Implicit Finite Difference Scheme Research Articles

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.

Variable thermal conductivity and mass diffusivity effects in a free convective flow of doubly stratified non-darcian porous medium over a vertical plate

The research in this article is carried out to study incompressible and unsteady free convective flow on a semi-infinite isothermal vertical plate in a doubly stratified non-Darcian porous media with variable mass diffusivity and variable thermal conductivity. The governing non-linear partial differential equations of flow were calculated by applying an implicit finite difference scheme of Crank-Nicolson type. Various parametric impacts on concentration profiles, temperature, velocity, as well Sherwood number, Nusselt number and skin friction, were examined and presented in graphs. It is examined that there exists a significant temperature decrease for high Darcy number in stratified fluids. Also, it is detected that the presence of stratification produces a considerable drop in skin friction while increases the mass and heat transfer rate. Comparison of current outcomes well agreed with the available solutions.

CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics

This paper proposes a Crank-Nicolson alternating direction implicit (CN-ADI) finite difference scheme for solving the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamic for the first time. Due to the weakly singular behavior of the exact solution near the initial time t=0, we use the non-uniform meshes to capture the rapid change of the solution at t=0. The Crank-Nicolson method and product-integration (PI) rule are proposed to approximate temporal derivative and the Riemann-Liouville (R-L) fractional integral term, respectively. The fully discrete scheme is obtained by the standard central finite difference method (FDM) in space. The stability in L2-norm and convergence of the CN-ADI difference scheme are strictly proved, where the convergence reached O(τ2+hx2+hy2+hz2). The ADI algorithm greatly reduces the computational cost of the three-dimensional problems in viscoelastic dynamics. At last, the results of numerical examples verify the correctness of the theoretical analysis and prove the effectiveness of the proposed method.

The Effect of Variable Heat Flux on Unsteady Laminar MHD Boundary Layer Flow and Heat Transfer Due to a Stretching Sheet

The purpose of this research is to look into the solution technique for obtaining MHD velocity and temperature profiles. In the presence of a changing heat flux, the unsteady laminar boundary layer flow and heat transfer of a viscous incompressible fluid across a stretching sheet are numerically investigated. The unsteadiness is thought to be generated by a sudden increase in the surface temperature and a time-dependent stretching velocity. The flow and heat transfer partial differential equations were numerically solved using an implicit finite difference scheme and a quasi-linearization technique. Both velocity and temperature rise with time and magnetic field, according to the findings. The computed results are compared to previous work that has been published. Variable heat flux (VHF) conditions have also been taken into account.

Analytical and Numerical Investigation of Two-Dimensional Heat Transfer with Periodic Boundary Conditions

A two-dimensional heat diffusion problem with a heat source that is a quasilinear parabolic problem is examined analytically and numerically. Periodic boundary conditions are employed. As the problem is nonlinear, Picard’s successive approximation theorem is utilized. We demonstrate the existence, uniqueness, and constant dependence of the solution on the data using the generalized Fourier method under specific conditions of natural regularity and consistency imposed on the input data. For the numerical solution, an implicit finite difference scheme is used. The results obtained from the analytical and numerical solutions closely match each other.

Numerical treatment for some abstract degenerate second-order evolutionary problem

This paper addresses the numerical analysis of a class of a degenerate second-order evolution equations. We employ a finite element method for spatial discretization and a family of implicit finite difference schemes for time discretization. Introducing a stabilization parameter, denoted by θ, we propose a well-posed fully-discrete scheme. Sufficient conditions for its well-posedness and for quasi-optimal error estimates are established. The abstract theory is illustrated through the application to the degenerate wave equation, and numerical results validate our theoretical findings.

Time-dependent simplified [formula omitted] approximation for neutron transport equation

Time-dependent transport is still too computationally expensive for various practical purposes. On the other hand, the diffusion approximation is unable to provide accurate results for many real problems. To establish a middle ground between the accuracy of the transport and the simplicity of the diffusion approximations, the simplified spherical harmonics (SPN) method was introduced. The present paper aims to improve the accuracy of the time-dependent SP3 method by employing the simplified double-P1 (SDP1) approximation, which is based on the one-dimensional double-spherical harmonics DP1 equations. Using the finite element method to treat the spatial dependence of the SDP1 equations leads to a set of first-order ordinary differential equations on the time variable. These equations are then converted into a set of linear algebraic equations by discretizing the time variable with the implicit backward finite difference scheme. Our numerical results indicate that the SDP1 approximation could compute more precise values than the widely used SP3 method in transient problems.

