Semi-implicit Finite Difference Method Research Articles

A novel semi-implicit WLS scheme for time-memory nonlinear behavior in 2D variable-order TF-NLSEs

In this paper, a novel hybrid semi-implicit meshless weighted least-squares (WLS) scheme H-SIFPM is developed for the first time by coupling the semi-implicit finite point-set method and finite difference method (FDM), and then applied to predict the time-memory nonlinear behavior dominated by a 2D variable-order time-fractional nonlinear Schrödinger equation (TF-NLSE). The proposed H-SIFPM for TF-NLSE is mainly derived from that: firstly, the variable-order time-fractional with Caputo derivative is approximated by a FDM scheme and then imposed to the approximation of WLS as a boundary condition; secondly, the hybrid process is implicitly discretized in temporal level, and an implicit meshless FPM scheme is developed; thirdly, an iterative technique is adopted to treat the implicit format. Subsequently, the estimated error and stability of the proposed scheme is tentatively proved. Moreover, the accuracy and capacity of the proposed H-SIFPM are also tested by solving several examples with constant/variable-order cases, in which a second-order numerical convergent rate is illustrated. We also demonstrate the advantages of the meshless method by solving the problem on complex irregular domain with local refinement. Finally, the nonlinear dispersion behavior dominated by 2D TF-NLSE/TF-GPE with different boundary conditions are predicted, and the FDM results are also presented for comparison. All numerical results show that the proposed H-SIFPM scheme for TF-NLSEs is stable, accurate and flexible.

MHD flow of third-grade fluid through a vertical micro-channel filled with porous media using semi implicit finite difference method

This study examines the complex dynamics of an incompressible, electrically conducting third-grade fluid (TGF) flow through a vertical micro-channel that is filled with porous media. The research utilizes asymmetrical convective cooling, a transverse magnetic field, and exothermic reactions to introduce a comprehensive model. The viscosity of the fluid, which is determined by the Naham-type law, changes significantly with temperature. Additionally, the convective heat transfer rates, which govern the asymmetric convective cooling at the surfaces of the micro-channel, are modeled using Newton's law of cooling, and Modified Darcy law is used to address the porous medium effect. To solve the complex system of nonlinear partial differential equations, we employ computational solutions derived from reliable and efficient finite difference methods (FDM). The method is tested for different time-steps and mesh sizes and it is found that the algorithms reliably produce the same results. Our investigation includes an evaluation of both spatial and temporal convergence of the FDM scheme. The study present qualitative discussions of graphical results, that highlight the impact of various flow parameters integrated into the system. The velocity and temperature profiles increases for higher values of thermal Grashoff number Gr and Reynolds number Re while opposite effect is noticed for increasing values of Hartman number M.

Impact of Climate Change on Maximum Sustained Wind Radius and its Associated Storm Surge Estimation along the Coast of Bangladesh

This research is a basic study on cyclones. The changing behaviour of storm surge in the Bay of Bengal due to the impact of climatic changes has been analyzed in this study. Certain characteristics of cyclones, such as the maximum sustained wind radius, have been analyzed, and their effect due to climatic change has been determined. The correlation between the maximum sustained wind radius and surge height was observed for this purpose. To accurately determine surge height, the vertically integrated shallow water wave model equation was employed, and it was solved using the semi-implicit finite difference method through the Arakawa-C grid. The surge model was performed by increasing and decreasing its wind radius by 10% to 20%, based on changes in the maximum sustained wind radius due to the effects of climate change. A strong conclusion was reached from the obtained results, indicating a significant effect of the maximum sustained wind radius on storm surge. But if it increases, there is a visible change in storm in some area of the coast of Bangladesh. For example, 1% increase in wind radius, the surge height increases by 0.032m, where the storm strikes. In some areas far away from where the storm hits the rising rate of the surge height is much lower. Finally, it may be stated that the surge height is affected by the maximum sustained wind radius and that it is altered by climatic impacts.

