Second-order Finite Difference Method Research Articles

Finite Element Method for Non-Newtonian Radiative Maxwell Nanofluid Flow under the Influence of Heat and Mass Transfer

The recent study was concerned with employing the finite element method for heat and mass transfer of MHD Maxwell nanofluid flow over the stretching sheet under the effects of radiations and chemical reactions. Moreover, the effects of viscous dissipation and porous plate were considered. The mathematical model of the flow was described in the form of a set of partial differential equations (PDEs). Further, these PDEs were transformed into a set of nonlinear ordinary differential equations (ODEs) using similarity transformations. Rather than analytical integrations, numerical integration was used to compute integrals obtained by applying the finite element method. The mesh-free analysis and comparison of the finite element method with the finite difference method are also provided to justify the calculated results. The effect of different parameters on velocity, temperature and concentration profile is shown in graphs, and numerical values for physical quantities of interest are also given in a tabular form. In addition, simulations were carried out by employing software that applies the finite element method for solving PDEs. The calculated results are also portrayed in graphs with varying sheet velocities. The results show that the second-order finite difference method is more accurate than the finite element method with linear interpolation polynomial. However, the finite element method requires less number of iterations than the finite difference method in a considered particular case. We had high hopes that this work would act as a roadmap for future researchers entrusted with resolving outstanding challenges in the realm of enclosures utilized in industry and engineering.

A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems

In this work, a novel second-order nonstandard finite difference (NSFD) method that preserves simultaneously the positivity and local asymptotic stability of one-dimensional autonomous dynamical systems is introduced and analyzed. This method is based on novel non-local approximations for right-hand side functions of differential equations in combination with nonstandard denominator functions. The obtained results not only resolve the contradiction between the dynamic consistency and high-order accuracy of NSFD methods but also improve and extend some well-known results that have been published recently in Kojouharov et al. (2021); Gupta et al. (2020); Wood and Kojouharov (2015). Furthermore, as a simple but important application, we apply the constructed NSFD method for solving the logistic, sine, cubic, and Monod equations. Consequently, the NSFD schemes constructed in the earlier work (Mickens, 1999) are improved significantly. Finally, we report some numerical experiments to support and illustrate the theoretical assertions as well as advantages of the constructed NSFD method.

Flow and transport in three-dimensional discrete fracture matrix models using mimetic finite difference on a conforming multi-dimensional mesh

We present a comprehensive workflow to simulate single-phase flow and transport in fractured porous media using the discrete fracture matrix approach. The workflow has three primary parts: (1) a method for conforming mesh generation of and around a three-dimensional fracture network, (2) the discretization of the governing equations using a second-order mimetic finite difference method, and (3) implementation of numerical methods for high-performance computing environments. A method to create a conforming Delaunay tetrahedralization of the volume surrounding the fracture network, where the triangular cells of the fracture mesh are faces in the volume mesh, that addresses pathological cases which commonly arise and degrade mesh quality is also provided. Our open-source subsurface simulator uses a hierarchy of process kernels (one kernel per physical process) that allows for both strong and weak coupling of the fracture and matrix domains. We provide verification tests based on analytic solutions for flow and transport, as well as numerical convergence. We also provide multiple expositions of the method in complex fracture networks. In the first example, we demonstrate that the method is robust by considering two scenarios where the fracture network acts as a barrier to flow, as the primary pathway, or offers the same resistance as the surrounding matrix. In the second test, flow and transport through a three-dimensional stochastically generated network containing 257 fractures is presented.

Micro-polar fluid flow over a unique form of vertical stretching sheet: Special emphasis to temperature-dependent properties

Convective micro-polar fluid movement through an impermeable nonlinear stretching sheet with changing thickness is investigated in this research. The governing equations for fluid flow are coupled with nonlinear partial differential equations transformed into ordinary differential equations using the similarity transformation and a numerical second-order finite difference method called the Keller-box scheme. The problem's findings show that microparticle suspensions considerably influence the flow field when different fluid properties are used. We investigate the impact of physical variables on the velocity, microrotation, temperature, skin friction coefficient, and heat transfer rate. The result shows that increasing the micropolar parameter widens the velocity boundary layer while decreasing the thermal boundary layer.

