Implicit Finite Difference Method Research Articles

Analytical – numerical estimation of temperature distribution in living tissue by the DPL bio-heat model

Here, we used mathematical modeling to predict and simulate the behavior and effect of temperature on human skin, together with studying the effects of the blood perfusion parameters, the time of tissue exposure to the laser, and the effect of the delay times on temperature distribution on living tissue. The model that has been used is a dual-phase lag DPL bioheat transfer model. The solution of the model has been obtained by using the analytical scheme represented by Laplace transforms, as well as the numerical scheme represented by the implicit finite difference method. The paper involves comparing analytical and numerical solutions, a comparison between the temperature distribution according to various bio-heat models and the resulting thermal damage. The results of the DPL bioheat model have been compared with the previous study and it showed. A favorable convergence results have been presented graphically and discussed in detail.

Efficient ADI schemes and preconditioning for a class of high-dimensional spatial fractional diffusion equations with variable diffusion coefficients

In this paper, alternating direction implicit (ADI) finite difference method and preconditioned Krylov subspace method are combined to solve a class of high-dimensional spatial fractional diffusion equations with variable diffusion coefficients. We prove the unconditional stability and convergence rate of the ADI finite difference method provided that the diffusion coefficients satisfy the given conditions. For the linear system in each spatial direction, we establish a circulant approximate inverse preconditioner to accelerate the Krylov subspace method. In addition, we also use matrix-free algorithms and fast Fourier transforms (FFT) to speed up the solution of linear systems. Numerical experiments show the utility of the ADI method and the preconditioner.

Numerical solution of the three-dimensional Burger’s equation by using the DQ-FD combined method in the determination of the 3D velocity of the flow

In this paper, the differential quadrature and the finite difference combined method (DQ-FDM) was applied to solve the three-dimensional Burger’s equation in the determination of the 3D velocity of the flow; so that spatial terms were discretized by the differential quadrature method, and the temporal term was discretized by the finite difference method, and the resulting nonlinear equations were solved using the Newton–Raphson method. All variables were considered as dimensionless in this equation. The solution results were compared with solution results of the two-dimensional equation in the two other numerical methods available in the literature which provided an acceptable accuracy. Also, the results of the mentioned numerical method were compared with those of the fully implicit finite difference method that was solved for larger than or equal viscosities of 0.1. The results showed that by increasing time and viscosity, the longitudinal, depth and transverse velocities were decreased. The occurrence of the upward flow was observed especially in the upsilon = 0.05 in the close of the bed, end of the length and width that in the presence of very fine particles of the clay and silt shows suspension of these particles in some spaces. The position of the longitudinal, depth and transverse velocities in the plan for the passing plates through the section depth for different viscosities and times showed that by increasing viscosity and time, the position of the maximum velocities became closer to the middle of the section width. Also, stream lines were plotted in all of sections and then analyzed.

The effect of single and hybrid nanofluids in the performance of Solar Water Heating System

In the present study, the feasibility of using single particle Cu and hybrid Cu/Al2O3 nanofluids as heat transfer fluids in a coupled solar parabolic trough collector-heat water storage tank for domestic absorption cooling systems. A computer program based on one dimensional implicit finite difference method and energy balance approach has been developed to investigate the behavior of the studied system under the real climate conditions of a typical summer day in Adrar city, Algeria. The simulation findings reveal that solar parabolic trough collector with small area of 6m² and storage tank of 0.3m3 can ensure higher storage tank temperature able to drive an absorption cooling machine. Solar system with hybrid nanofluid shows superior performance compared to single nanofluid and pure water. Furthermore, the effect of nanoparticle's volume fraction is evaluated. The heat storage tank temperature can attain starting operating chiller temperature more rapidly with small volume fraction equal to 0.2% in the case of Cu-Al2O3/Water hybrid nanofluid.