Modelling and Analysis of the Dispersal of a Polymeric Pollutant Injected into a Channel Flow of a Newtonian Liquid

The transient dynamics of nonlinear dispersion of a polymeric pollutant ejected by an external source into a laminar flow of a Newtonian liquid flowing through a rectangular channel is investigated. The Boussinesq approximation is assumed for the density variation with pollutant concentration. The governing equations of mass and momentum conservation are coupled to the pollutant concentration equation as well as to the viscoelastic constitutive model for the polymer stresses. The Oldroyd-B viscoelastic constitutive model is employed to model the deformation and characteristics of the polymer stresses. The coupled system of nonlinear partial differential equations is solved numerically using robust and efficient semi-implicit finite difference methods (FDM). Solutions are presented in graphical form for various parameter values. The model can be a useful tool in understanding the dynamics of domestic and industrial pollution situations that may arise from improper discharge of long-chain hydrocarbon products into, say, water drainage systems. The novelty of this investigation is in the modelling of the long-chain hydrocarbon-product pollutants via appropriate viscoelastic (polymeric) constitutive equations. In general, it is observed that parameters which increase (decrease) the flow velocity correspondingly increase (respectively decrease) the wall shear stress. Similarly, it is observed that parameters which increase (decrease) the polymer concentration correspondingly increase (respectively decrease) the mass transfer rates. The wall shear stress and mass transfer are measurable quantities. In this respect, our work offers such measurements as predictive tools to detect the scale of contamination.

Assessing the Impact of Extensive Severe Cyclonic Storm Fani on the Coastal Communities of Bangladesh: A Case Study

This research article presents a numerical simulation of the extensive severe cyclonic storm Fani and its impact along the coast of Bangladesh, which made landfall in the coastal region of Odisha, India, on May 3, 2019. A semi-implicit finite difference method in Cartesian coordinates was employed in this study to solve vertically integrated shallow water equations. The approach allowed for effective forecasting of the storm Fani's impact on the region of choice. Our considered physical domain is discretized with high-resolution gird to cover all the big and small offshore islands. The model predicted water levels for a total of sixteen coastal stations of the Bay of Bengal along the coast of Bangladesh (from 2 May to 03 May 2019, at 3-h interval). Simulated water levels are found closely co-related to the reported data. The simulation results demonstrate that the semi-implicit finite difference method is an effective tool for simulating the storm surge and flooding caused by any severe cyclonic storm. The study also shows that the storm surge and flooding caused by Fani were significant, with a maximum surge height of over 5 meters in some areas. The simulated results provide insights into the spatial and temporal evolution of the storm surge and flooding, which can be useful for designing and implementing appropriate disaster management strategies in the affected regions. Overall, this research article contributes to the scientific understanding of the behavior of severe cyclonic storms and the associated storm surge and flooding, and provides a valuable tool for policymakers and stakeholders to develop and implement effective disaster management strategies in cyclone vulnerable coastal regions.

An Analysis of Track Effects on Storm Surge Simulation

This study investigates the effects of cyclone tracks on water levels during storm surges along the coast of Bangladesh. A numerical hydrodynamical model in Cartesian coordinates has been developed, where the developed equations are solved by a semi-implicit finite difference method. The boundary fitted grid technique is employed to rectangularized the physical domain. Along the north-east corner of the model, the Meghna River fresh water discharge is taken into account. A stable tidal regime over the region of interest is generated with the help of four major tidal constituents, namely M_2 (principal lunar semi diurnal), S_2 (principal solar semi diurnal), O_1 (principal lunar diurnal) and K_1 (principal lunar diurnal) along the southern open boundary of the outermost model. This previously generated tidal regime is used as the initial state of sea in order to obtain water levels due to the nonlinear interaction of tide and surge. Numerical experiments are made with the severe storm April 1991 and some hypothetical storm tracks. The simulated results with the real track are found to be in a good agreement with some observed and some reported data. The water levels are found to be greatly influenced by the hypothetical cyclone tracks.

Numerical Treatment of Allen’s Equation Using Semi Implicit Finite Difference Methods

Numerical Treatment of Allen’s Equation Using Semi Implicit Finite Difference Methods

An energy stable finite difference method for anisotropic surface diffusion on closed curves

In this paper, we develop an energy stable finite difference method for the problem of curve motion under anisotropic surface diffusion. The motion of curve by anisotropic surface diffusion is governed by the fourth-order (highly nonlinear) geometric evolution equation. As in Li and Bao (2021), we first split the fourth-order evolution equation into two second-order equations, where the position vector of curve and the weighted curvature are treated as unknowns. Instead of using the arclength parameter, we introduce a Lagrangian coordinate parameter such that a closed curve can be parametrized over a fixed interval so that the equations can be represented using the new parameter. We then propose a linearly semi-implicit finite difference method to discretize these two equations, and prove that the scheme satisfies discrete energy dissipation so it is energy stable under suitable condition on the anisotropic surface energy. To show the applicability of our present method, we perform several numerical tests on various initial curves and different anisotropic energies. The numerical results show that our scheme is indeed energy dissipative and conserves even the total area well.

