Implicit Finite Difference Research Articles

Operator splitting method for numerical solving the atmospheric pollutant dispersion problem

The paper discusses the numerical modeling of transport and diffusion of air pollutants in the atmospheric boundary layer. There was developed a mathematical model of industrial emissions spread in the atmosphere taking into account the deposition velocity of fine particles. The model is described by multidimensional partial differential equations with appropriate initial and boundary conditions. The basic laws of hydrothermodynamics were used in deriving the model. In order to obtain the numerical solution of the problem, we used one of the splitting methods according to physical processes involved (transport, diffusion and absorption), as well as a second-order implicit finite-difference scheme in time. Analysis of numerical results showed that the developed computational algorithm provides sufficient accuracy of the problem solution compared with field measurement data and it has a certain advantage over other numerical methods. In the course of computational experiments, there was determined the degree of influence of such parameters as wind speed and direction, absorption coefficient and physicomechanical properties of particles on the process of atmospheric air pollutants dispersion.

Polymeric Dissipative Convection Flow from an Inclined Plane with Chemical Reaction: Numerical Study

An analytical model is developed to study the effects of viscous dissipation and chemical reaction in viscoelastic convection from an inclined plate as a simulation of electro-conductive polymer materials processing. The Jeffery’s viscoelastic model is deployed to describe the non-Newtonian characteristics of the fluid and provides a good approximation for polymers, which constitutes a novelty of the present work. The normalized nonlinear boundary value problem is solved computationally with the Keller-Box implicit finite-difference technique. Extensive solutions for velocity, surface temperature and concentration, skin friction, heat, and transfer rates are visualized numerically and graphically for various thermophysical parameters. Validation is conducted with earlier published work for the case of a vertical plate in the absence of viscous dissipation, chemical reaction, and non-Newtonian effects. The boundary layer flow is accelerated with increasing Deborah number whereas temperatures and concentrations are decelerated slightly. Temperatures and concentration are boosted with increasing inclination parameter whereas velocity is lowered. A reverse trend is seen for increasing Richardson number. Increasing chemical reaction reduces velocity and concentration whereas it enhances temperature. Increasing the viscous dissipation parameter is found to enhance velocity and temperature whereas it suppresses concentration.

Optimal Control Strategy for a Discrete Time to the Dynamics of a Population of Diabetics with Highlighting the Impact of Living Environment

Nowadays, Diabetes is one of the most common diseases, which has a huge and growing socio-economic burden affecting individuals, families, and the whole society. In this paper, we propose an optimal control approach modeling the evolution from pre-diabetes to diabetes with and without complications and the effect of living environment. We show the existence of an optimal control and then use a numerical implicit finite-difference method to monitor the size of population in each compartment.

Bingham fluid with viscosity and yield stress depending on the density

In this note we present a mathematical model for the channel flow of a Bingham fluid in which the rheological parameters are not constant. We consider two different situations: (i) the yield stress and Bingham viscosity depend on the density which is not constant; (ii) the yield stress and Bingham viscosity depend on a parameter that is not spatially uniform but the density of the fluid is constant. Model (i) can be used to describe the mud flow in coastal and estuarine waters. Model (ii) can be used to describe blood flow in vessels. After formulating the mathematical model, which turns out to be a moving boundary problem, we solve the problem numerically by means of an implicit finite-difference scheme, determining the evolution of the yield surface (free boundary). We perform numerical simulations for cases (i) and (ii) using experimental data that are available in the literature.

Fully Coupled Modeling of Dynamic Loading of the Wellbore

Summary Loadings acting on a wellbore are more realistically regarded as dynamic rather than static, and the wellbore response under dynamic loading can be different from that under static loading. Under dynamic loading, the inertia term should be considered and the changing rate of loading could induce a change in the mechanical properties of the wellbore, which might compromise wellbore stability and integrity. In this paper, a fully coupled poroelastodynamic model is proposed to study wellbore behavior. This model not only considers fully coupled deformation/diffusion effects, but also includes both solid and fluid inertia terms. The implicit finite-difference method was applied to solve the governing equations, which allows this model to handle all kinds of dynamic loading conditions. After modifying the existing code only slightly, our numerical solution can neglect inertia terms. The numerical results were validated by comparing them to the analytical solution with a simulated sinusoidal boundary condition. To understand this model better, a sensitivity analysis was performed, and the influence of inertia terms was investigated. After that, the model was applied to analyze wellbore stability under tripping operations. The results show that the inertial effect is insignificant for tripping and a fully coupled, quasistatic model is recommended for wellbore stability under tripping operations. The fully coupled poroelastodynamic model should be used for rapid dynamic loading conditions, such as earthquakes and perforations.