Effect of Ohmic heating on magnetohydrodynamic flow with variable pressure gradient: a computational approach

ABSTRACT Ohmic heating, which significantly affects magnetohydrodynamic flow, is one of the exciting effects to be experienced. This study looks at the significance of Ohmic heating on two-dimensional flow and dissipative heat transfer of a viscous fluid over a flat surface. The mainstream flow is subjected to a pressure gradient and externally oriented inclined magnetic field. An implicit finite difference method has been used to obtain the convergent solutions and the graphical results have been obtained using MATLAB software. A positive correlation is found between temperature and viscous dissipation parameter. Further, the flow encountered higher resistance due to the magnetic field, which is more prominent when the magnetic field is normal to the flow direction .

Computer Simulations of Silicide-Tetrahedrite Thermoelectric Generators

With global warming and rising energy demands, it is important now than ever to transit to renewable energy systems. Thermoelectric (TE) devices can present a feasible alternative to generate clean energy from waste heat. However, to become attractive for large-scale applications, such devices must be cheap, efficient, and based on ecofriendly materials. In this study, the potential of novel silicide-tetrahedrite modules for energy generation was examined. Computer simulations based on the finite element method (FEM) and implicit finite difference method (IFDM) were performed. The developed computational models were validated against data measured on a customized system working with commercial TE devices. The models were capable of predicting the TEGs' behavior with low deviations (≤10%). IFDM was used to study the power produced by the silicide-tetrahedrite TEGs for different ΔT between the sinks, whereas FEM was used to study the temperature distributions across the testing system in detail. To complement these results, the influence of the electrical and thermal contact resistances was evaluated. High thermal resistances were found to affect the devices ΔT up to ~15%, whereas high electrical contact resistances reduced the power output of the silicide-tetrahedrite TEGs by more than ~85%.

Dissipated-radiative compressible flow of nanofluids over unsmoothed inclined surfaces with variable properties

This study aims to examine the impacts of the variable nanofluid properties on the compressible flow over an inclined irregular surface. The density, dynamic viscosity and thermal conductivity of the nano liquid are assumed to be dependent on the temperature, and the viscous dissipation in this case is introduced. Sinusoidal formulations for the unsmoothed surface together with the governing system in case of are presented. The temperature on the edge is, also, variable and depending on the discerption function of the surface. The nonlinear radiation is taken into account, and the two-phase model for the nanofluid is applied. A fully implicit finite difference method (FDM) is used to solve the governing system. The major outcomes revealed that the skin friction coefficient and Nusselt number are rising as the compressibility parameter is altered. Also, the maximization of the thermal conductivity-variation is better for the velocity and temperature behaviors.

Eavesdropping and Anti-Eavesdropping Game in UAV Wiretap System: A Differential Game Approach

Despite its advantages of flexility and low-cost networking, unmanned aerial vehicle (UAV) communications face various attacks such as eavesdropping. Existing studies on secure UAV communications assume fixed-location eavesdroppers and rarely consider interactions between legitimate nodes and eavesdroppers. In this paper, we investigate eavesdropping and anti-eavesdropping interaction between a UAV-enabled eavesdropper (UAV-E) and a UAV-enabled base station (UAV-BS) in a downlink wiretap system. The UAV-E aims to wiretap downlink signals by adaptively adjusting its trajectory while the UAV-BS aims to maximize secrecy-sum-rate with minimum power consumption by jointly optimizing user scheduling, power control, and trajectory. Dynamic differential equations are formulated to characterize motions of UAVs, following which a zero-sum differential game is formulated to model the “pursuit-evasion” interaction between the UAV-BS and the UAV-E. Definition and existence of Nash equilibrium (NE) are provided. To obtain the NE, Pontryagins minimum principle is leveraged to solve the trajectory design problem. Further, Gauss-Seidel-like implicit finite-difference method is leveraged to obtain saddle-point strategies at NE. Finally, numerical results are provided to verify the effectiveness of the proposed game model. It is revealed that the differential game can well-characterize the strategy interactions between UAVs. Moreover, results show that the initial positions and weights of UAVs, the energy consumption factor, and the user scheduling have key impacts on motion interactions between the UAV-BS and the UAV-E and further on UAV-BS’s power control.

