Semi-implicit Finite Difference Research Articles

Numerical simulation of heat transfer and entropy generation due to the nanofluid natural convection with viscous dissipation in an inclined square cavity

In this work, the effects of viscous dissipation on heat transfer and entropy generation of the nanofluid natural convection in an inclined square cavity are numerically investigated. A fractional-step semiimplicit finite difference algorithm based on the projection method on a staggered grid is proposed to solve the laminar natural convection problem, which has the advantage of larger time steps. The square cavity is filled with a nanofluid composed of (Cu) copper nanoparticles and water. and the viscous dissipative behavior of the mixture flow is not negligible. The studied parameters are Rayleigh number Ra ( 10 4 and 10 5 ), Eckert number Ec ( 0 − 2 ), the volume fraction of solid particles ϕ ( 0 − 0.06 ), and inclination angle of square cavity α ( 0 − π / 2 ). The Prandtl number is fixed to Pr = 6.2 . The results show that at any inclination angle, the increase in viscous dissipation leads to weakened heat transfer on the hot wall, enhanced heat transfer on the cold wall, and weakened flow in the square cavity. For the base solution without nanoparticles, the Eckert number has the greatest impact, with its effects on the average Nusselt number on the hot wall, maximum streamfunction, and average Nusselt number on the cold wall being ∼13, 0.15, 18.69%, and at Ra = 10 4 , and 12, 0.5, 28.87% at Ra = 10 5 , respectively. Research on entropy generation shows that as Eckert number increases, the entropy generation due to heat transfer increases, while due to fluid friction decreases. As the Rayleigh number increases, the effect of viscous dissipation increases. As the inclination angle increases, the effects of volume fraction and Eckert number weaken. Adding solid particles can effectively weaken the effect of viscous dissipation.

Chaotic Communication Systems with Signal Modulation Based on Controlled Symmetry of Semi-Implicit Finite-Difference Models

The article is devoted to investigation coherent communication system model with a new method of signal modulation based on variable symmetry of finite-difference schemes with subsequent experimental analysis of the effectiveness of different modulation techniques. The aim of the study is to investigate a computer model of chaotic communication system with signal modulation based on variable symmetry of semi-implicit finite-difference schemes. Novelty: elements of scientific novelty have finite-difference models of receivers/transmitters, allowing to realize a new method of modulation of chaotic signals. Result: obtaining a simulation model of coherent chaotic communication systems with tools for covertness and noise immunity analyses. Practical relevance: The simulation model of chaotic communication system is a necessary tool for analyzing the performance of the system before its physical implementation.

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.

Unconditionally stable higher order semi-implicit level set method for advection equations

We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of advection by an external velocity and by a speed in the normal direction that are applicable in level set methods. The recommended numerical scheme is third order accurate for the linear advection in the 2D case with a space dependent velocity. Using a combination of analytical and numerical tools in the von Neumann stability analysis, the third order scheme is claimed to be unconditionally stable. We also present a simple high-resolution scheme that gives a TVD (Total Variation Diminishing) approximation of the spatial derivative for the advected level set function in the 1D case. In the case of nonlinear advection, a semi-implicit discretization is proposed to linearize the problem. The compact implicit part of the stencil of numerical schemes contains unknowns only in the upwind direction. Consequently, algebraic solvers like the fast sweeping method can be applied efficiently to solve the resulting algebraic systems. Numerical tests to evolve a smooth and non-smooth interface and an example with a large variation of the velocity confirm the good accuracy of the third order scheme even in the case of very large Courant numbers. The advantage of the high-resolution scheme is documented for examples where the advected level set functions contain large jumps in the gradient.

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.

The splitting-based semi-implicit finite-difference schemes for simulation of blood flow in arteries

This paper is devoted to construction and analysis of splitting-based finite-difference schemes for simulation of blood flow in applications of one-dimensional models. Proposed schemes are constructed for inviscid and viscid models of blood. The proposed approach is based on the modification of averaged incompressibility condition, rewritten for the square root of a cross-sectional area. It is approximated by an absolutely stable scheme with a skew-symmetric operator. The constructed schemes can be classified as semi-implicit. The stability of proposed schemes is analyzed by the method of energy inequalities. Sufficient stability conditions are obtained. For the problems with analytical solutions, it is demonstrated that a second-order convergence rate is achieved in practice.The proposed schemes are compared with the widely used explicit finite-difference schemes, such as Lax–Wendroff, Lax–Friedrichs and McCormack schemes. The benchmark problems for simulation of flows in the following vascular systems are considered: human carotid artery, human aorta with bifurcation, vessel with bifurcation and absorbing outflow conditions, and networks with 31, 63, and 127 vessels. It is demonstrated that the computations based on the splitting-based semi-implicit schemes can be performed much faster than for the explicit schemes with close values of relative errors.

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.

Fractional-order diffusion coupled with integer-order diffusion for multiplicative noise removal

The problem of multiplicative noise removal has attracted much attention in recent years. Unlike additive noise, multiplicative noise destroys almost all information of an image; therefore, it is more difficult to remove them from corrupted images. In this paper, we propose a denoising model of fractional-order diffusion coupled with integer-order diffusion for multiplicative gamma noise, which takes advantage of both texture-preserving property of fractional-order diffusion and edge-preserving property of integer-order diffusion. We explore the mutual mechanism of two diffusion equations in the diffusion process, i.e., mutual transfer of texture and edge information respectively from two filtering images, and mutual regularization on the diffusion coefficients in both equations. We design an alternating numerical scheme based on semi-implicit finite difference and discrete Fourier transform for solving the coupled diffusion system. The proposed model is tested on some commonly-used images and is compared with six state-of-the-art models, qualitatively and quantitatively. Experimental results show the superiority of the proposed model in reducing multiplicative gamma noise while preserving textures and edges.