Semi-implicit finite-difference methods to study the spin-orbit- and coherently-coupled spinor Bose–Einstein condensates

We develop time-splitting finite-difference methods, using an implicit Backward–Euler and a semi-implicit Crank–Nicolson discretization schemes, to study the spin-orbit-coupled (SO-coupled) spinor Bose–Einstein condensates with coherent coupling in quasi-one- and quasi-two-dimensional traps. The split equations involving kinetic energy and spin-orbit-coupling operators are solved using either a time implicit Backward–Euler or a semi-implicit Crank–Nicolson method. We explicitly develop the methods for pseudospin-1/2, spin-1 and spin-2 condensates. The results for ground states obtained with time-splitting Backward–Euler and Crank–Nicolson methods are in excellent agreement with time-splitting Fourier spectral method, which is one of the popular methods to solve the mean-field models for SO-coupled spinor condensates. We confirm the emergence of different phases in SO-coupled pseudospin-1/2, spin-1 and spin-2 condensates with coherent coupling.

Comparative Response of Newtonian and Non-Newtonian Fluids Subjected to Exothermic Reactions in Shear Flow

A comparative computational study of the thermal response of Newtonian and non-Newtonian fluids subjected to exothermic reactions is conducted in simple shear flow. The investigations conducted in this study are of fundamental importance to industrial and biological applications in which heat generation minimization is important, such as in heat exchangers, in lubrication, and in internal medicine. Specifically, the comparative investigations central to this study are conducted on four types of fluids, namely; Newtonian fluids, generalized Newtonian fluids, viscoelastic fluids, and generalized viscoelastic fluids. The Oldroyd-B constitutive model is used for the viscoelastic fluids and a Carreau viscosity constitutive model is used to describe the viscosity shear-rate dependence of the generalized fluids. Efficient semi-implicit finite difference methods are employed to obtain the numerical solutions to the governing systems of equations. The computational methodologies are implemented in the MATLAB software. The sensitivity of the fluid temperature and the polymer stresses to increases in shear-thinning characteristics as well as to increases in polymeric properties is investigated. In general, it is observed that, at comparative parameter values, the viscoelastic fluids give the best resistance to temperature increases, followed by the generalized viscoelastic fluids, followed in turn by the Newtonian fluids, and with the generalized Newtonian fluids recording the highest temperature increases.

Heat transfer in an unsteady vertical porous channel with injection/suction in the presence of heat generation

This paper studies convective heat transfer in an unsteady vertical porous channel with suction/injection in the presence of heat generation. A semi-implicit finite difference method was employed to solve the Cartesian coordinate forms of the governing equations. Subsequently, the entropy generation number for varying values of intrinsic parameters, as well as the temperature and velocity profile, Bejan number and irreversibility of the system were successfully calculated. We observed that the convective cooling by suction is more effective on thermal properties than injection convective cooling. Hence, the process of entropy can be enhanced by convective heat transfer and viscous dissipation.

Modelling of Shear Banding in Plane Couette Flow of Non-linear Viscoelastic Fluids

We investigate shear banding phenomena in the plane Couette flow of viscoelastic fluids.The viscoelastic fluids are modelled viathe Giesekus constitutive equations with stress diffusion. The nonlinear and coupled systems of equations, comprising the momentum equation and the constitutive equations, are solved numerically using semi-implicit finite difference methods. Theeffects of the fluid parameters on the flow variables are also investigated. Under certain values of the material parameters, we observe formation of shear bands in plane Couette flow.

Analysis of unsteady viscous dissipative poiseuille fluid flow of two-step exothermic chemical reaction through a porous channel with convective cooling

Abstract In this study, the viscous dissipation of heat transfer of poiseuille fluid flow of two-step exothermic chemical reaction through a porous channel is investigated under different chemical kinetics. The flow is pressure-driven past fixed plates with heat generation. The upper wall surface is exposed to heat exchange with the temperature of the ambient aligning with the Newton’s law of cooling. The modeled coupled partial differential equations are non-dimensional and solved using stable and unconditional semi-implicit convergent finite difference method. The results are represented graphical and discussed accordingly for the velocity and temperature profiles showing the effect of some embodiment parameters entrenched in the system. From the results it was noticed that an increase in Frank-Kamenetskii parameters produces significant rises in the temperature distribution.

