Explicit Finite Difference Method Research Articles

General wetting energy boundary condition in a fully explicit nonideal fluids solver.

We present an explicit finite-difference method to simulate the nonideal multiphase fluid flow. The local density and momentum transport are modeled by the Navier-Stokes equationsand the pressure is computed by the van der Waals equationof the state. The static droplet and the dynamics of liquid-vapor separation simulations are performed as validations of this numerical scheme. In particular, to maintain the thermodynamic consistency, we propose a general wetting energy boundary condition at the contact line between fluids and the solid boundary. We conduct a series of comparisons between the current boundary condition and the constant contact angle boundary condition as well as the stress-balanced boundary condition. This boundary condition alleviates the instability induced by the constant contact angle boundary condition at θ≈0 and θ≈π. Using this boundary condition, the equilibrium contact angle is correctly recovered and the contact line dynamics are consistent with the simulation by applying a stress-balanced boundary condition. Nevertheless, unlike the stress-balanced boundary condition for which we need to further introduce the interface thickness parameter, the current boundary condition implicitly incorporates the interface thickness information into the wetting energy.

A Comparative Study of the Explicit Finite Difference Method and Physics-Informed Neural Networks for Solving the Burgers’ Equation

The Burgers’ equation is solved using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). We compare our numerical results, obtained using the EFDM and PINN for three test problems with various initial conditions and Dirichlet boundary conditions, with the analytical solutions, and, while both approaches yield very good agreement, the EFDM results are more closely aligned with the analytical solutions. Since there is good agreement between all of the numerical findings from the EFDM, PINN, and analytical solutions, both approaches are competitive and deserving of recommendation. The conclusions that are provided are significant for simulating a variety of nonlinear physical phenomena, such as those that occur in flood waves in rivers, chromatography, gas dynamics, and traffic flow. Additionally, the concepts of the solution techniques used in this study may be applied to the development of numerical models for this class of nonlinear partial differential equations by present and future model developers of a wide range of diverse nonlinear physical processes.

Theoretical and experimental investigation of the steady-state power distribution in multimode step-index plastic optical fibers

In this work we have examined theoretically and experimentally the influence of mode coupling on steady-state distribution (SSD) in step-index plastic optical fiber (SI POF). Numerical and analytical solution of the PFE under steady-state conditions are compared to our experimental results. A numerical solution of the power flow equation (PFE) has been obtained by the explicit finite difference method (EFDM). We demonstrated that SSD in the SI POF is achieved at length of 140 m, which is much shorter length compared to the case with silica fibers. The reported results appear to be particularly important since none of the main optical properties of an optical fiber (attenuation, bandwidth) can be accurately determined until the transmitted optical power reaches its SSD, due to strong influence of mode coupling phenomena that occur during the transmission process. These results are important for employment of the SI POFs as a part of communication or sensory systems.

Excavation analysis of large-scale slope considering effects of folded structure and in-situ stress

A cut slope in Rajamandala, Indonesia exhibited unexpected large-scale deformation during the excavation process which was revealed by the clinometers attached to the preventive piles. A geological survey indicated that the mudstone layer composing the slope had been folded by tectonic movement and was highly weathered and eroded at the surface. The above internal factors could have resulted in the significant deformation of the slope, although few studies have focused on the influence of folded structures and horizontal in-situ stress on the deformation behavior of cut slopes. Therefore, the objective of this paper is to clarify the influence of the folded structure and anisotropic in-situ stress state induced by tectonic movement on the deformation behavior of the aforementioned cut slope. To achieve this clarification, numerical analyses were performed based on the explicit finite difference method included in the FLAC2D software using different geological structures and in-situ stress states. Consequently, it was revealed that such folded structures can lead to larger displacement in cut slopes when compressive stress concentration occurs at the bottom of the fold. Moreover, an anisotropic in-situ stress state can result in shear failure at the foot of the mudstone and reproduce the displacement of piles that possess shapes as close as possible to those observed at the site. These analytical results confirmed that the folded structure and anisotropic in-situ stress state were the inevitable factors in the large deformation of the cut slope in Rajamandala, Indonesia.