A NUMERICAL APPROACH OF FORCED CONVECTIVE MHD CASSON HYBRID NANOFLUID FLOWS EXPOSED TO JOULE HEATING AND VISCOUS DISSIPATION OVER DIVERGING CHANNEL

The present study deals with boundary layer flows of buoyancy-driven magnetohydrodynamic, chemical-radiative, and temperature-sensitive Casson hybrid nanofluid over diverging channel. Copper (Cu) and aluminum oxide (Al<sub>2</sub>O<sub>3</sub>) nanoparticles are suspended upon ethylene glycol-based non-Newtonian Casson fluid. The proposed model is applicable in power transmission systems the design of nuclear reactors where a moving plate is used as a control rod, and the design of compression molding processes. The boundary layer governing equations undergo nonsimilar transformations followed by a quasilinearization technique and an implicit finite difference scheme. Varga's algorithm is applied on the obtained block tri-diagonal system of equations. The study pertinent to dimensionless parameters like Reynolds number, Eckert number, Casson parameter, and Richardson's number on velocity, temperature, drag coefficient, and heat transfer profiles. Also surface plots are plotted for varied values of Casson parameter and magnetic parameter on skin friction and heat transfer coefficients. It is to be noted that for enhanced values of Casson parameter β, the velocity profile is augmented, and the temperature profile is declined. It is observed that the temperature profile is enhanced at the center of the channel for enhanced values of viscous dissipation parameter Ec.

Pinning-depinning transition of droplets on inclined substrates with a three-dimensional topographical defect.

Droplets on inclined substrates can depin and slide freely above a critical substrate inclination angle. Pinning can be caused by topographical defects on the substrate, and understanding the influence of defect geometry on the pinning-depinning transition is important for diverse applications such as fog harvesting, droplet-based microfluidic devices, self-cleaning surfaces, and inkjet printing. Here, we develop a lubrication-theory-based model to investigate the motion of droplets on inclined substrates with a single three-dimensional Gaussian-shaped defect that can be in the form of a bump or a dent. A precursor-film/disjoining-pressure approach is used to capture contact-line motion, and a nonlinear evolution equation is derived which describes droplet thickness as a function of the position along the substrate and time. The evolution equation is solved numerically using an alternating direction implicit finite-difference scheme to study how the defect geometry influences the critical inclination angle and the shape of a pinned droplet. It is found that the critical substrate inclination angle increases as the defect becomes taller/deeper or wider along the direction lateral to the droplet-sliding direction. However, the critical inclination angle decreases as the defect becomes wider along the sliding direction. Below the critical inclination angle, the advancing contact line of the droplet at the droplet centerline is pinned to the defect at the point having maximum negative slope. Simple scaling relations that reflect the influence of defect geometry on the droplet retention force arising from surface tension are able to account for many of the trends observed in the numerical simulations.

A unified approach to solving parabolic Volterra partial integro-differential equations for a broad category of kernels: Numerical analysis and computing

This work is concerned with solving parabolic Volterra partial integro-differential equations (PIDE) considering differentiable and singular kernels. The implicit finite difference scheme is implemented to approximate the differential operator, and the nonlocal term is discretized based on an open-type formula with two distinct time step sizes related to the nature of the time level to guarantee to avoid the singular terms at the endpoints and denominators. The properties of the plied scheme are investigated, more precisely, its stability and consistency. Four detailed examples are implemented to demonstrate the efficiency and reliability of the applied finite difference scheme.

A finite difference/Kansa method for the two-dimensional time and space fractional Bloch-Torrey equation

In this paper, an implicit finite difference scheme combined with Kansa meshless method is proposed for the two-dimensional time and space fractional Bloch-Torrey equation. The Caputo derivative with respect to time is discretized by finite difference method based on Alikhanov's super convergent approximation while Riesz derivative with respect to spatial variables by Kansa meshless method. The convergence and stability of the time semi-discretization are proven using energy method. The convergence order in time direction is proven to be 2. The implementation of the full discretization is discussed in detail. Especially, the issue of calculation of Riesz derivative of RBF is addressed. Numerical examples are given which verify the feasibility of the proposed method.

Thermo-mechanical behavior of 2D functionally graded porous-auxetic metamaterial rotating disk with an auxetic foundation

In the present study, the thermo-mechanical behavior of two-directional functionally graded porous-auxetic metamaterial rotating disk (2D-FGPAMRD) has been analyzed. The rotating disk subjected to combination of transient thermal and steady mechanical loads and supported by an auxetic foundation. Haber-Schaim (H-S) frictionless elastic foundation model is considered to drive the structure-structure interaction. The governing equations are obtained based on the thermoelasticity theory. The response of disk to mechanical and thermal loads has been obtained by solving nonhomogeneous governing equations. To this end, the alternating direction implicit (ADI) finite difference scheme and combination of state-space and differential quadrature (DQ) methods have applied to solve the Fourier heat conduction and thermoelasticity equations, respectively. In the numerical examples, the effects of porosity and auxeticity, as well as inhomogeneity indices of the disk on its thermo-mechanical behavior have been investigated for the first time. The results reveal that, the elastic field components of rapid heating disk are significantly affected by the nature of temperature diffusion, inhomogeneity indices and the auxeticity of the disk material. Besides, the changes of thermal expansion gradient indices provide a relaxation in the normal transverse stress.