Simulation of Sedimentation in Lake Taihu with Geostationary Satellite Ocean Color Data

In this study, the goal is to estimate the sedimentation on the bottom bed of Lake Taihu using numerical simulation combined with geostationary satellite ocean color data. A two-dimensional (2D) model that couples the dynamics of shallow water and sediment transport is presented. The shallow water equations are solved using a semi-implicit finite difference method with an Alternating Direction Implicit (ADI) method. Suspended sediment transport is simulated by solving the general convection-diffusion equation with resuspension and deposition terms using a second-order explicit central difference method in space and two-step Adams–Bashforth method in time. Moreover, the total suspended particulate matter (TSM) is retrieved by the world’s first geostationary satellite ocean color sensor Geostationary Ocean Color Imager (GOCI) using atmospheric correction algorithm for turbid waters using ultraviolet wavelengths (UV-AC) and regional empirical TSM algorithm. The 2D model and GOCI-retrieved TSM are applied to study the sediment transport and sedimentation in Lake Taihu. Validation results show rationale TSM concentration retrieved by GOCI, and the simulated TSM concentrations are consistent with GOCI observations. In addition, simulated sedimentation results reveal the dangerous locations that must be observed and desilted.

Error Estimates of a Regularized Finite Difference Method for the Logarithmic Schrödinger Equation

We present a regularized finite difference method for the logarithmic Schr\"odinger equation (LogSE) and establish its error bound. Due to the blow-up of the logarithmic nonlinearity, i.e. $\ln \rho\to -\infty$ when $\rho\rightarrow 0^+$ with $\rho=|u|^2$ being the density and $u$ being the complex-valued wave function or order parameter, there are significant difficulties in designing numerical methods and establishing their error bounds for the LogSE. In order to suppress the round-off error and to avoid blow-up, a regularized logarithmic Schr\"odinger equation (RLogSE) is proposed with a small regularization parameter $0<\varepsilon\ll 1$ and linear convergence is established between the solutions of RLogSE and LogSE in term of $\varepsilon$. Then a semi-implicit finite difference method is presented for discretizing the RLogSE and error estimates are established in terms of the mesh size $h$ and time step $\tau$ as well as the small regularization parameter $\varepsilon$. Finally numerical results are reported to confirm our error bounds.

A radial basis function based ghost cell method with improved mass conservation for complex moving boundary flows

A sharp interface immersed boundary method is presented for simulating flows around moving boundaries with arbitrary complex geometries. A time semi-implicit finite difference method is used to solve the incompressible Navier–Stokes equations on a fixed, staggered Cartesian grid. The boundary conditions at the immersed interface are enforced by a ghost cell method. Tracking complex moving boundaries and suppressing pressure oscillations are two major challenges in the sharp interface method. In this work, a polynomial radial basis function (PRBF) is introduced to the ghost cell method to implicitly represent and reconstruct the arbitrary immersed boundaries. In addition, a simple and robust signed identification strategy is used to determine the phase state of the grid cells. To suppress violent pressure oscillations on the moving boundaries, a fractional area representation (FAR) method, together with a mass force term, is introduced to the pressure Poisson equation. This FAR method not only retains the desirable property of consistent discretization in the ghost cell method but also takes advantage of the mass conservation property of the cut cell method. The proposed method is validated using five test cases, including the flow around a hydrofoil, in-line oscillation of a cylinder in a static fluid, uniform flows around a transversely oscillating cylinder, twin oscillating cylinders, and a pitching hydrofoil. The present results are in good agreement with the reference results, which validates the accuracy and capability of the proposed method.

