Explicit Finite Difference Method Research Articles

A structure-preserving explicit numerical scheme for the Allen–Cahn equation with a logarithmic potential

This paper presents a stability analysis of a structure-preserving explicit finite difference method (FDM) for the Allen–Cahn (AC) equation with a logarithmic potential that has two arguments. Firstly, we compute the temporal step constraint that guarantees that if the initial condition is bounded by the two arguments of the minimum, then the numerical solutions are always bounded by them, i.e., the explicit numerical scheme satisfies the maximum principle. Secondly, we compute the temporal step constraint that guarantees that the discrete total energy of the system is non-increasing over time. To validate the preservation of the maximum principle and the decrease in discrete total energy, we perform numerical experiments.

Comparative simulation of nonlinear radiative nano casson and maxwell fluids with periodic magnetic force and sensitivity analysis

This study investigated cyclic magneto-hydrodynamic radiative effects in Casson and Maxwell fluids, including nonlinear radiation and Arrhenius activation energy. It promotes non-Newtonian fluid use in diverse fields like industry, manufacturing, sciences, medicine, and engineering. Using boundary layer approximations, non-dimensional equations are formulated. For numerical solutions, widely recognized explicit finite difference method (EFDM) has been utilized. To ensure the robustness of EFDM results, stability and convergence tests are performed. Exploration involve a detailed sensitivity analysis by using RSM, offering a thorough understanding of influential parameters. These analyses explore complex interactions among physical parameters, affecting Nusselt number, skin friction, and Sherwood number. Maxwell fluid's velocity is more affected by periodic magnetic force than Casson fluid, during the presence of nonlinear radiation. Additionally, nonlinear thermal radiation has a greater impact on temperature and concentration profiles compared to linear radiation for both fluids. Moreover, Casson fluid has a stronger influence on the average heat transfer rate compared to Maxwell fluid with nonlinear thermal radiation which is 8.6 % greater than the Maxwell fluid. On the other hand, at constant thermal radiation (Ra), due to decrease of Brownian motion (Nb), the rate of heat transfer is reduced by 1.2 % and 0.3 % respectively for Maxwell and Casson fluid. Also, for thermophoresis parameter (Nt), this rate is reduced by 2 % and 1.6 % respectively. The investigation also revealed that the Ra exhibits a positive sensitivity towards average Nusselt number, while Nb and Nt are displayed a negative sensitivity.

Identification of spatially distributed rheological parameters for viscoplastic gravity currents

The rheology of a fluid is often difficult to measure in situ and in real time. This work investigates whether it is possible to recover rheological information from direct field observations. We present a closed-form analytical solution for inverse problem of identification of the mobility coefficient in gravity-driven fluid flowed down an inclined plane. The lubrication approximation is used to model the three-dimensional thin film of fluid and the governing equation is solved numerically by the explicit finite difference method to calculate the fluid thickness. We show in this study that using very straightforward measurements of fluid thickness at only a few points and only two successive instants, the mobility coefficient of fluid flow can be estimated accurately. Then as an important application in Bingham fluids we show that the knowledge of the mobility coefficient together with velocity measurement at only one point and instant enable us to estimate temperature-dependent viscosity and yield stress of the Bingham fluid. Unlike other methods, the proposed method allows the local identification of the rheology which is particularly useful when the rheological parameters are functions of the local temperature and/or composition. The proposed inverse analysis is a non-iterative approach with negligible computational cost since our analysis revealed closed form formulae relating the rheological parameters to the local flow depth, its derivatives, and the local free surface velocity.

Influence of induced magnetic field and chemically reacting on hydromagnetic Couette flow of Jeffrey fluid in an inclined channel with variable viscosity and convective cooling: A Caputo derivative approach