Second-order robust finite difference method for singularly perturbed Burgers' equation

In this paper, a second-order robust method for solving singularly perturbed Burgers' equation were presented. To find the numerical approximation, we apply the quasilinearization technique before formulation of the scheme. The obtained experimental results show that the presented method has better numerical accuracy and convergence as compared to some existing methods in the literature. Thus, the present method provides an accurate solution and efficient to solve the singularly perturbed Burgers' equation.

A numerical method for the solution of the three-dimensional acoustic wave equation in a marine environment considering complex sources

The need to understand how complex acoustic sources propagate noise in a realistic environment is of growing interest. In this work, we propose a numerical model for the simulation of noise generated by sources with directivity and propagating in ocean waveguides. The numerical model solves the wave equation for the acoustic pressure in the physical space using a second-order accurate finite difference method (FDTD). The source is implemented using an improved form of the hard-source method, which implicitly takes into consideration the reflection of acoustic waves associated to the presence of the ocean free surface, through the use of the method of images. This novel method is shown to improve the results with respect to the standard hard source implementation. We first validate the numerical method considering both analytical solutions and benchmark cases for the case of a monopole and then explore the acoustic energy patterns developed in the case of dipole and quadrupole sources. Specifically, the algorithm and the implementation of complex sources are evaluated first in a semi-infinite fluid layer and then considering two classical waveguides: the Ideal one and the Pekeris one. The comparison with analytical results shows the numerical method’s accuracy and that incorporating free surface effects in the hard source implementation improves results. In addition, the study shows that the acoustic response in the near field, of the order of few kilometers from the source, is strongly influenced by the source’s directivity and orientation relative to the free surface. The results of this paper have implications for future research aimed at characterizing and quantifying ship propeller noise in realistic waveguides. Indeed, this preliminary work is necessary to proceed to more complex numerical experiments, such as considering a real propeller signal as a source or considering stratification of the medium or propagation in confined domains such as experimental tanks.

Finite Volume Method for a Time-Dependent Convection-Diffusion-Reaction Equation with Small Parameters

Convection, diffusion, and reaction mechanisms are characteristics of transient mass-transfer phenomena that occur in natural and industrial systems. In this article, we contemplate a passive scalar transport governed by the convection-diffusion-reaction (CDR) equation in 2D flow. The efficiency of solving computationally partial differential equations can be illustrated by using a precise numerical method that yields remarkable precision at a low cost. The accuracy and computational efficiency of two second-order finite difference methods were investigated. The results were compared to a finite volume technique, which has a memory advantage and conserves mass, momentum, and energy even on coarse grids. For various diffusion coefficient values, numerical simulation of unsteady CDR equation are also performed. The techniques were examined for consistency and convergence. The effectiveness and accuracy of these approaches for solving CDR equations are demonstrated by simulation results. Efficiency is measured using L 2 and L ∞ , and the estimated results are compared to the corresponding analytical solution.

A time two-grid algorithm for the two dimensional nonlinear fractional PIDE with a weakly singular kernel

The main aim of this paper is to solve the two-dimensional nonlinear fractional partial integro-differential equation (PIDE) with a weakly singular kernel by using the time two-grid finite difference (FD) algorithm. The second-order backward difference formula (BDF) and L1 scheme are used in time. The time two-grid algorithm is constructed to improve the solving efficiency of nonlinear systems. The Newton iteration is used to solve nonlinear discrete system on the coarse grid, and then we apply Lagrangian linear interpolation to attain the function value used in constructing the difference scheme on the fine grid. The second-order finite difference method (FDM) is used in space. The unconditional stability and convergence are attained for the two-grid fully discrete system. Numerical experiments show that the used CPU time for the presented two-grid numerical algorithm is lower than the general finite difference method for solving the nonlinear system.