Channel Wall Cooling by Evaporative Falling Water–Ethanol and Water–Methanol Films

The thermal protection by evaporative cooling is among the most efficient cooling methods, which provides higher wall temperature reduction owing to the phase conversion of the heat-transfer fluid. Thus, the purpose of this study is to investigate the capacity of pure water, water–ethanol, and water–methanol liquid mixtures to thermally protect a channel wall. To achieve the goals, an implicit finite difference in house code is developed and exploited to solve the coupled governing equations in both liquid and gas phases. The influence of water mass fraction in binary mixture is studied. The results revealed the convenient liquid and water mass fraction to have low wall temperature. It was found that both latent and sensible heat are important in this process. In addition, the presence of alcohols in the mixture with high concentration ensures a better cooling and thermal protection of the wall. A maximum wall temperature drop of 12 K was achieved by pure methanol liquid film versus an elevation of 6 K by pure water film. Moreover, it is highlighted that a pure ethanol film can be avoided by using only 25% of methanol with water, which permitted keeping approximately the same temperature drop.

Structural and numerical analysis of an implicit logarithmic scheme for diffusion equations with nonlinear reaction

In this work, we investigate a numerical diffusion equation with nonlinear reaction, defined spatially over a closed and bounded interval of the real line. The partial differential equation is expressed in an equivalent logarithmic form, and initial and Dirichlet boundary data are imposed upon the problem. An implicit finite-difference discretization of this logarithmic model is proposed then. We show that the numerical scheme is capable of preserving the constant solutions of the continuous model. Moreover, we establish the existence of positive and bounded numerical solutions using analytical arguments. Finally, an extensive set of numerical simulations is provided in order to illustrate the performance of the scheme. The results verify that the logarithmic scheme converges to the exact solution of the continuous problem, with first order of convergence in time and second order in space. Moreover, we provide some comparisons on the efficiency of the implicit method against an explicit logarithmic scheme of the literature. The results show that the present discretization is a more efficient technique.

Compact difference scheme for two-dimensional fourth-order hyperbolic equation

In this paper, we mainly study an initial and boundary value problem of a two-dimensional fourth-order hyperbolic equation. Firstly, the fourth-order equation is written as a system of two second-order equations by introducing two new variables. Next, in order to design an implicit compact finite difference scheme for the problem, we apply the compact finite difference operators to obtain a fourth-order discretization for the second-order spatial derivatives and the Crank–Nicolson difference scheme to obtain a second-order discretization for the first-order time derivative. We prove the unconditional stability of the scheme by the Fourier method. Then a convergence analysis is given by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.

G-Jitter Free Convection Flow of Nanofluid in The Three-Dimensional Stagnation Point Region

The three dimensional free convection boundary layer flow near a stagnation point region is embedded in viscous nanofluid with the effect of g-jitter is studied in this paper. Copper (Cu) and aluminium oxide (Al2O3) types of water base nanofluid are cho- sen with the constant Prandtl number, Pr=6.2. Based on Tiwari-Das nanofluid model, the boundary layer equation used is converted into a non-dimensional form by adopting non- dimensional variables and is solved numerically by engaging an implicit finite-difference scheme known as Keller-box method. Behaviors of fluid flow such as skin friction and Nusset number are studied by the controlled parameters including oscillation frequency, amplitude of gravity modulation and nanoparticles volume fraction. The reduced skin friction and Nusset number are presented graphically and discussed for different values of principal curvatures ratio at the nodal point. The numerical results shows that, in- crement occurs in the values of Nusset number with the presence of solid nanoparticles together with the values of the skin friction. It is worth mentioning that for the plane stagnation point there is an absence of reduced skin friction along the y-direction where as for axisymmetric stagnation point, the reduced skin friction for both directions are the same. As nanoparticles volume fraction increased, the skin friction increased as well as the Nusset number. The results, indicated that skin frictions of copper are found higher than aluminium oxide.

