Explicit Finite Difference Technique Research Articles

A Two-dimensional Mathematical Model for Long-term Contaminated Groundwater Pollution Measurement around a Land Fill

A source of contaminated groundwater is governed by the disposal of waste material on a land fill. There are many people in rural areas where the primary source of drinking water is well water. This well water may be contaminated with groundwater from landfills. In this research, a two-dimensional mathematical model for long-term contaminated groundwater pollution measurement around a land fill is proposed. The model is governed by a combination of two models. The first model is a transient two-dimensional groundwater flow model that provides the hydraulic head of the groundwater. The second model is a transient twodimensional advection-diffusion equation that provides the groundwater pollutant concentration. The proposed explicit finite difference techniques are used to approximate the hydraulic head and the groundwater pollutant concentration. The simulations can be used to indicate when each simulated zone becomes a hazardous zone or a protection zone.

Study of buoyant force acting on different fluids moving over a horizontal plate due to forced convection

In this paper, the effect of buoyancy force on heat transfer over flat plate due to forced convection is investigated numerically. The governing equations namely continuity, Navier-stokes and energy equations were discretized by using an explicit finite difference technique. Code generation has been done in FORTRAN-95 programming language. The testing of FORTRAN code has been done for different fluids at different grid points. The length of horizontal flat plate (x) was taken as 0.5 m. The inlet velocity of fluid (u) at the edge of plate was taken as 3 m/s. The kinematic viscosity (υ) of air and water at 323 K were taken as 17×10-6 m2/s and 66×10-7 m2/s respectively. The surface temperature (Ts) of plate and flow temperature (To) of fluid were taken as 300 K and 323 K respectively. The grid was discretized by continuous method into 1001×101 (m×n) mesh points. The results show the boundary layer thickness, velocity and temperature evolutions, and heat transfer coefficient formed by different fluids. Also, the heat transfer coefficient to surrounding fluid across flat plate was found higher for air as compared to water. Results showed that buoyancy force acts from downward position so it was observed maximum at lower distance of plate. However, buoyancy effect reduces as the distance of x increases.

A Numerical Analysis of 2D Unsteady Magnetohydrodynamics Nanofluid Flow Through an Exponentially Accelerated Vertical Plate with Variable Viscosity by Using Explicit Finite Difference Method

This paper discusses the numerical investigation of viscous incompressible nanofluid on unsteady magnetohydrodynamics (MHD) chemically radioactive flow through an exponentially accelerated porous plate with variable viscosity. A system of governing nonlinear partial differential equations (PDEs) is transformed into ordinary differential equations (ODEs) by the as usual mathematical technique of transformation. Then, the obtained non-dimensional nonlinear partial differential equations (PDEs) are solved by using explicit finite difference technique (EFDM). Besides, the numerical results are calculated by FORTRAN programming language. In order to accuracy of numerical investigation a convergence stability test (NCT) has been carried out where we have obtained the convergence criteria of Prindtl number Pr ≥ 0.47; thermophoresis parameter Nt ≥ 0.80 and Lewis number Le ≥ 0.28 when initial boundary conditions U = T = C = 0 and for Δτ = 0.0005, ΔX = 0.50 and ΔY = 0.50. After those convergence criteria, the numerical solutions for velocity, temperature and concentration profiles are obtained for various dimensionless parameters. Those solutions are graphically discussed by using the tecplot-9 software. Moreover, the skin friction coefficient, Nusselt number, Sherwood number, Isotherms and Streamline are also investigated by tabular form.