Two linear finite difference schemes based on exponential basis for two-dimensional time fractional Burgers equation

In present paper, we propose tailored finite point method (TFPM) based on exponential basis for solving two-dimensional time fractional Burgers equation. We use the L1 and L1-2 discretization formulas for the Caputo fractional derivative in time direction and use the TFPM to approximate the spatial derivatives. This method with the help of exponential functions fit the properties of the local solution in time and space simultaneously which act as basis functions in the frame of TFPM. We focus on constructing two linear implicit finite difference schemes, i.e. a five-point node centered scheme and a four-point cell-centered scheme based on exponential functions for deriving the numerical solution of given problem. The stability and convergence of fully discrete schemes based on five-point node centered are discussed. At last, we use some numerical examples to show the accuracy and efficiency of the presented schemes.

Dynamic analysis of wave scenarios based on enhanced numerical models for the good Boussinesq equation

The good Boussinesq equation, a modification of the Boussinesq equation, aims to enhance predictions about shallow water wave behavior. This paper introduces two finite difference schemes for solving the good Boussinesq equation including linear and nonlinear implicit finite difference methods. Both schemes utilize the pseudo-compact difference approach, delivering second-order precision with an additional term to boost numerical simulation accuracy while maintaining the grid points of the standard scheme. These schemes rigorously preserve the critical physical characteristics of the good Boussinesq equation, ensuring more precise representation. We establish the existence of solutions with discrete differences and demonstrate, through the discrete energy method, their uniqueness, stability, and second-order convergence in the maximum norm. Furthermore, we propose an iterative algorithm tailored for the nonlinear implicit finite difference scheme, resulting in significant reductions in computational costs compared to the linear scheme. The results of our numerical experiments demonstrate that our methods are competitive and efficient when compared to difference schemes and previously used methods, while maintaining crucial physical qualities. Furthermore, we run relevant numerical simulations to demonstrate the accuracy of the current methods using evidence from the solitary wave interaction with the initial amplitudes of the wave. It is also suggested that the issue has a critical initial wave amplitude for the interaction of two solitary waves, where blow up occurs in a finite amount of time for initial wave amplitudes greater than the new blow-up criteria value.

Three-Dimensional Boundary-Layer Flow Across A Wedge: A Numerical Investigation

The present work investigates the flow of a continuous, laminar, 3-D boundary-layer over a fixed wedge, where the outside free-stream flows are approximately approximated by a power of distances. The controlling partial differential equations are transformed into coupled nonlinear ordinary differential equations with the necessary boundary conditions using surface similarity transformations. These equations involve two physical variables: pressure gradient and shear-to-strain rate. The resulting equations are numerically solved by the implicit finite-difference scheme known as the Keller-box approach. The obtained results are compared with those reported in the literature for a few special cases. Our numerical results indicate that the flow zone is divided into two areas: near-field (close to the wedge surface) and far-field field (mostly controlled by inviscid flow).

Unsteady magneto bioconvective Sutterby nanofluid flow: Influence of g-Jitter effect

This study is driven by the potential applications of a microgravity environment in various sectors and fields of engineering, especially in the manufacturing of semiconductors. An electrically conducting bioconvective Sutterby nanofluid flow was considered around an exponentially stretching surface through g-Jitter and nonlinear thermal radiation effects. The dimensional partial differential equations (PDEs) that govern the fluid flow system are changed into nondimensional PDEs through nonsimilar transformations. Once these PDEs have been linearized using Quasilinearization, the implicit finite difference scheme is used to solve them numerically. A MATLAB code was constructed for the numerical calculations and for analysing the results via graphs. These findings indicate that the greater values of amplitude ε1 result in a steep rise in the velocity and surface drag near the sheet's wall for assisting buoyancy. An enhancement of approximately 108 % in the drag coefficient is observed when ε1(the amplitude in g-Jitter) jumps to 1 from 0. The percentile increments in heat transport strength due to the rise of radiation Rd from 1 to 2 is 80 %, and due to the intensification of the temperature difference ratio φw from 1.5 to 2 is about 167 %. Elevating Peclet number Pe led to the depletion of the microbial boundary layer thickness. Oxytactic microbes' presence impacts the enhanced mass transport strength of nanoparticles. The current findings have been validated by comparing them to previously published results, and they have been shown to be in perfect agreement with those results.