Numerical assessment of a solar pond under transient state with realistic energy extraction from all possible zones

A solar pond from which thermal energy is extracted from its non-convective zone (NCZ), lower convective zone (LCZ) and the ground below is modelled in transient state. The novelty of the study lies in the fact that heat extraction in each of the three zones is considered to be realistic, taking into account temperature drop across the exchanger surfaces. The pertinent weather parameters are described by curve fitting as series of sinusoidal time functions. Using these expressions, implicit finite difference method is used to solve the five coupled partial differential equations involved in the analysis, and the scaled down model is validated with simpler models available in the literature. To quantify the practical utility of pond, the final outgoing water stream heated from the pond is fed to a large scale solar still to assess the usefulness in a desalination application. Annual distillate production of the still coupled to such a pond is calculated and its variation with various manually adjustable parameters is studied. A total of 9 such parameters have been investigated. It is observed that there are five pond parameters related to pond’s operation that possess optimum values that maximize annual distillate production. These are: NCZ and LCZ thicknesses, NCZ and LCZ exchanger pipes’ radii, and ground extraction mass flow rate. An increase in any of the remaining parameters: upper convective zone (UCZ) thickness, ground exchanger pipes’ radius and NCZ, LCZ extraction mass flow rates generate a decrease in distillate produced. It is also revealed that extraction should be carried from how many zones depends on the application coupled to the pond, and in this case, where the application is a solar still, LCZ with ground extraction turns out to be the best choice. Further, the assumption of ideal NCZ and ground extraction used in the literature is seen to overestimate system output by nearly 46%. Therefore, this model is a generalized one and various other single or dual zone extraction models available so far become subsets of it. This generalized model presented here can serve to design such a triple zone extraction based solar pond system in a realistic and practical manner and help evaluate suitable values of user-controlled variables that yield maximum output of the application to which the pond supplies energy.

A Time-Domain Finite-Difference Method for Bending Waves on Infinite Beams on an Elastic Foundation

To model the vibration and structure-borne sound excitation and propagation of a railway rail, it can be modeled as an infinite beam on an elastic foundation. Existing analytical or numerical models are either formulated in the frequency domain or consider only finite beams in the time domain. Therefore, a time-domain approach for bending wave propagation on an effectively infinite beam on an elastic foundation is proposed. The approach makes use of an implicit finite-difference method that allows for varying properties of the beam and the foundation along the length of the beam. Strategies for an efficient discretization are discussed. The method is validated against existing analytical models for a single layer and two layers, as well as continuous and discrete support. The results show very good agreement, and it can be concluded that the proposed method can be seen as a versatile method for simulating the behavior of a beam on different kinds of elastic foundations.

Non-similar solutions and sensitivity analysis of nano-magnetic Eyring–Powell fluid flow over a circular cylinder with nonlinear convection

The current model is carried out to examine the radiative nano-magnetic coating flow of Eyring–Powell fluid over a circular cylinder with nonlinear convection and ohmic heating. In addition, response surface methodology is executed to probe the sensitivity of effective flow field parameters on heat transfer rate. The Brownian motion and thermophoresis effects are accounted for in the Eyring–Powell nanofluid model. The non-similar transformation approach is operated to reduce the nonlinear partial differential equations (PDEs) into dimensionless PDEs. The converted dimensionless PDEs are resolved by manipulating the Keller box technique (implicit finite difference method). To exhibit notable features of the solutions, parametric examination of the emerging parameters like thermal convection, Brownian motion, magnetic field, thermophoresis, inclination, Eckert number is carried out, the numerical computations of the flow field and physical quantities are portrayed via graphs. It is worth noting that the nanofluid coating velocity escalates by mixed convection parameter promotion. Higher values of mixed convection parameter promote the heat and mass transfer rate. The lower thermophoresis with the higher Eckert number causes a better heat transfer. The maximum heat transfer rate of 0.5467 is found via response surface methodology by setting the Eckert number=0.8, Brownian motion=0.3 and thermophoresiss=0.1.

