Finite Difference Method Research Articles

NUMERICAL SOLUTION OF HYPERBOLIC GOURSAT PARTIAL DIFFERENTIAL EQUATIONS WITH HYBRID CENTRAL DIFFERENCE - TAYLOR SERIES EXPANSIONS METHOD

This paper investigates a new method for solving the Goursat partial differential equation (PDE) using a combination of the central finite difference method (FDM) and Taylor series expansion. The study evaluates the effectiveness and accuracy of this new approach, analyzing linear Goursat problems and conducting multiple numerical experiments. The simulation study demonstrates that the suggested approach surpasses the existing method in terms of performance and accuracy. Applying this proposed scheme will minimize the cost, especially for engineers that might apply this model in solving their real-life problems.

Graphics processing unit (GPU)-enhanced nonhydrostatic model with grid nesting for global tsunami propagation and coastal inundation

Nonhydrostatic models have proven their superiority in describing tsunami propagation over trans-oceanic distances and nearshore transformation because of their good dispersion and nonlinearity properties. The novel one-layer nonhydrostatic formulations proposed by Wang et al. [Phys. Fluids 35, 076610 (2023)] have been rederived in the spherical coordinate system incorporating Coriolis effects to enable the application of basin-wide tsunami propagation. The model was implemented using the fractional step method, where the hydrostatic step was solved by a Godunov-type finite-volume scheme, and the nonhydrostatic step was obtained with the finite-difference method. Additionally, a two-way grid-nesting scheme was employed to adapt the topographic features for efficient computation of tsunami propagation in deep ocean and coastal inundation. Furthermore, graphics processing unit (GPU)-parallelism technique was incorporated to further optimize the model performance. An idealized benchmark test as well as three experiments of regular and irregular waves, solitary, and N-waves transformations have been simulated to demonstrate the superior performance of the current GPU-accelerated grid-nesting nonhydrostatic model. Finally, the model has been applied to reproduce the 1964 Prince William Sound Tsunami, its propagation across the North Pacific and induced inundation in the Seaside.

Asymptotic Study of Flows Induced by Oscillations of Cylindrical Bodies

Hydrodynamic flows induced by translational oscillations of cylindrical bodies of various cross-sectional shapes are studied. The motion of fluid around oscillating bodies is described using the system of Navier–Stokes equations written in a generalized curvilinear coordinate system. Transition to a given body shape is implemented using a conformal mapping. The problem is solved using the method of successive asymptotic expansions under the assumption that the oscillation amplitudes are small. In each asymptotic approximation, the subproblems are solved numerically using the finite-difference method. Based on the results of the work, estimates of the hydrodynamic effect are obtained, the applicability of the high-frequency asymptotic approximation is estimated, and secondary stationary flows near cylinders are studied, in particular, the occurrence of directed stationary flows near an oscillating asymmetric body is considered with reference to the Joukowski airfoil.

Higher-order compact finite difference method for a free boundary problem of ductal carcinoma in situ

A higher-order compact finite difference scheme is established to investigate the model of ductal carcinoma in situ, a free boundary value problem. For the first time, to determine the growth of a tumor numerically, a compact finite difference method is applied in the spatial direction, and the Crank-Nicolson scheme is used in the temporal direction. Through stability analysis, it has been demonstrated that the proposed numerical scheme is unconditionally stable. The convergence of the numerical scheme is analyzed theoretically by von Neumann's method. It has been proved that the proposed numerical method is second-order convergent in time and fourth-order convergent in space in the L2-norm. Two numerical examples are provided to verify the scheme's efficiency and validate the theoretical results. The tabular and graphical representations confirm the high accuracy and adaptability of the scheme.

Time-dependent model for two-phase flow in ultra-high water-cut reservoirs: Time-varying permeability and relative permeability

