Implicit Finite Difference Method Research Articles

A finite difference method to analyze heat and mass transfer in kerosene based γ-oxide nanofluid for cooling applications

Researchers have been paying attention to the utilization of nanofluid since they have disclosed their potentiality as a magnificent working fluid in thermal energy systems. Also, metal-oxide nanoparticles are among the most widely used nanomaterials due to their distinct characteristics such as limited size and high density. Holding such interesting characteristics of oxide-nanoparticles in mind, this research considers heat and mass transport attributes of γ-oxide nanoparticles in chemically reacted kerosene based nanofluid over a vertical cone in the presence of heat generation/absorption. An implicit finite difference method is implemented to obtain the numerical solutions of a constructed mathematical model with the aid of MATLAB software. The effects of numerous parameters on the associated distributions are examined, and their results are illustrated graphically. It is culminated that the velocity and temperature of nanofluid are increased in the presence of heat generation. The thermal conductivity of GO/kerosene is 0.633%, 0.443%, 0.126%, and 0.063% greater than CuO/kerosene, Al2O3/kerosene, TiO2/kerosene, and Fe3O4/kerosene, respectively, at 3% Vol. concentration of nanoparticles. To confirm the authenticity of current research, the acquired results are contrasted with the results of earlier published articles under certain conditions, and an outstanding coexistence is attained.

A Study of Photo-Thermoelastic Wave in Semiconductor Materials with Spherical Holes Using Analytical-Numerical Methods

Analytical and numerical solutions are two basic tools in the study of photothermal interaction problems in semiconductor medium. In this paper, we compare the analytical solutions with the numerical solutions for thermal interaction in semiconductor mediums containing spherical cavities. The governing equations are given in the domain of Laplace transforms and the eigenvalues approaches are used to obtained the analytical solution. The numerical solutions are obtained by applying the implicit finite difference method (IFDM). A comparison between the numerical solutions and analytical solution are presented. It is found that the implicit finite difference method (IFDM) is applicable, simple and efficient for such problems.

Numerical Investigation of Fluid-Structure Thermal Coupling under the Transient Flow Effect

This work is based on a numerical study of the fluid-structure thermal coupling in a concrete slab intended for habitable heating. A rectangular cross-section pipe in which a hot fluid flow is installed in this concrete slab. The Navier-Stokes equations that govern this flow have been solved numerically. To this end, these equations have been discretized by an implicit finite difference method. The systems of algebraic equations thus obtained have been solved by the Gauss and Thomas algorithms. The conduction equation in the concrete slab was solved using the same methodology as that of flow. In fact, we have based on an algorithm that makes an unsteady solid medium interact with a fluid medium consisting of permanent states series while ensuring the equality of fluxes and temperatures on the common interface between both media at every moment. The numerical simulation of heat transfer and the thermal behavior of the heating slab were analyzed for different parameters influencing thermal diffusion. The results obtained by the numerical model adopted for the control of the fluid-structure coupling are in good agreement with those of the literature results.

Linear implicit finite difference methods with energy conservation property for space fractional Klein-Gordon-Zakharov system

In this article, novel linearized implicit difference schemes with energy conservation property for fractional Klein-Gordon-Zakharov system are constructed and analyzed. The important feature of the article is that new auxiliary equations ∂u∂t=−v and ∂ϕ∂t=∂2ψ∂x2 are introduced to transform the original fractional Klein-Gordon-Zakharov system into an equivalent system of equations exactly. Especially, two kinds of efficacious difference operators, the leap-frog and modified Crank-Nicolson methods are respectively utilized to establish the linearized implicit difference schemes with energy conservation property for simulating the propagation of transformed equations. And above all, by employing the discrete energy method, we have proven that the constructed difference algorithms enjoy the convergence order of O(Δt2+h2) and O(Δt2+h4) in L∞- and L2-norms, without imposing any restrictive conditions on the grid ratio compared with the existing literature. Two numerical examples are carried out to investigate the physical behaviors of the wave propagation and substantiate the effectiveness of the suggested schemes.

