐Nicolson schemes to solve both the one and two dimensional mixed‐space time PDE. For the two‐dimensional case we implement the Crank‐Nicolson scheme using a Peaceman‐Rachford ADI scheme.

For zero‐shear boundaries the Cattanneo representation of the model equation blows up whilst the representation derived by Rajagopal is stable and produces solutions which decay over time.

The use of a Peaceman‐Rachford ADI scheme to solve a mixed space‐time PDE is both novel and new.

To provide an analysis of transient heat conduction, which is solved using different iterative solvers for graduate and postgraduate students (researchers) which can help them develop their own research.

Three‐dimensional transient heat conduction in homogeneous materials using different time‐stepping methods such as finite difference (Θ explicit, implicit and Crank‐Nicolson) and finite element (weighted residual and least squared) methods. Iterative solvers used in the paper are conjugate gradient (CG), preconditioned gradient, least square CG, conjugate gradient squared (CGS), preconditioned CGS, bi‐conjugate gradient (BCG), preconditioned BCG, bi‐conjugate gradient stabilized (BCGSTAB), reconditioned BCGSTAB and Gaussian elimination with incomplete Cholesky factorization.

Provides information on which time‐stepping method is the most accurate, which solver is the fastest to solve a symmetric and positive system of linear matrix equations of all those considered.

Fortran 90 code given as an abstract can be very useful for graduate and postgraduate students to develop their own code.

This paper offers practical help to an individual starting his/her research in the finite element technique and numerical methods.

The purpose of this paper is to present a new method, namely, “Re-modified quintic B-spline collocation method” to solve the Kuramoto–Sivashinsky (KS) type equations. In this method, re-modified quintic B-spline functions and the Crank–Nicolson formulation is used for space and time integration, respectively. Five examples are considered to test out the efficiency and accuracy of the method. The main objective is to develop a method which gives more accurate results and reduces the computational cost so that the authors require less memory storage.

A new collocation technique is developed to solve the KS type equations. In this technique, quintic B-spline basis functions are re-modified and used to integrate the space derivatives while time derivative is discretized by using Crank–Nicolson formulation. The discretization yields systems of linear equations, which are solved by using Gauss elimination method with partial pivoting.

Five examples are considered to test out the efficiency and accuracy of the method. Finally, the present study summarizes the following outcomes: first, the computational cost of the proposed method is the less than quintic B-spline collocation method. Second, the present method produces better results than those obtained by Lattice Boltzmann method (Lai and Ma, 2009), quintic B-spline collocation method (Mittal and Arora, 2010), quintic B-spline differential quadrature method (DQM) (Mittal and Dahiya, 2017), extended modified cubic B-spline DQM (Tamsir et al., 2016) and modified cubic B-splines collocation method (Mittal and Jain, 2012).

The method presented in this paper is new to best of the authors’ knowledge. This work is the original work of authors and the manuscript is not submitted anywhere else for publication.

The purpose of this paper is to show how a system of differential equations of one‐dimensional transient cooling heat conduction of different multi‐layer slabs has been solved numerically. A simple deterministic filtering matrix has been developed to remove errors involved in the experimental temperature measurements.

The system of differential equations is solved through Crank‐Nicolson method using a developed computer code. The developed matrix is based on the available information about the system and is strong enough to detect and remove errors from the measured temperatures.

The filtering algorithm is very straightforward and easy to implement and needs to be developed once for a given system.

This paper shows how the governing equations of transient heat conduction of multi‐layer slabs have been solved through Crank‐Nicolson method using a developed computer code. The code is user friendly and solves a large system of simultaneous differential equations for any given composite slab configurations. Furthermore, a matrix filter has been suggested to remove experimental errors. The filter is based on the available information about the considered system. The developed matrix filter is shown to be a powerful technique to detect the noisy data and correct them while intelligent enough not to harm good data. The filtering algorithm is very straightforward and easy to implement and needs to be developed once for a given system.

Implicit one‐step methods for the system of differential equations arising from a space discretisation of the semiconductor equations are considered. It is shown that mere spectral conditions like A‐stability or L‐stability do not give a reliable answer to the behaviour of the numerical solution. Rather, positivity arguments for the corresponding rational matrix functions play an important role.

This paper aims to reconstruct the spatially varying orthotropic conductivity based on a two-dimensional inverse heat conduction problem described by a partial differential equation (PDE) model with mixed boundary conditions. The proposed discretization uses a highly accurate technique and allows simple implementations. Also, the authors solve the related inverse problem in such a way that smoothness is enforced on the iterations, showing promising results in synthetic examples and real problems with moving heat source.

The discretization procedure applied to the model for the direct problem uses a pseudospectral collocation strategy in the spatial variables and Crank–Nicolson method for the time-dependent variable. Then, the related inverse problem of recovering the conductivity from temperature measurements is solved by a modified version of Levenberg–Marquardt method (LMM) which uses singular scaling matrices. Problems where data availability is limited are also considered, motivated by a face milling operation problem. Numerical examples are presented to indicate the accuracy and efficiency of the proposed method.

The paper presents a discretization for the PDEs model aiming on simple implementations and numerical performance. The modified version of LMM introduced using singular scaling matrices shows the capabilities on recovering quantities with precision at a low number of iterations. Numerical results showed good fit between exact and approximate solutions for synthetic noisy data and quite acceptable inverse solutions when experimental data are inverted.

The paper is significant because of the pseudospectral approach, known for its high precision and easy implementation, and usage of singular regularization matrices on LMM iterations, unlike classic implementations of the method, impacting positively on the reconstruction process.

This paper aims to propose a mathematical model and numerical modeling to study the behavior of low conductive incompressible multicomponent hydrocarbon mixture in a channel under the influence of electron irradiation. In addition, it also aims to present additional mechanisms to study the radiation transfer and the separation of the mixture’s components.

The three-dimensional non-stationary Navier–Stokes equation is the basis for this model. The Adams–Bashforth scheme is used to solve the convective terms of the equation of motion using a fourth-order accuracy five-point elimination method, and the viscous terms are computed with the Crank–Nicolson method. The Poisson equation is solved with the matrix sweep method and the concentration and electron irradiation equations are solved with the Crank–Nicolson method too.

It shows high computational efficiency and good estimation quality. On the basis of numerical results of mathematical model, the effect of the separation of mixture to fractions with various physical characteristics was obtained. The obtained results contribute to the improvement of technologies for obtaining high-quality oil products through oil separation into light and heavy fractions. Mathematical model is approbated based on test problem, and has good agreement with the experimental data.

The constructed mathematical model makes developing a methodology for conducting experimental studies of this phenomenon possible.