Algebra preconditionings for 2D Riesz distributed‐order space‐fractional diffusion equations on convex domains

AbstractWhen dealing with the discretization of differential equations on non‐rectangular domains, a careful treatment of the boundary is mandatory and may result in implementation difficulties and in coefficient matrices without a prescribed structure. Here we examine the numerical solution of a two‐dimensional constant coefficient distributed‐order space‐fractional diffusion equation with a nonlinear term on a convex domain. To avoid the aforementioned inconvenience, we resort to the volume‐penalization method, which consists of embedding the domain into a rectangle and in adding a reaction penalization term to the original equation that dominates in the region outside the original domain and annihilates the solution correspondingly. Thanks to the volume‐penalization, methods designed for problems in rectangular domains are available for those in convex domains and by applying an implicit finite difference scheme we obtain coefficient matrices with a 2‐level Toeplitz structure plus a diagonal matrix which arises from the penalty term. As a consequence of the latter, we can describe the asymptotic eigenvalue distribution as the matrix size diverges as well as estimate the intrinsic asymptotic ill‐conditioning of the involved matrices. On these bases, we discuss the performances of the conjugate gradient with circulant and ‐preconditioners and of the generalized minimal residual with split circulant and ‐preconditioners and conduct related numerical experiments.

Radiative ternary nanoliquid flow over an exponentially stretching sheet with motile oxytactic microorganisms

This article examines the mixed convection ternary nanoliquid flow containing a suspension of oxytactic microorganisms under the impact of nonlinear thermal radiation. The potential applications include microbiology, biotechnology, biosensors, cell therapy, medication release system, and protein expression. The governing equations that handle the problem are partial differential equations, they are dimensional, nonlinear in nature, and are nondimensionalized by using nonsimilar transformations. Further, for the computation of mathematical analysis, the Quasilinearization technique and numerical method such as the implicit finite difference scheme are used. The impacts of varying relevant factors on velocity, temperature, and microbe density are thoroughly examined. Results designate that the radiation factor favorably affects the velocity and temperature profiles. The rate of mass transport of liquid oxygen rises by about 45% when the Schmidt number grows from 1 to 2. The coefficient of skin friction is found to be more for ternary nanofluid than hybrid nanoliquid and nanoliquid. The current analysis compares the solution to earlier literature and finds that there is a good agreement.

Thermal analysis of bioconvective nanofluid flow over a sphere in presence of multiple diffusions and a periodic magnetic field

The current work intends to describe the heat and mass transfer in the mixed bioconvective flow of nanofluid along the sphere. This analysis also takes into account the influences of a periodic magnetic field and the diffusions of species concentrations such as liquid ammonia and liquid hydrogen. The physical problem is represented as a collection of connected nonlinear partial differential equations with appropriate boundary constraints using the Boussinesq approximation. The Buongiorno two-phase model is utilized to examine the nanoparticle's effect on the fluid. The governing equations are represented in the dimensionless form by using nonsimilar transformations. Further, the implicit finite difference scheme and the technique of quasilinearization are employed for numerical computation. Graphs are used to discuss the findings. The skin friction coefficient increases by about 27%, and the heat transfer rate to the fluid from the surface decreases by about 30% when the periodic magnetic field escalates from 0 to 1. The nanoparticle's mass transfer rate increases by about 35% when the Lewis number grows from 5 to 10. Higher Peclet numbers are favorable to the skin friction coefficient and heat transfer rate, but not the microorganism's density number. The present numerical results obtained in this analysis were compared to earlier published work and found to be in excellent agreement.

Nonparametric probabilistic forecasting based stochastic optimal scheduling of integrated electricity and gas systems

The volatility and sporadic availability of renewable energy create significant challenges to the optimal scheduling of integrated electricity and gas systems (IEGS). This paper develops a nonparametric probabilistic forecasting based stochastic scheduling approach of IEGS. The quantile at a series of quantile levels can be generated by direct quantile regression method. Given the set of predicted quantiles, a set of representative scenarios for wind power uncertainty can be obtained by using Monte Carlo simulation method and scenario reduction approach. Based on the implicit finite difference scheme, the original partial differential equations of the gas network are discretized to establish an algebraic model, which provide possibility for efficient solution. Then, the nonconvexity caused by the momentum equation is eliminated by the second-order core relaxation. Finally, the stochastic optimal scheduling model is reformulated as a second-order core programming problem. Numerical simulations are performed to showcase the superiority of the established stochastic optimal scheduling model.