A fast, efficient, and explicit phase-field model for 3D mesh denoising

In this paper, we propose a fast and efficient explicit three-dimensional (3D) mesh denoising algorithm that utilizes the Allen–Cahn (AC) equation with a fidelity term. The phase-field model is used to describe the characteristics of both the surface and interior of an object, allowing us to represent the 3D mesh model with noise using a phase-field function. By using the phase separation property of the AC equation and the fidelity term, the model can effectively preserve the original structures and features during the smoothing process, even in the presence of noise in various regions. The modified AC equation is numerically discretized using the explicit finite difference method, where the values at neighboring grid points are used as Dirichlet boundary conditions. Because the algorithm is local and explicit, it guarantees both effective denoising of 3D mesh models and rapid implementation speed. To validate the efficacy of the proposed algorithm, we conduct various computational experiments. Furthermore, we propose an implicit-explicit numerical scheme using the Crank–Nicolson method to address the denoising problem of 3D mesh models and perform related experiments.

Simulation of Heat Storage and Heat Regeneration in Phase Change Material

The present study explores numerically the energy storage and energy regeneration during Melting and Solidification processes in Phase Change Materials (PCM) used in Latent Heat Thermal Energy Storage (LHTES) systems. Transient two-dimensional (2-D) conduction heat transfer equations with phase change have been solved utilizing the Explicit Finite Difference Method (FDM) and Grid Generation technique. A Fortran computer program was built to solve the problem. The study included four different Paraffin's. The effects of container geometrical shape, which included cylindrical and square sections of the same volume and heat transfer area, the container volume or mass of PCM, variation of mass flow rate of heat transfer fluid (HTF), and temperatures difference between PCM and HTF were all investigated. Results showed that the PCMs in a cylindrical container melt and solidify quicker than the square container. The increase in mass flow rate and/or temperature difference decreases the time required for complete phase change. Paraffin's solidify quicker than they melt and store more energy than they release

Numerical analysis for fractional Bratu type equation with explicit and implicit methods

Bratu‐type equation has wide applications in physics, finance, and other fields. Also, fractional analysis has been a popular topic recently. It is difficult to find the analytical solution once we add the fractional operator on the left side of the equation. In this work, we apply explicit and implicit finite difference methods to solve the fractional Bratu equation. From the numerical examples, we will see that the explicit solution is not accurate, and the implicit solution based on Newton's method is efficient in some sense. Some theoretical results are mentioned in the paper.

Heat Transfer in a Silo Containing Rice: Solution by the Explicit Method of the Finite Differences with Approximations of Order 2 and 4 for a One-Dimensional Transient Model

Rice is one of the most consumed cereals in the world, and the cultivation of this crop has significant relevance in the southern region of Brazil. When subjected to inadequate conditions of temperature and humidity, rice becomes susceptible to attacks from pests and fungi and, therefore, care in the storage process is of paramount importance, since this is largely responsible for the quality of the harvest. Such care allows the food to arrive without harm to the consumer. In this sense, the mathematical modeling, among numerous possibilities, allows for evaluating the internal temperature of a silo and, through this, taking preventive measures so that the grains maintain their quality. The objective of this work is to model the heat transfer process in a silo prototype containing rice in husk through the explicit finite difference method for a one-dimensional and transient model considering two approaches centered on the spatial derivative: error of order 2 and 4. In addition, the thermal diffusivity of the grain with average value was analyzed. The results obtained by the solutions were analyzed through graphs and statistical indexes comparing with the experimental data of the literature, and the computer simulation was performed through the Google Colab platform. The chosen methodology proved effective for the work, and the predicted temperatures for the approximations of order 2 and 4 denote similarities both graphically and in the precision of the statistical indexes.