Numerical models of nitrogen compound measurements in a stream with removal mechanism using Saulyev technique with cubic spline interpolation

Nutrient pollution negatively affects ecology, environment and human health, due to excess contaminated nitrogen in aquatic ecosystems. Nitrogen has various forms, such as organic nitrogen, ammonia, nitrite, and nitrate. Nitrogen removal mechanisms are important for nitrogen compound pollutant control. Microbial agents provide an alternative for nitrogen removal and are implemented by natural removal mechanisms. For pollution level measurements, data collection from actual measurement processes can be only limitedly collected in some situations. The mathematical models play a role in data approximation that can describe relations of processes and reactions. Six numerical models in one-dimensional nitrogen pollutant and nitrogen remover measurement in a stream are proposed. The proposed models are combined with total nitrogen with a removal mechanism, organic nitrogen, ammonia, nitrite, nitrate, and nitrogen remover dispersion models. These models are implemented through use of an efficient numerical technique, which is the unconditionally explicit Saulyev finite difference technique, associated with the cubic spline interpolation to initial and left boundary conditions through practical simulations. All six models related to nitrogen compound measurements in a stream with removal mechanism models give accurately approximated solutions. The proposed numerical models can be applied to other realistic problems.

Modified Explicit Finite-Difference Time-Domain Method for Nonparaxial Wave Scattering From Electromagnetic Metasurfaces

An updated method for simulating the scattered fields from a dispersive Huygens’ metasurface using the explicit finite-difference time-domain technique has been proposed and numerically demonstrated. The method uses a spatial–temporal averaging of the electric and magnetic fields, using virtual sources in a standard Yee cell grid. This creates a nonparaxial implementation of the generalized sheet transition conditions in the time domain and rigorously solves the scattered fields in reflection and transmission regions separately. The metasurfaces are solved for Lorentzian susceptibilities, and the proposed method is successfully demonstrated using two examples: a uniform metasurface with a strongly divergent beam and a space-modulated metasurface emulating a diffraction grating.

A one-dimensional mathematical simulation to salinity control in a river with a barrage dam using an unconditionally stable explicit finite difference method

Salinity refers to the amount of salt in rivers, where the salt can be in many different forms. There are two main methods of defining the concentration of salt in water such as the total dissolved solid measurement (TDS) and the electrical conductivity measurement (EC). The salinity is measured by evaporating water to dryness and weighing the solid residue. The electrical conductivity measurement is measured by passing an electric current through the water and measuring how readily the current flows. The total amount of salt in the water can affect the taste of water. The World Health Organization’s guideline on water palatability is that water with a salinity level of less than about 0.50–0.60 g/L is generally considered to be of a standard level. The drinking-water becomes significantly and increasingly unpalatable at salinity levels greater than about 1.0 g/L. In this research, a one-dimensional mathematical model of salinity measurement in a river is proposed. A modified model of salinity control in a river with a barrage dam is also introduced. An unconditionally stable explicit finite difference technique is used to approximate the salinity level under several conditions from the proposed model. The proposed computational technique gives good agreement results in realistic scenarios for water supply processes.

High-order exponential spline method for pricing European options

ABSTRACTThe key purpose of the present work is to constitute an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing European option pricing. The method is based on exponential spline spatial discretization and an explicit finite-difference time-stepping technique. We establish the convergence and an error bound for the solutions of the fully discretized system. The numerical and graphical results elucidate that the suggested approach is very straightforward and accurate.

Double-diffusive natural convection in an inclined enclosure with heat generation and Soret effect