Entropy generation for MHD two phase blood flow through a curved permeable artery having variable viscosity with heat and mass transfer

Present study deals with the analysis of entropy generation of MHD blood flow through a stenosed permeable curved artery with heat source and chemical reaction. Blood flow is considered in two-phases; core and plasma region, respectively. Viscosity of the core region is considered as temperature-dependent, while constant viscosity is considered in plasma region. The governing equations of the proposed two-phase blood flow model are considered in the toroidal coordinate system. The second-order finite difference method is adopted to solve governing equations with 10−6 tolerance in the iteration process. A comparative study of darcy number (Da) is performed to understand the influence of permeable and impermeable wall conditions. The effect of various physical parameters such as magnetic field (M), viscosity variation parameter (λ1), Darcy number (Da), Brinkman number (Br), heat source (H), chemical reaction parameter (ξ) etc. are displayed graphically on the flow velocity, temperature, concentration, wall shear stress (WSS), frictional resistance profile and entropy generation profiles. A comparison with published work has also been displayed through the graph to validate the present model, and it is in fair agreement with the existing work. The present study suggested that the curvature and permeability of the arterial wall raise the risk of atherosclerosis formation, while the implication of heat source on the blood flow lower this risk. Clinical researchers and biologists can use the current mathematical research to lower the risk of lipid deposition and predict the risk of cardiovascular disease, and then diagnose patients based on risk.

Hemodynamical analysis of MHD two phase blood flow through a curved permeable artery having variable viscosity with heat and mass transfer.

A numerical investigation of MHD blood flow through a stenosed permeable curved artery has been done in this study. Blood flow is considered in two-phases; core and plasma region, respectively. Viscosity of the core region is considered as temperature-dependent, while constant viscosity is considered in plasma region. The governing equations of the proposed two-phase blood flow model are considered in the toroidal coordinate system. The second-order finite difference method is adopted to solve governing equations with [Formula: see text] tolerance in the iteration process. A comparative study of Darcy number (Da) is performed to understand the influence of permeable and impermeable wall conditions. The effect of various physical parameters such as magnetic field (M), viscosity variation parameter ([Formula: see text]), Darcy number (Da), heat source (H) and chemical reaction parameter ([Formula: see text]) is displayed graphically on the flow velocity, temperature, concentration, wall shear stress and frictional resistance profiles. A comparison with published work has also been displayed through the graph to validate the present model, and it is in fair agreement with the existing work. The present study suggested that the curvature and permeability of the arterial wall raise the risk of atherosclerosis formation, while the implication of heat source on the blood flow lower this risk.

Numerical Solutions of Two-Dimensional Navier–Stokes Equations Based on a Generalized Harmonic Polynomial Cell Method With Non-Uniform Grid

Abstract It is essential for a Navier–Stokes equations solver based on a projection method to be able to solve the resulting Poisson equation accurately and efficiently. In this paper, we present numerical solutions of the 2D Navier–Stokes equations using the fourth-order generalized harmonic polynomial cell (GHPC) method as the Poisson equation solver. Particular focus is on the local and global accuracy of the GHPC method on non-uniform grids. Our study reveals that the GHPC method enables the use of more stretched grids than the original HPC method. Compared with a second-order central finite difference method (FDM), global accuracy analysis also demonstrates the advantage of applying the GHPC method on stretched non-uniform grids. An immersed-boundary method is used to deal with general geometries involving the fluid–structure interaction problems. The Taylor–Green vortex and flow around a smooth circular cylinder and square are studied for the purpose of verification and validation. Good agreement with reference results in the literature confirms the accuracy and efficiency of the new 2D Navier–Stokes equation solver based on the present immersed-boundary GHPC method utilizing non-uniform grids. The present Navier–Stokes equations solver uses second-order central FDM and Quadratic Upstream Interpolation for Convective Kinematics scheme for the discretization of the diffusion term and advection term, respectively, which may be replaced by other higher-order schemes to further improve the accuracy.

Evaluating multifunctional efficiency of a structural battery composite via thermo-electro-chemical modeling