Dynamics of Trapped Solitary Waves for the Forced KdV Equation

The forced Korteweg-de Vries equation is considered to investigate the impact of bottom configurations on the free surface waves in a two-dimensional channel flow. In the study of shallow water waves, the bottom topography plays a critical role, which can determine the characteristics of wave motions significantly. The interplay between solitary waves and the bottom topography can exhibit more interesting dynamics of the free surface waves when the bottom configuration is more complex. In the presence of two bumps, there are multiple trapped solitary wave solutions, which remain stable between two bumps up to a finite time when they evolve in time. In this work, various stationary trapped wave solutions of the forced KdV equation are explored as the bump sizes and the distance between two bumps are varied. Moreover, the semi-implicit finite difference method is employed to study their time evolutions in the presence of two-bump configurations. Our numerical results show that the interplay between trapped solitary waves and two bumps is the key determinant which influences the time evolution of those wave solutions. The trapped solitary waves tend to remain between two bumps for a longer time period as the distance between two bumps increases. Interestingly, there exists a nontrivial relationship between the bump size and the time until trapped solitary waves remain stable between two bumps.

Thermal Stability Analysis in a Two-Step Reactive Cylindrical Stockpile

Thermal stability analysis in a cylindrical stockpile of reactive material undergoing a two-step low-temperature oxidation reaction is studied in this article. The reactant consumption in this case is neglected and heat transfer with thermal stability are investigated by application of the energy equation. The complicated combustion process results with nonlinear interactions and hence the nonlinear partial differential equation governing the problem is tackled numerically using the semi-implicit Finite Difference Method (FDM). Kinetic parameters embedded on the governing partial differential equation, are varied to study the behavior of the temperature during the combustion process. The results are depicted graphically and adequately discussed, to bring an understanding of heat transfer and thermal stability during the combustion process. The study in this paper is relevant to multistep combustion process analogous to the fuel combustion in automobiles. The results, in general, show that thermal stability is more for sensitized chemical kinetics than for bimolecular type of kinetics.

Thermodynamic Second Law Analysis of Hydromagnetic Gravity-Driven Two-step Exothermic Chemical Reactive Flow with Heat Absorption Along a Channel

This study examines the second law of thermodynamic gravity-driven viscous combustible fluid flow of two-step exothermic chemical reaction with heat absorption and convective cooling under bimolecular kinetic. The flow is acted uponby periodic changes in the axial pressure gradient and time along the axis of the channel with the existence of magnetic field. The heat convection at the channel surfaces with the environment are the same and satisfies Newtons law of cooling. The dimensionless main equations of the flow are solved using a convergent and stable semi-implicit finite difference method. The effect of some fluid parameters associated with the problem on momentum and temperature are obtained. The expression for irreversibility ratio, volumetric entropy generation and Bejan number along with the graphically results are presented and quantitatively discussed

Simulation of coupled hydro-mechanical-chemical phenomena in hydraulically fractured gas shale during fracturing-fluid flowback

Fracturing-fluid flowback in hydraulically fractured gas shale is a complicated transport behavior involving hydrodynamic (H), mechanical (M) and chemical (C) processes. Although, many flowback models have been published, none of the models flly coupled the transient fluid flow modeling with chemical-potential equilibrium and fracture closure phenomena.In this paper, a coupled hydro-mechanical-chemical (HMC) model based on hydrodynamics, linear-elastic fracture mechanics, and chemical-potential equations is presented to simulate the fracturing-fluid flowback behavior in hydraulically fractured gas shale. The HMC model takes into account a gas-liquid two-phase flow and a triple-porosity medium, which includes primary hydraulic fractures, secondary induced fractures and shale matrix. The flowback simulation with the HMC model accounts for all the important processes in fractured shale system, including (1)water transport driven by hydraulic, capillary and osmotic convections, (2)gas transport induced by both hydraulic pressure driven convection and adsorption, and (3)fracture closure considered as an elastic deformation. The fluid transport, coupled with rock deformation, are described by a set of partial differential equations. The semi-implicit finite-difference method is used to solve these equations.The evolution of pressure, saturation, salinity and aperture profiles of hydraulic fractures, induced fractures and matrix is calculated, revealing the multi-field coupled flowback behavior in fractured gas shale. The sensitivity analysis is also performed to investigate the physiochemical properties of shale on the water load recovery and gas production rate during the fracturing-fluid flowback. The results indicate that the capillarity has the most effect followed by chemical osmosis and induced fracture closure. The sub-irreducible water saturation has the weakest effect. Moreover, the matrix-related physiochemical parameters cause opposite influences on water load recovery and gas production rate. While, the influence of induced fracture-related parameters on both water load recovery and gas production rate is consistent. Results from this study are expected to explain which is the predominant mechanism for fracturing-fluid retention and the inherent relationship between fracturing-fluid flowback and gas production for different initial physiochemical properties of shale.