Heat Transfer by Natural Convection in a Square Enclosure Containing PCM Suspensions

Research on using phase change material (PCM) suspension to improve the heat transfer and energy storage capabilities of thermal systems is booming; however, there are limited studies on the application of PCM suspension in transient natural convection. In this paper, the implicit finite difference method was used to numerically investigate the transient and steady-state natural convection heat transfer in a square enclosure containing a PCM suspension. The following parameters were included in the simulation: aspect ratio of the physical model = 1, ratio of the buoyancies caused by temperature and concentration gradients = 1, Raleigh number (RaT) = 103–105, Stefan number (Ste) = 0.005–0.1, subcooling factor (Sb) = 0–1.0, and initial mass fraction (or concentration) of PCM particles (ci) = 0–0.1. The results showed that the use of a PCM suspension can effectively enhance heat transfer by natural convection. For example, when RaT = 103, Ste = 0.01, ci = 0.1, and Sb = 1, the steady-state natural convection heat transfer rate inside the square enclosure can be improved by 70% compared with that of pure water. With increasing Sb, the Nusselt number can change nonlinearly, resulting in a local optimal value.

Non-linear radiation effect on dusty fluid flow near a rotating blunt-nosed body

A two-phase, dusty boundary-layer flow over an isothermally heated rotating hemisphere is studied for the laminar, incompressible fluid. The contribution of non-linear thermal radiation is analyzed by using the Rosseland diffusion approximation model. The difficulty of having a unified mathematical treatment of the above problem is due to the complex nature of the governing partial differential equations, which arises from the transverse curvature of the round-nosed body, buoyant force, and thermal radiation. Therefore, this study presents a theoretical analysis of a dusty flow and heat transfer over a rotating axisymmetric round-nosed body (for example, hemisphere). In industries, for instance, vehicle industry, blunt-nosed shapes (or round-nosed body) are preferred because it provides better thermal management. The governing equations, along with their appropriate boundary conditions, are numerically solved using the iterative, implicit finite difference method. Numerical solutions are illustrated in the form of streamwise velocity distribution, spanwise velocity distribution, temperature distribution, and skin friction and heat transfer coefficients. The numerical scheme is validated through comparison with the benchmark solutions available in the literature. Solutions obtained here are discussed for i) pure and dusty air and ii) pure and dusty water. A wide range of thermal radiation parameter is tested for the above two cases, and it is found that skin friction coefficient achieves asymptotic behavior for [Formula: see text]; however, heat transfer coefficient increases considerably as much as we strengthen the parameter Rd. Further, momentum and thermal boundary layer thicknesses also increase for [Formula: see text]. Computations are also performed for the buoyancy parameter, λ, ranging from 0 to [Formula: see text], which respectively covers the solution of pure forced convection ( λ = 0) and pure natural convection ([Formula: see text]) boundary layer on a rotating hemisphere.

The studies of the linearly modified energy-preserving finite difference methods applied to solve two-dimensional nonlinear coupled wave equations

In this paper, four linearly energy-preserving finite difference methods (EP-FDMs) are designed for two-dimensional (2D) nonlinear coupled sine-Gordon equations (CSGEs) and coupled Klein-Gordon equations (CKGEs) using the invariant energy quadratization method (IEQM). The 1st EP-FDM is designed by first introducing two auxiliary functions to rewrite the original problems into the new system only including the 1st-order temporal derivatives, and then applying Crank-Nicolson (C-N) method and 2nd-order centered difference methods for the discretizations of temporal and spatial derivatives, respectively. The 2nd EP-FDM is directly devised based on the uses of 2nd-order centered difference methods to approximate 2nd-order temporal and spatial derivatives. The 1st and 2nd EP-FDMs need numerical solutions of the algebraic system with variable coefficient matrices at each time level. By modifying the 2nd EP-FDM, the 3rd EP-FDM, which is implemented by computing the system of algebraic equations with constant coefficient matrices at each time level, is developed. Finally, an energy-preserving alternating direction implicit (ADI) finite difference method (EP-ADI-FDM) is established by a combination of ADI method with the 3rd EP-FDM. By using the discrete energy method, it is shown that they are all uniquely solvable, and their solutions have a convergent rate of ${\mathscr{O}}(\varDelta t^{2}+{h^{2}_{x}}+{h^{2}_{y}})$ in H1-norm and satisfy the discrete conservative laws. Numerical results show the efficiency and accuracy of them.