For the ultra-high water-cut reservoirs, after long-term water injection exploitation, the physical properties of the reservoir change and the heterogeneity of the reservoir becomes increasingly severe, which further aggravates the spatial difference of the flow field. In this study, the displacement experiments were employed to investigate the variations in core permeability, porosity, and relative permeability after a large amount of water injection. A relative permeability endpoint model was proposed by utilizing the alternating conditional expectation (ACE) transformation to describe the variation in relative permeability based on the experimental data. Based on the time dependent models for permeability and relative permeability, the traditional oil-water two-phase model was improved and discretized using the mimetic finite difference method (MFD). The two cases were launched to confirm the validation of the proposed model. The impact of time-varying physical features on reservoir production performance was studied in a real water flooding reservoir. The experimental results indicate that the overall relative permeability curve shifts to the right as water injection increases. This shift corresponds to a transition towards a more hydrophilic wettability and a decrease in residual oil saturation. The endpoint model demonstrates excellent accuracy and can be applied to time-varying simulations of reservoir physics. The impact of variations in permeability and relative permeability on the reservoir production performance yields two distinct outcomes. The time-varying permeability of the reservoir results in intensified water channeling and poor development effects. On the other hand, the time-varying relative permeability enhances the oil phase seepage capacity, facilitating oil displacement. The comprehensive time-varying behavior is the result of the combined influence of these two parameters, which closely resemble the actual conditions observed in oil field exploitation. The time-varying simulation technique of reservoir physical properties proposed in this paper can continuously and stably characterize the dynamic changes of reservoir physical properties during water drive development. This approach ensures the reliability of the simulation results regarding residual oil distribution.

Numerical investigation of turbulence generation using Zakharov-like model equation

This study investigates the turbulence generation behavior with a Zakharov-like (ZL) equation in a fluid system. The model equation is derived using conservation equations (mass and momentum conservation), and the source of nonlinearity is the high amplitude of the acoustic wave. The Zakharov-like equation has been derived and then solved numerically, then turned into a modified nonlinear Schrödinger equation. Furthermore, modulation instability, or Benjamin–Feir instability, of the model equations, which leads to the emergence of Akhmediev breathers, is discussed. The numerical simulation uses a finite difference method for temporal evolution and a pseudo-spectral approach to determine spatial regimes. The outcomes indicate that the situation involving the nonlinear Schrödinger equation case displays a periodic pattern in space and time. The findings also demonstrate that the localization of structure and the Fermi, Pasta, and Ulam (FPU) recurrences are disrupted for the modified nonlinear Schrödinger equation and Zakharov-like equation cases. The energy spectrum exhibits a power law behavior that approximately follows k−1.65 in the ZL model equation case, and it is steeper than Kolmogorov's spectrum within the inertial sub-range.

Heat Generation/Absorption in MHD Double Diffusive Mixed Convection of Different Nanofluids in a Trapezoidal Enclosure

Numerical simulation of MHD double-diffusive mixed convection flow of different nanofluids in a trapezoidal enclosure is performed with an internal heat generation/absorption source inside the enclosure. The nondimensional momentum, heat and mass equations are solved numerically by using the finite difference method. The present study focused mainly on the increment of the rate of heat and mass transfer using internal heat generation or absorption sources inside a lid-driven trapezoidal cavity. Considering numerous governing parameters (Q = −5 to 5, Ha = 0 to 30, Ri = 0.01 to 100) the flow velosity, temperature and concentration profiles are calculated for various nanofluids. Graphs and numerical tables are utilized to examine how different physical entities affect the distribution of flow, temperature and concentration. It is noted that enhancing values of Ha reduces the mass and heat transfer rate. It is observed that heat generation/absorption significantly affect the heat transfer rate as internal heat generation source increases heat transmission rather than mass transfer. The involvement of heat generation/absorption source significantly affects the heat transfer rate. By considering Al2O3-water nanofluid, the solid volume percentage has an accelerating effect on the Nusselt and Sherwood numbers as compared to the other nanofluids in the study.

Numerical analysis of natural convection in a porous circular bend

The study of natural convection in porous circular bends is pivotal due to its applications in various cooling system designs. This paper focuses on a circular bend with a heated inner wall and a colder outer wall, employing the finite difference method to solve the governing equations. The research encompasses Rayleigh numbers ranging from 20 to 750 and an aspect ratio (defined as R=R2−R1R1) from 1.0 to 2.0. The findings highlight that variations in Rayleigh numbers and the circle’s radius significantly enhance heat transfer rates via convection, underscoring their critical roles in influencing flow and heat transfer within this configuration. It is also observed that theNuAvg on the heated wall increase with the increase in R. It is observed that for R = 0.75, NuAvg is 8.8 and for R = 1.5, NuAvg is increase to 9.7.

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.