Underground Storage of Natural Gas in Hydrate State: Numerical Experiment

The paper is devoted to simulation of the initial stage of natural gas hydrate underground storage: gas injection into aquifer just below permafrost rocks. It is based on the mathematical model of multiphase non-isothermal real gas and water flow in porous media. The model takes into account the transformation of gas and water into hydrate at certain temperature which depends on gas flow pressure. The dynamics of hydrate and water saturation as well as the pressure and temperature fields in a reservoir with given porosity, permeability and initial values of pressure, temperature and water saturation have been studied. An implicit finite-difference scheme is used to approximate the original boundary-value problem. The finite-difference equations have been solved using simple iteration and sweeping algorithms. Several examples of calculations corresponding to real cases are given. Calculations have revealed that the final result strongly depends on the combination of porosity and permeability of a reservoir.

A numerical method for a time-fractional advection–dispersion equation with a nonlinear source term

In this paper we propose an implicit finite-difference scheme to approximate the solution of an initial-boundary value problem for a time-fractional advection–dispersion equation with variable coefficients and a nonlinear source term. The time fractional derivative is taken in the sense of Caputo. The method is unconditionally stable and convergent. Some numerical examples are included and the results confirm the theoretical analysis. One of the examples is the fractional Fisher equation of mathematical biology.

Dynamic loading of crack-weakened complex-shaped bodies

Dynamic elasticity problems in the 2D formulation are solved by the method of lines using the Crank-Nicolson implicit finite-difference scheme for time-based integration and the finite element method for solving boundary problems. The applied variant of the finite element method includes two types of elements: isoparametric elements for modeling curvilinear boundaries and cohesion elements ensuring the compliance with Barenblatt’s postulates. It is shown how to apply the method to the strength calculations by the Petrov-Morozov method.

Анализ устойчивости неявных конечно-разностных решеточных схем Больцмана с направленными разностями

Статья посвящена анализу устойчивости неявных конечно-разностных схем для системы кинетических уравнений, применяемых для проведения гидродинамических расчетов в рамках метода решеточных уравнений Больцмана. Представлены семейства двухслойных и трехслойных схем с направленными разностями первого-четвертого порядков аппроксимации по пространственным переменным. Важной особенностью схем является то, что конвективные слагаемые аппроксимируются одной конечной разностью. Показано, что в выражении для аппроксимационной вязкости схем высоких порядков отсутствуют фиктивные слагаемые, что позволяет применять их во всем диапазоне значений времени релаксации. Анализ устойчивости проводится по линейному приближению с использованием метода Неймана. Получены приближенные условия устойчивости в виде неравенств на значения параметра Куранта. При расчетах показано, что площади областей устойчивости в пространстве параметров у двухслойных схем больше, чем у трехслойных. Исследованные схемы могут применяться при расчетах как непосредственно, так и в методах типа предиктор-корректор. The paper is devoted to the stability analysis of the implicit finite-difference schemes for the system of kinetic equations used for the hydrodynamic computations in the framework of the lattice Boltzmann method. The families of two- and three-layer upwind schemes of the first to fourth approximation orders on spatial variables are considered. An important feature of the presented schemes is that the convective terms are approximated by one finite difference. It is shown that, for the high-order schemes, in the expression for the current viscosity there are no fictitious terms, which makes it possible to perform computations in the whole range of relaxation time values. The stability analysis is based on the application of the von Neumann method to the linear approximations of the schemes. The stability conditions are obtained in the form of inequalities imposed on the Courant number values. It is also shown that the areas of stability domains for the two-layer schemes are greater than for the three-layer schemes in the parameter space. The considered schemes can be used as the fully implicit schemes in computational algorithms directly or in the predictor-corrector methods.