The emerging research field of structural batteries aims to combine the functions of load bearing and energy storage to improve system-level energy storage in battery-powered vehicles and consumer products. Structural batteries, when implemented in electric vehicles, will be exposed to greater temperature fluctuations than conventional batteries in electric vehicles. However, there is a lack of knowledge in public domains and scientific literature regarding how these thermal boundary conditions impact power capabilities of the structural batteries. To fill this gap, the present work simulates the transient temperature-dependent specific power capabilities of a high aspect ratio structural battery composite by solving the one-dimensional heat transfer equation with heat source terms and convective boundary conditions. Equivalent circuit modeling of resistivity-induced losses is used with a second-order finite difference method to examine battery performance. More than 60 different run configurations are evaluated in this work, examining how thermal boundary conditions and internal heat generation influence power capabilities and multifunctional efficiency of the structural battery. The simulated structural battery composite is shown to have good specific Young’s modulus (79.5%–80.3% of aluminum), a specific energy of 158 Wh kg−1, and specific power of 41.2–55.2 W kg−1, providing a multifunctional efficiency of 1.15–1.17 depending on configuration and thermal loading conditions and demonstrating the potential of load-bearing structural batteries to achieve mass savings. This work emphasizes the dependency of power efficiency on cell design and external environmental conditions. Insulating material is shown to improve multifunctional efficiency, particularly for low ambient temperatures. It is demonstrated that as cell temperature increases due to high ambient temperature or heat generation in the battery, the specific power efficiency increases exponentially due to a favorable nonlinear relation between ionic conductivity and cell temperature. The simulations also demonstrate a thermal feedback loop where resistivity-induced power losses can lead to self-regulation of cell temperature. This effect reduces run-averaged losses, particularly at low ambient temperatures.

Two-phase experimental and numerical studies on scouring at the toe of vertical seawall

This paper presents experimental and numerical modeling of scouring erosion at the toe due to wave impact on a highly reflective vertical seawall. For the experimental study, new experiments and analyses are provided in the light of advanced measurement techniques. They show the complex nature of the triple wave–structure–sediment interactions with their non-linearities, asymmetries, and hysteresis. For the numerical study, a two-dimensional x/z two-phase Eulerian model is introduced using a new fluid–solid particle interaction. The numerical model is also based on a projection technique, σ-transformation, and a second-order hybrid finite-difference and finite-volume method. A good agreement is obtained between the numerical and experimental results. Both models show that the wave run-up or down on vertical wall conducts to the formation of ascending or impinging vertical jets, which leads to the seawall scour.

Numerical Solution of Diffusion Equation using a Method of Lines and Generalized Finite Differences

One of the greatest challenges in the area of applied mathematics continues to be the design of numerical methods capable of approximating the solution of partial differential equations quickly and accurately. One of the most important equations, due to the hydraulic and transport applications it has, and the large number of difficulties that it usually presents when solving it numerically is the Diffusion Equation. In the present work, a Method of Lines applied to the numerical solution of the said equation in irregular regions is presented using a scheme of Generalized Finite Differences. The second-order finite difference method uses a central node and 8 neighbor points in order to address the spatial approximation. A series of tests and numerical results are presented, which show the accuracy of the proposed method.

Cross-Gradient Joint Inversion of Gravity and Seismic Data With Triangular Grid Division by the Second-Order Finite-Difference Method

Cross-Gradient Joint Inversion of Gravity and Seismic Data With Triangular Grid Division by the Second-Order Finite-Difference Method

A Novel Second-Order Adaptive Grid Method for Singularly Perturbed Convection–Diffusion Equations

In this paper, a singularly perturbed convection–diffusion equation is studied. At first, the original problem is transformed into a parameterized singularly perturbed Volterra integro-differential equation by using an integral transform. Then, a second-order finite difference method on an arbitrary mesh is given. The stability and local truncation error estimates of the discrete schemes are analyzed. Based on the mesh equidistribution principle and local truncation error estimation, an adaptive grid algorithm is given. In addition, in order to calculate the parameters of the transformation equation, a nonlinear unconstrained optimization problem is constructed. Numerical experiments are given to illustrate the effectiveness of our presented adaptive grid algorithm.