Control and analysis of floating ability of aeronautical cylindrical spiral groove gas seal based on F-K model

The slip flow phenomenon caused by the gas floating seal in ultra-thin gas films and high-altitude rarefied gas environments occurs frequently. This study represents the first attempt to calculate the gas film floating force by considering the coupling relationship among the slip flow effect, the surface micro-grooves, and the eccentricity using a high-precision eight-node finite difference method based on the linearized Boltzmann equation for a spiral-grooved cylindrical gas seal. Furthermore, the influence of slip flow on the operational and groove parameters of the spiral-grooved cylindrical gas seal is investigated and discussed. Results show that the velocity gradient of the lubrication gas is reduced and the effect of the fluid hydrodynamic pressure is weakened because of slip flow, particularly in high-speed, low-pressure, and high-eccentricity fields. However, increases in groove depth, number, and length improve the gas film floating force, which strengthens the slip flow response in the grooves. Therefore, the slip flow reduces the gas film floating force, but when the groove depth exceeds 32 μm and the groove length is greater than 45 mm, the slip flow may become negligible. The results presented here provide a theoretical basis to broaden the application scope of dynamic seals in the aerospace field.

Highly efficient nonuniform finite difference method for three-dimensional electrically stimulated liquid crystal photonic devices

Liquid crystal (LC) photonic devices have attracted intensive attention in recent decades, due to the merits of tunability, cost-effectiveness, and high efficiency. However, the precise and efficient simulation of large-scale three-dimensional electrically stimulated LC photonic devices remains challenging and resource consuming. Here we report a straightforward nonuniform finite difference method (NFDM) for efficiently simulating large-scale LC photonic devices by employing a spatially nonuniform mesh grid. We show that the NFDM can be further accelerated by approximately 504 times by using the improved successive over-relaxation method (by 12 times), the symmetric boundary (by 4 times), the momentum gradient descent algorithm (by 3.5 times), and the multigrid (by 3 times). We experimentally fabricated the large-scale electrically stimulated LC photonic device, and the measured results demonstrate the effectiveness and validity of the proposed NFDM. The NFDM allocates more grids to the core area with steep electric field gradient, thus reducing the distortion of electric field and the truncation error of calculation, rendering it more precise than the finite element method and traditional finite difference method with similar computing resources. This study demonstrates an efficient and highly reliable method to simulate the large-scale electrically stimulated LC photonic device, and paves the way for customizing a large-scale LC photonic device with designable functionalities.

Nonlinear Convection Flow of a Micropolar Nanofluid Past a Stretching Sphere with Convective Heat Transfer

An incompressible, steady combined nonlinear convective transport system on a micropolar nanofluid through a stretching sphere with convective heat transfer was investigated. The conservation equations corresponding to momentum, microrotation, thermal energy, and concentration particles have been formulated with suitable boundary constraints. By using the required non-dimensional variables, the conservation equations have been turned into a set of high-order standard differential equations. Then, an implicit finite difference method, also known as the Keller-Box Method (KBM), was used to numerically solve the flow problem. The obtained outcomes are displayed through graphs and tables to explain the impact of various governing variables over velocity, temperature, concentration, number of skin friction, wall coupled stress, Nusselt number, and Sherwood number. The findings demonstrate that increasing the convective heat parameter Bi enhances the factor of skin friction, local Nusselt number, Sherwood number, velocity field, and temperature profile while lowering the wall-coupled stress. It is observed that for high values of the material parameter β, the fluid velocity and the spin of the micro-elements both increase, which causes the dynamic viscosity and microrotation velocity to decrease. In addition, as the rates of magnetic constant Ma, thermophoresis Nt and Brownian movement Nb rise, the thermal distribution and its thickness of boundary layer increase. However, it decline along the enlarging quantities of nonlinear convection parameter λ, Prandtl number Pr, material parameter β, and solutal Grashof number Gm, which agrees to increase fluid density. When the range of thermophoresis Nt surges, it causes an increment in the nanoparticle species, but the opposite behavior have seen in the case of Brownian number Nb, and Lewis number Le. The comparison made with the related published paper achieves a significant agreement. The numerical result is generated through the implementation of the computational software MATLAB R2023a.