The unsteady nonlinear convective flow of a nanofluid due to impulsive motion: Triple diffusion and magnetic effects

• The unsteady nonlinear convection due to impulsive motion is investigated. • Governing equations have been transformed using Mangler's transformations. • Quasi-linearization technique is utilized for solution of the problem. • The surface drag coefficient increases with magnetic parameter. • Heat transport rate rises only when the Eckert number is positive. By a semi-similar approach, the paper aims to study the unsteady nonlinear convection due to impulsive motion with triple diffusion and magnetic effects. Since the mainstream has been set into impulsive motion, the fluid motion is unsteady. Simultaneously, the temperature of the surface is abruptly changed from its ambient fluid temperature. The influence of chemical reactions and convective boundary constraints are also considered in this investigation. The nonlinear coupled partial differential equations (PDEs) with appropriate boundary conditions that govern the given problem are transformed into dimensionless equations by using Mangler's transformations. For mathematical computations, the technique of Quasi-linearization and second order implicit finite difference method are utilized, and so obtained results are semi-similar solutions that are discussed with the aid of graphs. The results reveal that for the case of aiding buoyancy, the surface drag coefficient increases about 5% when the nonlinear convection parameter increases from 0 to 3 and about 61% when the magnetic parameter increases from 0 to 0.5. The heat transfer rate enhances about 65% for variation of the magnetic parameter from 0 to 0.3 for a positive Eckert number. The mass transfer rate is pronounced more for linear chemical reaction than the nonlinear chemical reaction for the constructive chemical reaction parameter. It is found at about 7% for Schmidt number 240. The current findings are compared to previously reported literature and are found to be in good agreement.

MAGNETOHYDRODYNAMIC FREE CONVECTION FLOW OF FLUID OF A VERTICAL PLANE WITH VISCOSITY AND THERMAL CONDUCTIVITY DEPENDING ON TEMPERATURE

A two-dimensional natural convection flow of a viscous incompressible and electrically conducting fluid with both the viscosity and the thermal conductivity depending on temperature past a vertical impermeable flat plate is considered in presence of a uniform transverse magnetic field. The governing equations are reduced to non-similar boundary layer equations by introducing coordinate transformations appropriate to the cases (i) near the leading edge (ii) in the region far away from the leading edge and (iii) for the entire regime from leading edge to down stream. The governing equations for the flow in the upstream regime are investigated by perturbation method for smaller values of x, the stream-wise distributed magnetic field parameter. The equations governing the flow for large x and for all x, have been investigated by employing the implicit finite difference method with Keller box scheme. The effects of the viscosity variation parameter, e and the thermal conductivity variation parameter g, on the skin friction as well as the rate of heat transfer for the fluid for low Prandtl number are shown graphically.

Continuous vs. discrete time: Some computational insights

Solving a workhorse incomplete markets model in continuous time is many times faster compared to its discrete time counterpart. This paper dissects the computational discrepancies and identifies the key bottlenecks. The implicit finite difference method – a commonly used tool in continuous time – accounts for a large share of the difference. This method is shown to be a special case of Howard’s improvement algorithm, efficiently implemented by relying on sparse matrix operations. By representing the policy function with a transition matrix it is possible to formulate a similar procedure in discrete time, which effectively eliminates the differences in run-times entirely.

A numerical solution of 2-D single-phase-lag (SPL) bio-heat model using alternating direction implicit (ADI) finite difference method

A numerical solution of 2-D single-phase-lag (SPL) bio-heat model using alternating direction implicit (ADI) finite difference method

A Hybrid Finite Difference Method for Singularly Perturbed Delay Partial Differential Equations with Discontinuous Coefficient and Source

The article presents a hybrid finite difference scheme to solve a singularly perturbed parabolic functional differential equation with discontinuous coefficient and source. The simultaneous presence of deviating argument with a discontinuous source and coefficient makes the problem stiff. The solution of the problem exhibits turning point behaviour across discontinuity as tends to zero. The hybrid scheme presented is a composition of a central difference scheme and a midpoint upwind scheme on a specially generated mesh. At the same time, an implicit finite difference method is used to discretize the time variable. Consistency, stability, and convergence of the presented numerical approach have been investigated. The presented method converges uniformly independent of the perturbation parameter. Numerical results have been presented for two test examples that verify the effectiveness of the scheme.