Stability of Finite Difference Solution of Time-Dependent Schrodinger Equations

In this paper, the stability of finite difference methods for time-dependent Schrodinger equation with Dirichlet boundary conditions on a staggered mesh was considered with explicit and implicit discretization. Using the matrix representation for the numerical algorithm, it is shown that for the explicit finite difference method, the solution is conditionally stable while it becomes unconditionally stable for implicit finite difference methods. A 1D Harmonic Oscillator problem shall be used to illustrate this behaviour.

High bandwidth performance of multimode graded-index microstructured polymer optical fibers

The investigation of the bandwidth in multimode graded-index microstructured polymer optical fiber (GI mPOF) with a solid core is proposed using a modal diffusion approach. For a variety of launch radial offsets of multimode GI mPOF, bandwidth is reported by numerically solving the time-dependent power flow equation (TD PFE) using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). The decline in bandwidth with fiber length becomes slower at fiber lengths close to the coupling length Lc at which an equilibrium mode distribution (EMD) is attained, showing that mode coupling enhances bandwidth at longer fiber lengths. As fiber length is increased, bandwidth approaches complete independence from radial offset, suggesting the steady-state distribution (SSD) has been reached. We compare multimode GI mPOF performance in terms of bandwidth with that of traditional multimode GI POFs made of the same material. Higher bandwidth performance and quicker bandwidth improvement are displayed by the GI mPOF. To enhance fiber performance in GI mPOF links, such a fiber characterization can be used.

Effects of Thermodiffusion and Chemical Reaction on Magnetohydrodynamic-Radiated Unsteady Flow Past an Exponentially Accelerated Inclined Permeable Plate Embedded in a Porous Medium

A finite difference computational study is conducted to assess the influence of thermodiffusion and chemical reaction on unsteady free convective radiated magnetohydrodynamic flow past an exponentially accelerated inclined permeable plate embedded in a saturated porous medium of uniform permeability with variable temperature and concentration. The governing nondimensional set of coupled nonlinear partial differential equations with related initial and boundary conditions are solved numerically by using the accurate and efficient DuFort–Frankel’s explicit finite difference method. The physical features of fluid flow, heat, and mass transfer under the influence of the magnetic field, angle of inclination, plate acceleration, radiation, heat source/sink, thermodiffusion, chemical reaction, and time are scrutinized by plotting graphs and then discussed in detail. It was found that the effective magnetic field and angle of inclination tend to decline the fluid motion, whereas the reverse result is detected by increasing the porosity parameter and plate acceleration. The velocity and temperature of the fluid lessen with increasing the radiation parameter. The effect of thermodiffusion raises the fluid velocity and concentration, whereas a chemical reaction has the opposite impact. The Nusselt number increases with increasing the radiation parameter and time. Increasing chemical reaction and time causes to improve the Sherwood number. This kind of problem finds momentous industrial applications such as food processing, polymer production, inclined surfaces in a seepage flow, and design of fins.

Qualitative Numerical Analysis of a Free-Boundary Diffusive Logistic Model

A two-dimensional free-boundary diffusive logistic model with radial symmetry is considered. This model is used in various fields to describe the dynamics of spreading in different media: fire propagation, spreading of population or biological invasions. Due to the radial symmetry, the free boundary can be treated by a front-fixing approach resulting in a fixed-domain non-linear problem, which is solved by an explicit finite difference method. Qualitative numerical analysis establishes the stability, positivity and monotonicity conditions. Special attention is paid to the spreading–vanishing dichotomy and a numerical algorithm for the spreading–vanishing boundary is proposed. Theoretical statements are illustrated by numerical tests.