PurposeThe purpose of this paper is to study numerically thermosolutal natural convection within an inclined rectangular cavity in the presence of Soret effect and heat generation. The enclosure is heated and salted from its long sides with constant but different temperatures and concentrations. The study focuses on the effects of three main parameters which are, the Soret parameter (Sr = 0 and –0.5), the internal to external Rayleigh numbers ratio 0 ≤ R ≤ 80 and the cavity inclination γ, varied from 0° (vertical position) to 60°. The combined effects of these parameters on fluid flow and heat and mass transfer characteristics are examined for the external Rayleigh number RaE = 105, the Prandtl number Pr = 0.71, the buoyancy ratio N = 1, the Lewis number Le = 2 and the aspect ratio of the cavity A = 2.Design/methodology/approachA hybrid lattice Boltzmann-finite difference method (LBM-FD) was used to tackle the problem under consideration. The LBM with the simple relaxation time was used for the fluid flow in the presence of the gravity force, while the temperature and concentration equations were solved separately using an explicit finite-difference technique at the Boltzmann scale.FindingsThe monocellular nature of the flow, obtained for R = 0 is not destroyed by varying the cavity inclination and the Soret parameter but rather by the increase of the parameter R. The Soret parameter and the cavity inclination become perceptible at high values of R. The inclination γ = 60° leads to high mean temperatures compared to the other inclinations. The effect of R on mean concentration is amplified in the presence of Soret effect but limited in the absence of the latter. The negative Soret parameter combined with high internal heat generation and a relatively high inclination is important when the objective is to maintain the fluid at a high concentration of species. The presence of bicellular flow combined with the important elevation undergone by the fluid temperature, makes both the cold and hot walls playing a cooling role with the most important exchanges taking place at the upper part of these walls. The analysis of the mean mass transfer shows that the increase of the inclination may lead to an increase or a decrease of the mass transfer depending on the range of R, in the case of Sr = 0. However, for Sr = −0.5, it is observed that the increase of γ is generally accompanied by a reduction of the mass transfer.Originality/valueTo the best of the authors’ knowledge, the hybrid LBM-FD was not used before to study such a problem. Combined effect of R and inclination may be useful in charging the fluid with species when the objective is to maintain high concentrations in the medium.

Effects of nanoparticles (Cu, TiO[formula omitted], Al[formula omitted]O[formula omitted]) on unsteady blood flow through a curved overlapping stenosed channel

In this letter, we have explored the effects of nanoparticles on unsteady pulsatile blood flow through a curved overlapping stenosed channel. The governing differential equations are derived for the given physical problem and this contains momentum and energy equations along with suitable boundary conditions. An explicit finite difference technique has been used to calculate the numerical results of the given differential equations. The numerical results for different values of curved parameter (Rc), nanoparticles (Cu, TiO2, Al2O3) and Grashof numbers (Gr) are calculated in order to analyze the effects of these parameters on blood flow pattern. A notable deviation in the results of blood flow pattern has been observed with the comparison of pure blood with nano-fluids. It is further noted that the curved parameter (Rc) of a channel has a great influence on velocity profile, flow rate and impedance/resistance to flow. It is also extracted from the velocity profiles that the shape of the velocity curves reduces to symmetric pattern for the higher values of the curved parameters (Rc). Similarly, the magnitude of temperature profiles is constricted due to the insertion of nanoparticles within blood. Moreover, streamlines are also calculated in order to analyze the effects of nanoparticles inside the circulating regions of the flow field.

Effects of Hall Current and Chemical Reaction on Magnetohydrodynamics Unsteady Heat and Mass Transfer of Casson Nanofluid Flow Through a Vertical Plate

In this paper, we have reported the effects of Hall current on magnetohydrodynamics (MHD) unsteady heat and mass transfer of Casson nanofluid flow through a vertical plate in the presence of radiation and chemical reaction. The model equations have been used for the Casson nanofluid and they include the effects of thermophoresis and Brownian motion. Then, the obtained model equations have been transformed into nondimensional form by the usual mathematical procedure of transformation and the resultant nondimensional couple of partial differential equations are solved by explicit finite difference technique. Then, the obtained results are plotted after stability test by using the graphical software tecplot-9 and these results indicate the fluid flow, temperature, and concentration distributions which are significantly invaded by the variation of different dimensionless parameters such as magnetic parameter, Schmidt number, thermal Grashof number, Lewis number, Prandtl number, mass Grashof number, Dufour number, thermophoresis parameter, Brownian motion parameter, chemical reaction, and radiation parameter on velocity, temperature, and concentration along with the skin friction coefficient, Nusselt number, and Sherwood number. Further, the results have been discussed also with the help of graphs. Furthermore, it is observed that with the increase of the Casson parameter, velocity puts down, whereas by increasing the heat generation parameter, the temperature profiles are decreased.