Effect of sediment transport, flow depth and infiltration on soil moisture profiles in irrigation furrows

A numerical model was used to analyze the effect of sediment transport and flow depth on soil moisture profiles from a simulation of the hypothetical irrigation furrows. The Saint Venant and sediment continuity equations were solved by using MacCormack scheme based on finite difference method to analyze the overland flow and sediment transport, and Richards equation was used to analyze the subsurface flow, which was solved by using a mass conservative fully implicit finite difference method. Flow depth significantly affected the moisture profiles in irrigation channels while the effect of sediment-laden water on the moisture profiles was observed insignificant. The model was used to study the effect of infiltration on moisture profiles and soil water retention curve (SWRCs) in irrigation furrows. It was observed that infiltration significantly affected and altered the moisture profiles and soil water retention curve (SWRC). Among the different soil parameters (av and nv ) and saturated hydraulic conductivity () significantly affected the moisture profiles, while and influenced the soil water retention curve (SWRC) as compared to other retention parameters.

Fully Implicit Form of Differential Quadrature Method for Multi-Species Solute Transport in Porous Media

Solute transport problems, including sequential multi-species transport phenomena, frequently occur in soil systems. The goal of this paper is to present a novel one-dimensional numerical model with a fully implicit form of differential quadrature method for solving multi-species solute transport equations. The analytical results of three multi-species solute dispersion problems with three- and four-chain members are used to analyse the developed model. Simultaneously, the outcomes of the developed model are compared with the performance of the fully implicit fourth-order finite difference method. Finally, the accuracy of the established model is discussed and evaluated. According to the numerical experiments, the derived model is very useful and widely applicable.

Differential Game Approach for Attack-Defense Strategy Analysis in Internet of Things Networks

Internet of Things (IoT) is vulnerable to various cyber attacks due to the massive deployment of IoT devices and the openness of wireless environments. In this article, taking IoT devices as the network resources competed between an attacker and a defender, we study the modeling and analysis of network resource competition in an attack-defense game. The attacker and defender inject different competition strength in each IoT device as their strategies. As a result, the security state of each IoT device will change, which is captured by differential equations. To study the interaction between the attacker and defender and the evolution of the system security states, a zero-sum differential game is formulated by modeling the competition of IoT devices. To achieve the equilibrium of the formulated differential game, optimal control theory is employed to solve the optimization problems of players. Further, a Gauss–Seidel-like implicit finite-difference method is utilized to obtain the saddle point strategy. Finally, numerical results are provided to demonstrate the evolution of network resource competition between the attacker and defender. The results show that our formulated model can effectively and accurately characterize the evolution of the system security states with strategic interactions between the attacker and defender.

NUMERICAL STUDY OF THE INFLUENCE OF TEMPERATURE-DEPENDENT VISCOSITY ON THE UNSTEADY LAMINAR FLOW AND HEAT TRANSFER OF A VISCOUS INCOMPRESSIBLE FLUID DUE TO A ROTATING DISC

In this article, the effect of temperature-dependent viscosity (TVD) on the unsteady laminar flow and heat transfer (HT) of a viscous incompressible fluid due to a rotating disc (RD) has been investigated numerically by exploiting an in-house numerical code. A set of time-dependent, axisymmetric, and non-linear partial differential equations which govern the fluid flows and heat transfer are reduced to non-linear local non-similarity ordinary differential equations by introducing a newly developed group of transformations for different time regimes. Three different solution methods, such as, (i) perturbation solution method for small t, (ii) asymptotic solution method for large t, and (iii) implicit finite difference method for the entire t regime, have been applied to solve the resulting equations treating t as the time-dependent rotating parameter. The local radial skin friction, tangential skin friction and the heat transfer are computed at the surface of the disc for different numerical parameters, such as, Prandtl number, Pr and the viscosity-variation parameter, e. Besides, the key dimensionless quantities such as velocity and temperature profiles, which are inherently linked with the boundary layer thickness, are presented graphically for different values of e while Pr = 0.72. It is found that the dimensionless radial, tangential and axial velocity profiles decrease as e increases, and consequently, the momentum boundary layer thickness is decreased. On the other hand, the non-dimensional temperature profiles are increased owing to the increasing values of e, and this effect eventually leads to a small increment in the thermal boundary layer thickness.