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An efficient and robust time discretization procedure of theta type is proposed in the frame of the finite element‐circuit equation coupling for electronic circuits with switches, i.e. the use of diodes, thyristors and transistors. This procedure enables the use of the Crank‐Nicolson scheme whatever the circuit and its working conditions by eliminating the undesirable oscillations of the solution peculiar to this scheme. It is based on the accurate determination of the switching instants and on a local modification of the theta parameter.

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.

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 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 develop a time implicit discontinuous Galerkin method for the simulation of two‐dimensional time‐domain electromagnetic wave propagation on non‐uniform triangular meshes.

The proposed method combines an arbitrary high‐order discontinuous Galerkin method for the discretization in space designed on triangular meshes, with a second‐order Cranck‐Nicolson scheme for time integration. At each time step, a multifrontal sparse LU method is used for solving the linear system resulting from the discretization of the TE Maxwell equations.

Despite the computational overhead of the solution of a linear system at each time step, the resulting implicit discontinuous Galerkin time‐domain method allows for a noticeable reduction of the computing time as compared to its explicit counterpart based on a leap‐frog time integration scheme.

The proposed method is useful if the underlying mesh is non‐uniform or locally refined such as when dealing with complex geometric features or with heterogeneous propagation media.

The paper is a first step towards the development of an efficient discontinuous Galerkin method for the simulation of three‐dimensional time‐domain electromagnetic wave propagation on non‐uniform tetrahedral meshes. It yields first insights of the capabilities of implicit time stepping through a detailed numerical assessment of accuracy properties and computational performances.

In the field of high‐frequency computational electromagnetism, the use of implicit time stepping has so far been limited to Cartesian meshes in conjunction with the finite difference time‐domain (FDTD) method (e.g. the alternating direction implicit FDTD method). The paper is the first attempt to combine implicit time stepping with a discontinuous Galerkin discretization method designed on simplex meshes.

The purpose of this paper is to study haemodynamic flow characteristics and entropy analysis in a bifurcated artery system subjected to stenosis, magnetohydrodynamic (MHD) flow and aneurysm conditions. The findings of this study offer significant insights into the intricate interplay encompassing electro-osmosis, MHD flow, microorganisms, Joule heating and the ternary hybrid nanofluid.

The governing equations are first non-dimensionalised, and subsequently, a coordinate transformation is used to regularise the irregular boundaries. The discretisation of the governing equations is accomplished by using the Crank–Nicolson scheme. Furthermore, the tri-diagonal matrix algorithm is applied to solve the resulting matrix arising from the discretisation.

The investigation reveals that the velocity profile experiences enhancement with an increase in the Debye–Hückel parameter, whereas the magnetic field parameter exhibits the opposite effect, reducing the velocity profile. A comparative study demonstrates the velocity distribution in Au-CuO hybrid nanofluid and Au-CuO-GO ternary hybrid nanofluid. The results indicate a notable enhancement in velocity for the ternary hybrid nanofluid compared to the hybrid nanofluids. Moreover, an increase in the Brinkmann number results in an augmentation in entropy generation.

This study investigates the flow characteristics and entropy analysis in a bifurcated artery system subjected to stenosis, MHD flow and aneurysm conditions. The governing equations are non-dimensionalised, and a coordinate transformation is applied to regularise the irregular boundaries. The Crank–Nicolson scheme is used to model blood flow in the presence of a ternary hybrid nanofluid (Au-CuO-GO/blood) within the arterial domain. The findings shed light on the complex interactions involving stenosis, MHD flow, aneurysms, Joule heating and the ternary hybrid nanofluid. The results indicate a decrease in the wall shear stress (WSS) profile with increasing stenosis size. The MHD effects are observed to influence the velocity distribution, as the velocity profile exhibits a declining nature with an increase in the Hartmann number. In addition, entropy generation increases with an enhancement in the Brinkmann number. This research contributes to understanding fluid dynamics and heat transfer mechanisms in bifurcated arteries, providing valuable insights for diagnosing and treating cardiovascular diseases.

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.

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 find the numerical solution of planar and non-planar Burgers’ equation and analysis of the shock behave.

First, the authors discritize the time-dependent term using Crank–Nicholson finite difference approximation and use quasilinearization to linearize the nonlinear term then apply Scale-2 Haar wavelets for space integration. After applying this scheme on partial differential, the equation transforms into a system of algebraic equation. Then, the system of equation is solved using Gauss elimination method.

Present method is the extension of the method (Jiwari, 2012). The numerical solutions using Scale-2 Haar wavelets prove that the proposed method is reliable for planar and non-planar nonlinear Burgers’ equation and yields results better than other methods and compatible with the exact solutions.

The numerical results for non-planar Burgers’ equation are very sparse. In the present paper, the authors identify where the shock wave and discontinuity occur in planar and non-planar Burgers’' equation.