A Front-Fixing Implicit Finite Difference Method for the American Put Options Model

In this paper, we present an implicit finite difference method for the numerical solution of the Black–Scholes model of American put options without dividend payments. We combine the proposed numerical method by using a front-fixing approach where the option price and the early exercise boundary are computed simultaneously. We study the consistency and prove the stability of the implicit method by fixing the values of the free boundary and of its first derivative. We improve the accuracy of the computed solution via a mesh refinement based on Richardson’s extrapolation. Comparisons with some proposed methods for the American options problem are carried out to validate the obtained numerical results and to show the efficiency of the proposed numerical method. Finally, by using an a posteriori error estimator, we find a suitable computational grid requiring that the computed solution verifies a prefixed error tolerance.

Transient Convective Heating Transport of the Micropolar Fluid Flow Between Asymmetric Channel with Activation Energy

Here modelling and computations are presented to introduce the concept of activation energy involved in the micropolar fluid between the porous vertical channel with the boundary conditions of third kind. The fluid is considered to be gray, absorbing-emitting but non scattering medium. The resulting system of equations are solved numerically by Crank-Nicolson implicit finite difference method. The effect of various physical parameter such as activation energy, micropolar parameter, reaction rate, Reynolds number, Schmidt number, heat and mass transfer Biot numbers on the velocity, temperature and concentration field are discussed by means of pictorial representation.

BPM-Matlab: an open-source optical propagation simulation tool in MATLAB.

We present the use of the Douglas-Gunn Alternating Direction Implicit finite difference method for computationally efficient simulation of the electric field propagation through a wide variety of optical fiber geometries. The method can accommodate refractive index profiles of arbitrary shape and is implemented in a tool called BPM-Matlab. We validate BPM-Matlab by comparing it to published experimental, numerical, and theoretical data and to commercially available state-of-the-art software. It is user-friendly, fast, and is available open-source. BPM-Matlab has a broad scope of applications in modeling a variety of optical fibers for diverse fields such as imaging, communication, material processing, and remote sensing.

Development of a computational approach for a space–time fractional moving boundary problem arising from drug release systems

This paper presents an iterative procedure based on an implicit finite difference method to solve a mathematical model of drug delivery from a planar matrix with a moving boundary condition. This model includes the diffusion equation with space–time fractional-order derivatives. We establish the stability and convergence analysis of the method. We compare the numerical results with the scale-invariant and the homotopy perturbation solutions for different space–time-fractional orders and the problem parameters.

Reconstruction of the local volatility function using the Black–Scholes model

In this paper, we propose a robust and accurate numerical algorithm to reconstruct a local volatility function using the Black–Scholes (BS) partial differential equation (PDE). Using the BS PDE and given market data, option prices at strike prices and expiry times, a time-dependent local volatility function is computed. The proposed algorithm consists of the following steps: (1) The time-dependent volatility function is computed using a recently developed method; (2) A Monte Carlo simulation technique is used to find the effective region which has a strong influence on option prices; and we partition the effective domain into several sub-regions and define a local volatility function based on the time-dependent volatility function on the sub-regions; and (3) Finally, we calibrate the local volatility function using the fully implicit finite difference method and the conjugate gradient optimization algorithm. We demonstrate the robustness and accuracy of the proposed local volatility reconstruction algorithm using manufactured volatility surface and real market data.

Evolution of corrosion damage in an aluminum alloy: an experimental and numerical study

In this paper, the results of a recent study on the environment-induced damage behavior of aluminum (Al) alloy 2024-T3, which is often a preferred choice for use in a spectrum of aircraft structural applications, are presented and discussed. Accelerated corrosion tests by way of exposure of test specimens of the alloy to an aggressive aqueous environment were carried out at ambient temperature (27°C). A numerical approach, based on the meshless peridynamic method, was developed to study the morphology of damage caused by the aqueous environment by way of pitting. To improve the numerical accuracy while concurrently reducing the solution time, an implicit finite-difference method was developed to solve the peridynamic control equation. The numerical algorithm was implemented in a Matlab program and was applied to simulate the evolution of pits resulting as a direct consequence of sustained exposure to the aqueous environment. The numerical predictions accord reasonably well with the experimental findings and results, thereby providing adequate proof that the simulation algorithm based on the meshless peridynamic method is effective at predicting both the nucleation and growth of the damage in the form of pits arising as a direct consequence of exposure to an aggressive aqueous environment.