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

The advection-diffusion-reaction (ADR) equation is a fundamental mathematical model used to describe various processes in many different areas of science and engineering. Due to wide applicability of the ADR equation, finding accurate solution is very important to better understand a physical phenomenon represented by the equation. In this study, a numerical scheme for solving two-dimensional unsteady ADR equations with spatially varying velocity and diffusion coefficients is presented. The equations include nonlinear reaction terms. To discretize the ADR equations, the Crank–Nicolson finite difference method is employed with a uniform grid. The resulting nonlinear system of equations is solved using Newton’s method. At each iteration of Newton’s method, the Gauss–Seidel iterative method with sparse matrix computation is utilized to solve the block tridiagonal system and obtain the error correction vector. The consistency and stability of the numerical scheme are investigated. MATLAB codes are developed to implement this combined numerical approach. The validation of the scheme is verified by solving a two-dimensional advection-diffusion equation without reaction term. Numerical tests are provided to show the good performances of the proposed numerical scheme in simulation of ADR problems. The numerical scheme gives accurate results. The obtained numerical solutions are presented graphically. The result of this study may provide insights to apply numerical methods in solving comprehensive models of physical phenomena that capture the underlying situations.

Determination of Timewise-Source Coefficient in Time-Fractional Reaction-Diffusion Equation from First Order Heat Moment

This article aims to determine the time-dependent heat coefficient together with the temperature solution for a type of semi-linear time-fractional inverse source problem by applying a method based on the finite difference scheme and Tikhonov regularization. An unconditionally stable implicit finite difference scheme is used as a direct (forward) solver. While by the MATLAB routine lsqnonlin from the optimization toolbox, the inverse problem is reformulated as nonlinear least square minimization and solved efficiently. Since the problem is generally incorrect or ill-posed that means any error inclusion in the input data will produce a large error in the output data. Therefore, the Tikhonov regularization technique is applied to obtain stable and accurate results. Finally, to demonstrate the accuracy and effectiveness of our scheme, two benchmark test problems have been considered, and its good working with different noise levels.

Design and analysis of Si photonic NaCl sensors based on suspended microdisk resonators

In this paper, two optical refractive index sensors, respectively based on a suspended microdisk resonator (SMDR) and a sub-wavelength Bragg grating suspended microdisk resonator (SWG-SMDR) have been designed on a silicon on insulator (SOI) platform, and their optical and sensing performances have been analyzed using the three-dimensional finite-difference time-domain method. It has been shown that the Bragg grating increases the optical confinement factor in the sensing region and, consequently, improves the light–matter interaction and sensitivity. Considering concentrations of 0 to 25% for the NaCl dissolved in the deionized water, the enhanced sensitivity and FOM of the SWG-SMDR sensor with the miniaturized optimal geometrical dimensions are 163.77 nm RIU−1 and 10.81 RIU−1, respectively. This compact optical sensor enhances the sensitivity and FOM by combining effects of the Bragg grating and suspension configuration in improving the light–matter interaction.

A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

Simulation of heat transfer in Poiseuille pipe flow via generalized finite difference method with a space stepping algorithm

The aim of this paper is to propose an efficient numerical scheme to deal with heat transfer problems in pipe flow with a large length/diameter ratio. The generalized finite difference method (GFDM) is combined with a space stepping algorithm (SSA) for the solution. The SSA divides the solution domain into a number of overlapping sections in the length direction of the pipe. Due to the large Peclet number, the heat transport process is dominated by advection, which allows an approximate boundary condition to be applied at the downstream cross section. The problem is then solved section by section. Using the uniform distributed points, the solution matrix for each section does not change, and only the right hand side needs to be refreshed for the changing boundary conditions. This leads to a highly efficient scheme for problems with a large pipe length. To show the accuracy of the numerical scheme, a problem with available analytical solution is studied. Then the method is applied to problems of single pipe with different pipe wall temperature and flow Reynolds number, and concentric annular pipe. Numerical results confirm the stability and efficiency of the GFDM-SSA. The method can be applied to many real world transient heat transfer problems in which the pipe is heated or cooled along its length with arbitrary wall temperature for heat exchange and other purposes.

A mathematical‐boundary‐recognition domain‐decomposition Lattice Boltzmann method combined with large eddy simulation applied to airfoil aeroacoustics simulation