Theory of Second Order Numerical Simulation Method of Enhanced Oil Production

A kind of second-order implicit upwind fractional steps finite difference method is presented in this paper to numerically simulate the coupled system of enhanced (chemical) oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and optimal order error estimates in l2 norm are derived.

Numerical Solution of MRLW Equation with İmplicit Finite-Difference Approximation

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation has been showed using the classical implicit finite-difference. Error norms........ DOI :10.7176/JSTR/5-3-09

Dynamic Wellbore Stability Analysis Under Tripping Operations

The loadings which act on the wellbore are more frequently dynamic than static, such as the surge/swab pressure caused by tripping operations. The changing rate of loading could induce a change in wellbore stress and result in wellbore instability. The conventional Kirsch solution to calculate wellbore stress is only applicable to the steady state without considering the coupled deformation–diffusion effect. To account for these deficiencies, this paper introduces a coupled poroelastodynamics model to obtain wellbore stress distribution under dynamic loading. The model is solved by the implicit finite-difference method with the surge/swab pressure caused by tripping operations taken as the specific dynamic loading source. Then, failure criteria are applied to analyze dynamic wellbore stability and the hemisphere plots of minimum mud density (MMD) to avoid wellbore collapse are generated. The effect of in-situ stress regimes, failure criteria, and permeable properties on the instability of the borehole has been investigated, and the applicability and accuracy of the conventional failure criteria for dynamic loading conditions have been studied by the extensive triaxial compression tests performed on Bedford limestone under different strain rates. Furthermore, two specific cases have been analyzed and results show that compared with the poroelastodynamics model, the conventional method either overestimates or underestimates the MMD, the difference of which can be as significant as 0.11 g/cm3. Not limited to the tripping operations, the developed poroelastodynamics model can be applied to any specific dynamic loading conditions with the fluid inertia neglected.

Numerical Model of Time-Dependent Gas Flows Through Bed of Granular Phase Change Material

A novel mathematical model and original numerical method for investigating time-dependent gas flows through a bed of granular phase change material (PCM) are proposed and described in detail. Such material is modeled as a porous medium, and continua mechanics method are used for constructing the mathematical model. The numerical method is based on a combination of explicit and implicit finite-difference schemes. Comparison of calculation results with known experimental data demonstrates a very good coincidence. The results of the study can be applied in modeling the thermal energy storage with granular PCM in advanced adiabatic compressed air energy storage and other heat storage devices.

Numerical solution of a parabolic optimal control problem with time-fractional derivative

The article deals with a linear-quadratic optimal control problem governed by a parabolic equation with time-fractional derivative taken in Caputo definition. We impose pointwise constraints on state and control functions. The derivative in time of state function is taken in the Caputo derivative form of fractional order α ϵ (0,1). The mesh approximation is based on the using of finite-difference implicit scheme for the state equation. The Lagrange function’s saddle point satisfies the saddle point problem. Uzawa-type iterative method for numerical solution is used. The results of numerical tests are presented.

High-order temporal and implicit spatial staggered-grid finite-difference operators for modelling seismic wave propagation

High-order temporal and implicit spatial staggered-grid finite-difference operators for modelling seismic wave propagation

Radiative Flow of Third Grade Non-Newtonian Fluid From A Horizontal Circular Cylinder

Abstract In this article, we study the nonlinear steady thermal convection of an incompressible third-grade non-Newtonian fluid from a horizontal circular cylinder. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-differences Keller Box technique. The influence of a number of emerging non-dimensional parameters, namely the third-grade fluid parameter (ϕ), the material fluid parameters (ϵ1, ϵ2), Prandtl number (Pr), Biot number (y), thermal radiation (F) and dimensionless tangential coordinate (ξ) on velocity and temperature evolution in the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. Validation with earlier Newtonian studies is presented and excellent correlation is achieved. It is found that the velocity, skin friction and Nusselt number (heat transfer rate) reduce with increasing third grade fluid parameter (ϕ), whereas the temperature is enhanced. Increasing material fluid parameter (ϵ1) reduces the velocity and heat transfer rate but enhances the temperature and skin friction. The study is relevant to chemical materials processing applications and low density polymer materials processing.