Mixed convective flow and heat transfer of hybrid nanofluid impinging obliquely on a vertical cylinder

Flow and heat transportation in hybrid nanofluid, impinging obliquely on a vertical cylinder, are investigated numerically in this communication. Silver and aluminium oxide nanoparticles having stable and relatively high thermal conductivity, with pure water as base fluid, are used in the analysis. The non-linear coupled partial differential equations are transformed into ordinary differential equations using suitable transformations. An implicit finite difference method is used to approximate the velocities and temperature for particular values of the various parameters and their effects are emphasised graphically. The tabular data are documented to compare the heat transfer rate for pure water silver/water nanofluid, aluminium/water nanofluid and silveraluminiumwater hybrid nanofluid. It is observed that heat transfer rate increases by increasing the curvature of the stretching cylinder and solid volume fraction of nanoparticles and for hybrid nanofluid it has a larger magnitude than that of water, nanofluid and nanofluid.

A robust numerical solution to a time-fractional Black\u2013Scholes equation

Dividend paying European stock options are modeled using a time-fractional Black–Scholes (tfBS) partial differential equation (PDE). The underlying fractional stochastic dynamics explored in this work are appropriate for capturing market fluctuations in which random fractional white noise has the potential to accurately estimate European put option premiums while providing a good numerical convergence. The aim of this paper is two fold: firstly, to construct a time-fractional (tfBS) PDE for pricing European options on continuous dividend paying stocks, and, secondly, to propose an implicit finite difference method for solving the constructed tfBS PDE. Through rigorous mathematical analysis it is established that the implicit finite difference scheme is unconditionally stable. To support these theoretical observations, two numerical examples are presented under the proposed fractional framework. Results indicate that the tfBS and its proposed numerical method are very effective mathematical tools for pricing European options.

Variable-order optimal implicit finite-difference schemes for explicit time-marching solutions to wave equations

The time-space-domain finite-difference method (FDM) is widely used in forward modeling of wave equations. Conventional explicit FDMs (EFDMs) (with high order in space and the second order in time) would result in apparent temporal and spatial dispersion for high frequencies and large time steps. Moreover, the saturation effect of Taylor expansion seriously restricts the improvement of bandwidth coverage and efficiency of EFDMs. We have developed a variable-order optimization scheme of the implicit FDM (IFDM) to improve the computational efficiency and numerical accuracy of forward modeling. Then, we applied time-dispersion transforms (TDTs) to filter out the temporal dispersion generated by the second-order temporal approximations. Our method greatly alleviates the saturation effect of the high-order spatial finite-difference (FD) operators. Dispersion analysis indicates that the optimized coefficients of the IFDM based on the Remez algorithm can achieve the widest bandwidth (close to the Nyquist wavenumber), which corresponds to the shortest length of the spatial FD operator under a given error threshold. Our method has great potential to approach the highest spectral accuracy but with minimal increase in computational cost. Numerical experiments indicate that the combination of the variable-order optimization of IFDM and TDTs can significantly reduce numerical dispersion. Compared with traditional methods, our scheme is more advantageous for numerical simulation of large-scale geologic models because it has the least amount of calculation burden under the same accuracy requirements.