Mathematical analysis of two-phase blood flow through a stenosed curved artery with hematocrit and temperature dependent viscosity

A two-phase blood flow model is considered to analyze the fluid flow and heat transfer in a curved tube with time-variant stenosis. In both core and plasma regions, the variable viscosity model (Hematocrit and non linear temperature-dependent, respectively) is considered. A toroidal coordinate system is considered to describe the governing equations. The perturbation technique in terms of perturbation parameter ε is used to obtain the temperature profile of blood flow. In order to find the velocity, wall shear stress and impedance profiles, a second-order finite difference method is employed with the accuracy of 10−6 in the each iteration. Under the conditions of fully-developed flow and mild stenosis, the significance of various physical parameters on the blood velocity, temperature, wall shear stress (WSS) and impedance are investigated with the help of graphs. A validation of our results has been presented and comparison has been made with the previously published work and present study, and it revels the good agreement with published work. The present mathematical study suggested that arterial curvature increase the fear of deposition of plaque (atherosclerosis), while, the use of thermal radiation in heat therapies lowers this risk. The positive add in the value of λ 1 causes to increase in plasma viscosity; as a result, blood flow velocity in the stenosed artery decreases due to the assumption of temperature-dependent viscosity of the plasma region. Clinical researchers and biologists can adopt the present mathematical study to lower the risk of lipid deposition, predict cardiovascular disease risk and current state of disease by understanding the symptomatic spectrum, and then diagnose patients based on the risk.

A third-order two-step numerical scheme for heat and mass transfer of chemically reactive radiative MHD power-law fluid

A two-stage third-order numerical scheme is proposed for solving ordinary differential equations. The scheme is explicit and implicit type in two stages. First, the stability region of the scheme is found when it is applied to the linear equation. Further, the stability conditions of the scheme are found using a linearized homogenous set of differential equations. This set of equations is obtained by applying transformations on the governing equations of heat and mass transfer of incompressible, laminar, steady, two-dimensional, and non-Newtonian power-law fluid flows over a stretching sheet with effects of thermal radiations and chemical reaction. The proposed scheme with an iterative method is employed in two different forms called linearized and non-linearized. But it is found that the non-linearized approach performs better than the linearized one when residuals are compared through plots. Additionally, the proposed scheme is compared to the second-order central finite difference method for second-order non-linear differential equations and the Keller-Box/trapezoidal method for a linear differential equation. It is determined that the proposed scheme is more effective and computationally less expensive than the standard/classical finite difference methods. Moreover, the impact of magnetic parameter, radiation parameter, modified Prandtl and Schmidt numbers for power-law fluid, and chemical reaction rate parameter on velocity, temperature, and concentration profiles are displayed through graphs and discussed. The power-law fluid’s heat and mass transfer simulations are also carried out with varying flow behavior index, sheet velocity, and mass diffusivity. We hoped that this effort would serve as a guide for investigators tasked with resolving unresolved issues in the field of enclosures used in industry and engineering.

Stability and error analysis for a second-order approximation of 1D nonlocal Schrödinger equation under DtN-type boundary conditions

A second-order Crank-Nicolson finite difference method is designed to solve a 1D nonlocal Schrödinger equation on the whole real axis. We employ an asymptotically compatible scheme to discretize the spatially nonlocal operator, and apply the Crank-Nicolson scheme in time to achieve a fully discrete infinite system. An iterative technique for the second-order matrix difference equation is then developed to obtain Dirichlet-to-Dirichlet (DtD)-type artificial boundary conditions (ABCs) with the application of z z -transform for the resulting fully discrete system. After that, with the aid of discrete nonlocal Green’s first identity, we derive Dirichlet-to-Neumann (DtN)-type ABCs from DtD-type ABCs. The resulting DtN-type ABCs are available to reduce the infinite discrete system to a finite discrete system on a truncated computational domain, and make it possible to perform stability and convergence analysis for the reduced problem. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Estimation of the Frequency Response Function of the Rotational Degree of Freedom

One of the factors that influence the dynamic characteristics of machining systems is the cutting tool. Cutting tools are very diverse, and receptance coupling substructure analysis (RCSA) is essential for analyzing the dynamic characteristics of each tool. For RCSA, a full receptance matrix of the equipment and tools is essential. In this study, rotational degree-of-freedom receptance was estimated and analyzed using translational receptance. Displacement/moment receptance was analyzed according to the distance of the response point using the first-and second-order finite difference methods. The rotation/moment receptance was estimated according to the distance of the response point. Rotation/moment receptance was analyzed using Schmitz’s method and compensation strategies. The limitations of these strategies were analyzed, and the rotation/moment receptance for the beam under free-free boundary conditions was predicted using the second compensation strategy.