Finite-Difference Time-Domain Modeling of Space–Time-Modulated Metasurfaces

A finite-difference time-domain modeling of finite-size zero thickness space–time-modulated Huygens’ metasurfaces based on generalized sheet transition conditions is proposed and numerically demonstrated. A typical all-dielectric Huygens’ unit cell is taken as an example and its material permittivity is modulated in both space and time, to emulate a traveling-type spatio-temporal perturbation on the metasurface. By mapping the permittivity variation onto the parameters of the equivalent Lorentzian electric and magnetic susceptibility densities, $\chi _{\text {ee}}$ and $\chi _{\text {mm}}$ , the problem is formulated into a set of second-order differential equations in time with nonconstant coefficients. The resulting field solutions are then conveniently solved using an explicit finite-difference technique and integrated with a Yee-cell-based propagation region to visualize the scattered fields taking into account the various diffractive effects from the metasurface of finite size. Several examples are shown for both linear and space–time varying metasurfaces which are excited with normally incident plane and Gaussian beams, showing detailed scattering field solutions. While the time-modulated metasurface leads to the generation of new collinearly propagating temporal harmonics, these harmonics are angularly separated in space, when an additional space modulation is introduced in the metasurface.

A local adaptive method for the numerical approximation in seismic wave modelling

Abstract We propose a new numerical approach for the solution of the 2D acoustic wave equation to model the predicted data in the field of active-source seismic inverse problems. This method consists in using an explicit finite difference technique with an adaptive order of approximation of the spatial derivatives that takes into account the local velocity at the grid nodes. Testing our method to simulate the recorded seismograms in a marine seismic acquisition, we found that the low computational time and the low approximation error of the proposed approach make it suitable in the context of seismic inversion problems.

Finite-Difference Modeling of Broadband Huygens’ Metasurfaces Based on Generalized Sheet Transition Conditions

An explicit time-domain finite-difference technique for modeling zero-thickness Huygens’ metasurfaces based on generalized sheet transition conditions (GSTCs) is proposed and demonstrated using full-wave simulations. The Huygens’ metasurface is modeled using electric and magnetic surface susceptibilities, which are found to follow a double-Lorentz dispersion profile. To solve zero-thickness Huygens’ metasurface problems for general broadband excitations, the double-Lorentz dispersion profile is combined with GSTCs, leading to a set of first-order differential fields equations in time domain. Identifying the exact equivalence between Huygens’ metasurfaces and coupled RLC oscillator circuits, the field equations are then subsequently solved using standard circuit modeling techniques based on a finite-difference formulation. Several examples, including generalized refraction, are shown to illustrate the temporal evolution of scattered fields from the Huygens’ metasurface under plane-wave normal incidence, in both harmonic steady-state and broadband regimes. In particular, due to its inherent time-domain formulation, a significant strength of the methodology is its ability to model time-varying metasurfaces, which is demonstrated with a simple example of a pumped metasurface leading to new frequency generation and wave amplification.

Numerical Simulation to Air Pollution Emission Control near an Industrial Zone

A rapid industrial development causes several environment pollution problems. One of the main problems is air pollution, which affects human health and the environment. The consideration of an air pollutant has to focus on a polluted source. An industrial factory is an important reason that releases the air pollutant into the atmosphere. Thus a mathematical model, an atmospheric diffusion model, is used to estimate air quality that can be used to describe the sulfur dioxide dispersion. In this research, numerical simulations to air pollution measurement near industrial zone are proposed. The air pollution control strategies are simulated to achieve desired pollutant concentration levels. The monitoring points are installed to detect the air pollution concentration data. The numerical experiment of air pollution consisted of different situations such as normal and controlled emissions. The air pollutant concentration is approximated by using an explicit finite difference technique. The solutions of calculated air pollutant concentration in each controlled and uncontrolled point source at the monitoring points are compared. The air pollutant concentration levels for each monitoring point are controlled to be at or below the national air quality standard near industrial zone index.