Comparison of third-order fractional partial differential equation based on the fractional operators using the explicit finite difference method

In this research paper, the third-order fractional partial differential equation (FPDE) in the sense of the Caputo fractional derivative and the Atangana-Baleanu Caputo (ABC) fractional derivative is investigated for the first time. The importance of the proposed problem is more general than the third-order linear time-varying systems model. This fractional partial differential equation is converted to the abstract form of the ordinary differential equation at β = 1. A first-order finite difference scheme is created for the converted problem. The stability inequality of the (FDS) is demonstrated for this abstract form problem using Cardano's relation method in Hilbert space. The explicit finite difference method (EFDM) is used to obtain an approximate solution of the third-order fractional partial differential equation (FPDE) based on the Caputo fractional derivative and the (ABC) fractional derivative of order β∈(0,1]. Concerning these equations, the obtained approximate solutions using the proposed method are contrasted with the exact solution. Finally, the simulation for the third-order (FPDE) based on the fractional derivatives of Caputo and (ABC) for both the exact and approximate solutions is given.

Simulation of hydrogen diffusion along the axial direction in zirconium cladding tube during dry storage

This study presents an updated mHNGD (modified hydrogen nucleation-growth-dissolution) model by improving atomic fraction of solid solution hydrogen with the use of instantaneous solid solution concentration and carefully selecting hydrogen solubility curves specific to reactor cladding materials. An explicit finite difference method was applied to simultaneously solve diffusion and phase change kinetics and ensure stable solution. The code validation with past experiments demonstrate improved accuracy. A prototypic hottest fuel pin in assembly average discharge burnup of 50 MWd/kgU of CASTOR® V/21 dry storage cask was simulated. The result of the reference case shows that the increase in the hydrogen amount at the upper end of the cladding due to hydrogen migration after ∼20 years of dry storage was limited to ∼50 wt.ppm with the initial Peak Cladding Temperature (PCT) of 394 ℃. The aspirational increase in discharge burnup to 70 MWd/kgU results in the upper end hydrogen increase to ∼80 wt.ppm with the same initial PCT. This study demonstrates that hydrogen migration along the axial direction during dry storage practically poses no safety challenge as its amount is limited and can be further alleviated by extending the wet storage period.

Numerical Solution for Fuzzy Time-Fractional Cancer Tumor Model with a Time-Dependent Net Killing Rate of Cancer Cells.

A cancer tumor model is an important tool for studying the behavior of various cancer tumors. Recently, many fuzzy time-fractional diffusion equations have been employed to describe cancer tumor models in fuzzy conditions. In this paper, an explicit finite difference method has been developed and applied to solve a fuzzy time-fractional cancer tumor model. The impact of using the fuzzy time-fractional derivative has been examined under the double parametric form of fuzzy numbers rather than using classical time derivatives in fuzzy cancer tumor models. In addition, the stability of the proposed model has been investigated by applying the Fourier method, where the net killing rate of the cancer cells is only time-dependent, and the time-fractional derivative is Caputo's derivative. Moreover, certain numerical experiments are discussed to examine the feasibility of the new approach and to check the related aspects. Over and above, certain needs in studying the fuzzy fractional cancer tumor model are detected to provide a better comprehensive understanding of the behavior of the tumor by utilizing several fuzzy cases on the initial conditions of the proposed model.

Approximating the Navier-Stokes Equations with the Explicit Finite Difference Method to Solve the Lid-driven Cavity Flow Problem

The Navier-Stokes equations, one of the seven Millennium Problems, are a set of 2nd-order partial differential equations that are near impossible to solve. Instead of solving it, this paper uses the finite difference method to convert the differential terms in the Navier-Stokes Equations into algebraic equations to generate approximate solutions. The procedure to use the explicit finite difference method is outlined and proved in the context of solving the lid-driven cavity flow problem. The solution discussed is then applied in Mat lab and its accuracy and speed is evaluated. The finite difference method is then evaluated and its applications are outlined.

Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation

In this study, we present the stability analysis of a fully explicit finite difference method (FDM) for solving the Allen–Cahn (AC) equation. The AC equation is a second-order nonlinear partial differential equation (PDE), which describes the antiphase boundaries of the binary phase separation. In the presented stability analysis, we consider the explicit Euler method for the temporal derivative and second-order finite difference in the space direction. The explicit scheme is fast and accurate because it uses a small time step, however, it has a temporal step constraint. We analyze and compute that the explicit time step constraint formula guarantees the discrete maximum principle for the numerical solutions of the AC equation. The numerical stability of the explicit scheme automatically holds when we use the time satisfying the discrete maximum principle. The computational numerical experiments demonstrate the stability, discrete maximum principle, and accuracy of the explicit scheme for the constrained time step. Furthermore, it is shown that the time step obtained is not severe restriction when we consider the temporal accuracy.

Thermo-Mechanical Modeling of High-Strength Concrete Column Subjected to Moderate Case Heating Scenario in a Fire

This paper presents a numerically developed computer model to simulatethe thermal behavior and evaluate the mechanical performance of a fixedend loaded loaded High Strength Concrete Column (HSCC), subjectedto Moderate Case Heating Scenario (MCHS), in a hydrocarbon fire. Thetemperature distribution within the mid-height cross-sectional area of thecolumn was obtained to determine the thermal and mechanical responsesas a function of temperature. The governing two-dimensional transient heattransfer partial differential equation (PDE), was converted into a set of ordinary algebraic equations, subsequently, integrated numerically by usingthe explicit finite difference method, (FDM). A computer program, VisualBasic for Applications (VBA), was then developed to solve the set of ordinary algebraic equations by implementing the boundary as well as initialconditions. The predictions of the model were validated against experimental data from previous studies. The general behavior of the model as wellas the effect of the key model parameters were investigated at length in thereview. Finally, the reduction in the column’s compression strength and themodulus of elasticity was estimated using correlations from existing literature. And the HSCC failure load under fire conditions was predicted usingthe Rankine formula. The results showed that the model predictions of thetemperature distribution within the concrete column are in good agreementwith the experimental data. Furthermore, the increase in temperature ofthe reinforced concrete column, (RCC), due to fire resulted in a significantreduction in the column compression strength and considerably acceleratesthe column fire failure load.

Explicit Richardson extrapolation methods and their analyses for solving two-dimensional nonlinear wave equation with delays

<abstract><p>In this study, we construct two explicit finite difference methods (EFDMs) for nonlinear wave equation with delay. The first EFDM is developed by modifying the standard second-order EFDM used to solve linear second-order wave equations, of which stable requirement is accepted. The second EFDM is devised for nonlinear wave equation with delay by extending the famous Du Fort-Frankel scheme initially applied to solve linear parabolic equation. The error estimations of these two EFDMs are given by applying the discrete energy methods. Besides, Richardson extrapolation methods (REMs), which are used along with them, are established to improve the convergent rates of the numerical solutions. Finally, numerical results confirm the accuracies of the algorithms and the correctness of theoretical findings. There are few studies on numerical solutions of wave equations with delay by Du Fort-Frankel-type scheme. Therefore, a main contribution of this study is that Du Fort-Frankel scheme and a corresponding new REM are constructed to solve nonlinear wave equation with delay, efficiently.</p></abstract>

Bubble rising and interaction in ternary fluid flow: a phase field study.

Bubble-droplet interaction is essential in the gas-flotation technique employed in wastewater treatment. However, due to the limitations of experimental methods, the details of the fluid flow involved have not been fully understood. Therefore, a phase field model for a three-phase flow was developed to study the rise of a single bubble and bubble-droplet interactions. The fluid-fluid interfaces are tracked by the Cahn-Hilliard equation, which is coupled with the Navier-Stokes equations with an equivalent volumetric force substituted for interfacial tensions. The model was discretized using an explicit finite difference method on a half staggered grid, and the pressure velocity coupling was tackled using the projection method. The in-house code was written in Fortran and run with the help of OpenMP, a shared memory parallelism. The model was validated against experiments with gratifying agreement achieved. Bubble-droplet interaction was simulated in two distinct situations: the first features a gas bubble crossing the interface between two other phases, and the second features a gas bubble chasing from behind an oil droplet in a surrounding fluid of the third phase.