AbstractBeing a direct computational aeroacoustics method, Lattice Boltzmann method (LBM) has great potential and broad application perspective in the field of numerical simulation of aerodynamic noise due to its low dispersion and low dissipation. A series of numerical algorithms and the related improvements based on the standard LBM method are proposed and developed in this paper to adapt to the airfoil noise calculation with complex grid at middle‐high Reynolds number. First, a new mathematical‐boundary‐recognition algorithm based on Green's formula is proposed to deal with complex curved geometric models, which is validated by three‐element airfoil 30P30N benchmark. Then, in order to reduce grid redundancy and improve computing efficiency, the grid refinement technique of domain decomposition model (DDM) is adopted and also improved, which is verified by calculating the flow and sound fields around 2D and 3D cylinders at Reynolds number equal to 90,000. Finally, three different LES turbulence models are combined with the standard MRT‐LBM method, where different finite difference schemes are used to solve Reynolds stress tensor which is different from the traditional one. Through the direct acoustic numerical simulation of NACA0012 airfoil at Reynolds number equal to 200,000, the effects of Smagorinsky models and Wall‐adapting local eddy‐viscosity (WALE) model on aerodynamic noise prediction are compared and analyzed. Overall, the proposed methodology is shown to be appropriate for predicting the aerodynamic noise at low Mach number and can successfully simulate the generation and propagation of far field acoustics.

Mathematical modeling of functionally graded porous geometrically nonlinear micro/nano cylindrical panels

Relevance. The study investigates the problem of stress-strain state and stability of porous functional-gradient size-dependent cylindrical panels. The composition and properties of alloys can differ and significantly affect the performance characteristics of products. Therefore, the research of material properties is relevant and contributes to the creation of new types of products demanded by the oil and gas industry. Aim. Development of a new model and creation of accurate methods for analyzing the stress-strain state of porous functional-gradient size-dependent micro/nano cylindrical panels taking into account geometrical nonlinearity. Methods. The method of variational iterations – the extended Kantorovich method is used to analyze the stress-strain state of cylindrical panels. The validity of the results is ensured by comparing the solutions obtained by the method of variational iterations in the first and second approximations with the solutions obtained by the authors, by the Bubnov–Galerkin method in higher approximations, by the finite difference method of the second order of accuracy, for which their convergence is investigated depending on a number of partitions of the integration area in the finite difference method and the number of series terms in the expansion of the basic functions in the Bubnov–Galerkin method. The results obtained by these methods are compared with the solutions obtained by other authors. It should be noted that the solutions obtained by the method of variational iterations for bending of functionally graded cylindrical panels under the action of transverse uniformly distributed load can be considered accurate. Results and conclusions. The authors have constructed the model of porous functional-gradient size-dependent cylindrical panels. Its use will allow studying the properties of alloys for producing drill pipes. The influence of material porosity type, porosity index, functional-gradient index, boundary conditions, size-dependent parameter, curvature parameters on the stress-strain state of cylindrical panels was analyzed using the developed method of variational iterations.

ANALYTICAL SOLUTION OF THE DIFFERENTIAL EQUATION OF HEAT CONDUCTIVITY FOR DAMAGED THERMAL INSULATION OF PIPELINES

For heat supply systems of Ukraine, the determination and forecasting of thermal energy losses during the transportation of heat carrier is an urgent problem. The thermal lines of the heating networks are long and its insulation is damaged. Installation of heat energy metering devices at all sources and all consumers (i.e. buildings) without exception allows determining and forecasting real heat losses in the heat network, but this is a difficult task, not all consumers and sources are actually covered by heat consumption accounting. The task of determining heat losses is also relevant for energy management systems of heat supply systems, energy companies and industry. The solution to the problem is proposed by a separate mathematical modeling of the temperature state of the section of the damaged insulating layer with the determination of the heat flow through it. It is proposed to solve the problem by the method of analytical solution of the differential equation of heat conduction with boundary conditions of the third kind. With the uniform distribution of characteristic damages along the total length of the pipeline, knowing the limits of damage influence, the share of insulation damage and the number of damages on the pipeline, it is possible to determine the real heat flow from the outer surface of the pipelines, including coefficient of increase of heat losses on the section of the heat pipe in relation to those determined by regulatory documents. Traditionally, one-dimensional models or determinations of the two-dimensional temperature field and actual heat fluxes in a cross-section to the axis of a characteristic damaged section of insulation are considered. But this does not take into account the two-dimensionality of the temperature field of the damaged layer of insulation along the length (that is, along the axis) of the pipeline. Therefore, the purpose of this work is to develop a methodology for determining heat losses through pipelines, taking into account the damage to their insulation and the distribution of characteristic damages along the length. The following simulation was carried out for one of the areas. Also, experimental and numerical studies using the finite difference method in combination with the method of running variable directions for similar models confirmed the coincidence of the analytical solution of the proposed model and the finite difference model within the permissible error.