Inverse estimation of the time-dependent wall temperature in stagnation region of an annular jet on a cylinder rod using Levenberg–Marquardt method

For the first time, a numerical solution code, based on Levenberg–Marquardt method is presented for solving non-linear problem of inverse heat transfer in axisymmetric stagnation flow impinging on a cylinder rod to determine time-dependent wall temperature by temperature distribution at a specific point in the fluid region. Also, the effect of noisy data on the final result has been studied. For this purpose, the numerical solution of the dimensionless temperature and the convective heat transfer in a radial incompressible flow on a cylinder axis is carried out as a direct problem. In the direct problem, the free stream is steady with an initial flow strain rate of $$\overline{k}$$ . Using similarity variable and appropriate transformations, momentum and energy equations are converted into semi-similar equations. The new equation systems are then discretized using an implicit finite difference method and solved by applying the tridiagonal matrix algorithm (TDMA). The wall temperature is then estimated by applying the Levenberg–Marquardt parameter estimation approach. This technique is an iterative approach based on minimizing the least-square summation of the error values, the error being the difference between the estimated and measured temperatures. Results of the inverse analysis indicate that the Levenberg–Marquardt algorithm is an efficient and acceptably stable technique for estimating wall temperature in axisymmetric stagnation flow. The maximum value of the sensitivity coefficient is related to the estimation of polynomial wall temperature and its value is 0.1952 also the minimum value of the sensitivity coefficient is 8.62 × 10–6 which is related to the triangular wall temperature. The results show that the parameter estimation error in calculating the triangular and trapezoidal wall temperature is greater than the others because the maximum value of RMS error is obtained for these two cases, which are 0.451 and 0.479, respectively, the reason for the increase in error in estimating these functions is the existence of points where the first derivative of the function does not exist. This method also exhibits considerable stability for noisy input data.

Micropolar medium in a funnel-shaped crusher

In this paper, the solution to a coupled flow problem for a micropolar medium undergoing structural changes is presented. The structural changes occur because of a grinding of the medium in a funnel-shaped crusher. The standard macroscopic equations for mass and linear momentum are solved in combination with a balance equation for the microinertia tensor containing a production term. The constitutive equations of the medium describe a linear viscous material with a viscosity coefficient depending on the characteristic particle moment of inertia, the so-called microinertia. A coupled system of equations is presented and solved numerically in order to determine the distribution of the fields for velocity, pressure, viscosity coefficient, and microinertia in all points of the continuum. The numerical solution to this problem is found by using the implicit finite difference method and the upwind scheme.

Numerical Investigation of the Unsteady Flows in Hydropower Plants

One of the most important hazards that threatens the stability of power plant buildings is the phenomenon of water hammer, which can occur in the Penstock pipe of a turbine due to the rapid opening and closing of a valve. Fluid Descriptive Equations in this situation, there are two hyperbolic partial nonlinear partial differential equations that are very difficult and complex to solve analytically and are possible only for very simple conditions. In this study, by examining the two numerical methods of characteristic lines and implicit finite difference with Verwy & Yu schema, which are widely used in the analysis of instabilities, their disadvantages and advantages are clearly clarified and a suitable comparison basis for use. They should be provided in different conditions in hydropower plant. The results of the characteristic method in terms of maximum and minimum pressure show more and less values than the implicit finite difference method. In the characteristic method, perturbations and fast wave fronts are presented with more accuracy than the implicit finite difference method. At points near the upstream, downstream and middle boundaries, the accuracy of the characteristic method in presenting pressure and flow fluctuations is higher than the implicit finite difference method. In the characteristic method, it is recommended not to use certain time steps and try as much as possible avoid interpolation by selecting the appropriate time step. The results of examining the amount of changes in coefficient of friction in both methods show that it is not correct to take its value constant (proportional to the value obtained in stable conditions) and coefficient of friction should be calculated in proportion to changes in velocity at different times and used in the governing equation.

A photo-thermal interaction in semi-conductor medium with cylindrical cavity by analytical and numerical methods

In this work, we compare the analytical solutions with the numerical solutions for photothermal interactions in semiconductor medium containing cylindrical cavity. This paper is devoted to a study of the photothermal interactions in semiconductor medium in the context of the coupled photo-thermal model. The basic equations are formulated in the domain of Laplace transform and the eigenvalue scheme are used to get the analytical solutions. The numerical solution is obtained by using the implicit finite difference method (IFDM). A comparison between the analytical solution and the numerical solutions are obtained. It is found that the implicit finite difference method (IFDM) is applicable, simple and efficient for such problems.