In this study, the influence of induced magnetic field and chemically reacting on hydromagnetic generalized Couette flow of Jeffrey fluid in an inclined channel through a porous medium with variable viscosity and convective cooling has been investigated using the Caputo fractional order derivative operator. The mathematical formulation used for the hydromagnetic Couette flow of Jeffrey fluid takes into account the effects of viscous dissipation, Soret, and Dufour. The system of nonlinear partial differential equations governing the flow were solved numerically using the explicit finite difference method. The numerical results for the behavior of various physical parameter on the flow variables are obtained and represented graphically. Moreover, effects of the flow parameters on heat and mass transfer rates are obtained and discussed numerically through tabular forms. The graphical findings show that velocity, concentration, induced magnetic field, and thermal field profiles decline with progressively increment of Jeffrey parameter. The velocity, and temperature of the fluid decline with higher values of fluid viscosity parameter. An increase in the chemical reaction parameter recede the concentration field profiles while increase with raises the values of Soret number. Increasing Biot, and Dufour numbers significantly grows the thermal field profiles. Induced magnetic field grows with larger values of fluid viscosity parameter. The findings of this study are important due to its application in magnetohydrodynamics pumps, polymer manufacturing, fins designs, and food processing.

Optimal harvesting policy for biological resources with uncertain heterogeneity for application in fisheries management

AbstractConventional harvesting problems for natural resources often assume physiological homogeneity of the body length/weight among individuals. However, such assumptions generally are not valid in real‐world problems, where heterogeneity plays an essential role in the planning of biological resource harvesting. Furthermore, it is difficult to observe heterogeneity directly from the available data. This paper presents a novel optimal control framework for the cost‐efficient harvesting of biological resources for application in fisheries management. The heterogeneity is incorporated into the resource dynamics, which is the population dynamics in this case, through a probability density that can be distorted from reality. Subsequently, the distortion, which is the model uncertainty, is penalized through a divergence, leading to a nonstandard dynamic differential game wherein the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation has a unique nonlinear partial differential term. Here, the existence and uniqueness results of the HJBI equation are presented along with an explicit monotone finite difference method. Finally, the proposed optimal control is applied to a harvesting problem with recreationally, economically, and ecologically important fish species using collected field data.

Simplified implicit finite-difference method of spatial derivative using explicit schemes with optimized constant coefficients based on L1 norm

The explicit finite-difference (EFD) method is widely used in numerical simulation of seismic wave propagation to approximate spatial derivatives. However, the traditional and optimized high-order EFD methods suffer from the saturation effect, which seriously restricts the improvement of numerical accuracy. In contrast, the implicit FD (IFD) method approximates the spatial derivatives in the form of rational functions and thus can obtain much higher numerical accuracy with relatively low orders; however, its computational cost is expensive due to the need to invert a multidiagonal matrix. We derive an explicit strategy for the IFD method to reduce the computational cost by constructing the IFD method with the discrete Fourier matrix; then, we transform the inversion of the multidiagonal matrix into an explicit matrix multiplication; next, we construct an objective function based on the [Formula: see text] norm to reduce approximation error of the IFD method. This explicit strategy of the IFD method can avoid inverting the multidiagonal matrix, thus improving the computational efficiency. This constant coefficient optimization method reduces the approximation error in the medium-wavenumber range at the cost of tolerable deviation (smaller than 0.0001) in the low-wavenumber range. For the 2D Marmousi model, the root-mean-square error of the numerical results obtained by this method is one-fifth that of the traditional IFD method with the same order (i.e., 5/3) and one-third that of the traditional EFD method with much higher orders (i.e., 72). The significant reduction of numerical error makes the developed method promising for numerical simulation in large-scale models, especially for long-time simulations.

An explicit fourth-order accurate compact method for the Allen-Cahn equation

<abstract><p>In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.</p></abstract>

Improved Finite Difference Technique via Adomian Polynomial to Solve the Coupled Drinfeld’s–Sokolov–Wilson System

This study presents a new algorithm for effectively solving the nonlinear coupled Drinfeld’s–Sokolov–Wilson (DSW) system using a hybrid explicit finite difference technique with the Adomian polynomial (EFD-AP). The suggested approach addresses the problem of accurately solving the DSW system. Numerical results are obtained by comparing the exact solution with absolute and mean square errors using a test problem to assess the EFD-AP accuracy against the exact solution and the conventional explicit finite difference (EFD) method. The results exhibit excellent agreement between the approximate and exact solutions at different time values, and the results showed that the proposed EFD-AP method achieves superior accuracy and efficiency compared to the EFD method, which makes it a promising method for solving nonlinear partial differential systems of higher order.

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.