Unsteady Free Convection Fluid Flow over an Inclined Plate in the Presence of a Magnetic Field with Thermally Stratified High Porosity Medium

MHD free convection over an inclined plate in a thermally stratified high porous medium in the presence of a magnetic field has been studied. The dimensionless momentum and temperature equations have been solved numerically by explicit finite difference technique with the help of a computer programming language Compaq Visual Fortran 6.6a. The obtained results of these studies have been discussed for the different values of well known parameters with different time steps. Also, the stability conditions and convergence criteria of the explicit finite difference scheme has been analyzed for finding the restriction of the values of various parameters to get more accuracy. The effects of various governing parameters on the fluid velocity, temperature, local and average shear stress and Nusselt number has been investigated and presented graphically.

Unsteady MHD Casson Fluid Flow through a Parallel Plate with Hall Current

Unsteady MHD Casson fluid flow through a parallel plate with hall current is investigated. The uniform magnetic field is applied perpendicular to the plates and the fluid motion is subjected to a uniform suction and injection. The lower plate is stationary and the upper plate is moving. Explicit Finite Difference technique has been used to solve the momentum and energy equations. The effect of pressure gradient, the Hall parameter and other parameters describing in the equations are shown graphically. Effect of decaying parameter with different Casson number on primary velocity, secondary velocity and temperature distributions are illustrated in the form of the graph.

MHD Free Convection Fluid Flow Through Parallel Plates with Hall Current in a Rotating System

An attempt has been made to study the influence of inclined magnetic field with hall current of a viscous incompressible electrically conducting free convection fluid flow through the parallel plates in a rotating system. The channel is rotating with uniform angular velocity about an axis normal to the plates. For numerical solution, the problems have been transferred into dimensionless form and then the explicit finite difference technique has been used. The stability conditions and convergence criteria of the explicit finite difference scheme are established for finding the restriction of the values of various parameters to get more accuracy. The effects of Rotation parameter, Hall parameter, Magnetic parameterand Angle of inclination on primary and secondary velocities have been shown in graphical representations. Finally the effects of these parameters have been analyzed by the figures of local shear stress and average current density at the moving plates of channel.

Investigate Micropolar Fluid Behavior on MHD Free Convection and Mass Transfer Flow with Constant Heat and Mass Fluxes by Finite Difference Method

Micropolar fluid behavior on MHD free convection and mass transfer with constant heat and mass fluxes is studied numerically. Finite difference technique is used as the main tool for the numerical approach. Micropolar fluid behavior on MHD steady free convection and mass transfer with constant heat and mass fluxes have been considered and its similarities solution have been obtained. Similarity equations of the corresponding momentum, angular momentum, temperature and concentration equations are derived by employing the usual similarity technique. The dimensionless similarity equations for momentum, angular momentum, temperature and concentration equations solved numerically by explicit finite difference technique. With the help of graphs the effects of the various important parameters entering into each of the problems on the velocity, microrotation, temperature and concentration profiles within the boundary layer are separately discussed.

Free Convection Fluid Flow In the Presence of A Magnetic Field with Thermally Stratified High Porosity Medium

MHD free convection over an inclined plate in a thermally stratified high porous medium in the presence of a magnetic field has been studied. The dimensionless momentum and temperature equations have been solved numerically by explicit finite difference technique with the help of a computer programming language Compaq Visual Fortran 6.6a. The obtained results of these studies have been discussed for the different values of well known parameters with different time steps. Also, the stability conditions and convergence criteria of the explicit finite difference scheme has been analyzed for finding the restriction of the values of various parameters to get more accuracy. The effects of various governing parameters on the fluid velocity, temperature, local and average shear stress and Nusselt number has been investigated and presented graphically.

Diffusion-Thermo and Thermal-diffusion Effects on MHD Visco-Elastic Fluid Flow over a Vertical Plate

Unsteady MHD visco-elastic fluid flow has been studied numerically under the action of transverse magnetic field with diffusion-thermo and thermal diffusion for small magnetic Reynolds number. The governing equations are non-dimensionalized by usual non-dimensional variables. The obtained equations are solved by explicit finite difference technique. The solutions of the dimensionless velocity, temperature and concentration equations are shown graphically. The effects of parameters on the shear stress, Nusselt number and Sherwood number are discussed in graphical form. Finally, a qualitative comparison with previous work is tabulated.