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.

Numerical Investigation of Natural Convention to a Pseudoplastic Fluid in a Long Channel using a Semi-Implicit Scheme

We develop and computationally analyze a mathematical model for natural convection to a non-Newtonian fluid in a long and thin channel. The channel is bounded by antisymmetric heated and cooled walls and encloses a non-Newtonian pseudoplastic fluid. The flow and heat transfer characteristics are investigated subject to the prevailing buoyancy forces resulting from the combined natural convection and gravitational effects. An efficient and accurate semi-implicit finite difference algorithm is implemented in time and space to analyse the model equations. In the case when the fluid flow and heat transfer are sustained for a long enough time to allow for steady states to develop, the model equations would reduce to a boundary value problem. Even in such cases, we demonstrate that, by recasting the problem as an initial boundary value problem, our numerical algorithms would still converge in time to the relevant, steady-state solutions of the original boundary value problem. We also demonstrate the dependence of solutions on the embedded parameters at a steady state.

High resolution finite difference schemes for a size structured coagulation-fragmentation model in the space of radon measures.

In this paper, we develop explicit and semi-implicit second-order high-resolution finite difference schemes for a structured coagulation-fragmentation model formulated on the space of Radon measures. We prove the convergence of each of the two schemes to the unique weak solution of the model. We perform numerical simulations to demonstrate that the second order accuracy in the Bounded-Lipschitz norm is achieved by both schemes.

Development of a Regional Numerical Tidal Forecast Model Along the Coast of Bangladesh and Its Associated Tidal Height Estimation

A two-dimensional shallow water equation in the Cartesian coordinate model is developed to estimate the tide along the coast of Bangladesh. The model was developed and discretized by a finite difference method. The model equation was discretized by the Finite Difference Method (FDM) considering the forwarding of time and central in space. In our model, the coastal boundaries, islands, and some small rivers are approximated by an exact stair step representation and solved by a conditional stable semi-Implicit Finite Difference Technique in a Structured Arakawa C-Grid system. Ignoring the wind stress created stable tidal conditions along the southern open boundary of the ocean which was the astronomical M2 component. The desired results are obtained after running the model for a fixed period of time since the steady state. The model results give a reasonable agreement with the observed data. The root means square error of the mean also shows good reasonable agreement.

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.

Effect of Coriolis Force on the Numerical Estimation of Water Level Elevation Due to a Catastrophic Cyclone along the Coast of Bangladesh

A two-dimensional vertically integrated shallow water equation in the Cartesian coordinate model is used to estimate the water level considering the impact of Coriolis force. The shallow water model equation was discretized by a finite difference method (FDM). Consider the forwarding of time and central space as a moderator of this discretization. The model approximates coastal boundaries, small islands, small rivers and complex tributaries by an accurate stair step representation. The model equations are solved by a static semi-implicit finite difference technique where a structured Arakawa C-grid system is used as the condition. A one-way nested scheme technique is used to incorporate complex land-sea interfaces such as small offshore islands and water depths with sufficient accuracy as well as decreasing the computational cost. A stable tidal condition was created by applying M2 tidal forcing with the largest tide along the southern open boundary of the Bay of Bengal. The model uses the Coriolis force as an external force that can affect water buoyancy. The main task is to analyse the effect of Coriolis force on water buoyancy. The described model was applied to simulate sea-surface elevation associated with the severe cyclone in April 1991 that strike on the east coast of Bangladesh. We have found a significant impact of Coriolis force on surge height. However, the model gives an accurate numerical estimate of surge height.

Positivity-preserving and energy-dissipative finite difference schemes for the Fokker–Planck and Keller–Segel equations

Abstract In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker–Planck equation and the Keller–Segel equations. The two proposed schemes are first-order accurate in time, explicitly solvable, and second-order and fourth-order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proved to be positivity preserving and energy dissipative: the second-order scheme can achieve so unconditionally while the fourth-order scheme only requires a mild time step and mesh size constraint. In particular, the fourth-order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions.

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.

Numerical investigation of the onset of three-dimensional characteristics in flow past a pair of square cylinders at various arrangements

An incompressible Navier–Stokes solver has been developed to analyze the three-dimensional viscous flow around a pair of square cylinders at a low Reynolds number (Re = 100) at various arrangements. A semi-implicit finite difference scheme was employed on a staggered grid, which facilitated the easier application of boundary conditions while achieving satisfactory accuracy and stability. A third-order upwind (Kawamura–Kuwahara) scheme was applied to discretize the convective terms and a second-order central difference scheme to discretize the diffusive terms. Time integration was performed using a second-order accurate Adams–Bashforth scheme. An iterative pressure–velocity correction was used to obtain a divergence-free velocity field. Aspects of the flow such as vorticity profiles, the time evolution of lift and drag coefficients, and three-dimensionality appearance were studied. Three-dimensional characteristics ceased to appear in a tandem arrangement, irrespective of the gap ratio. In contrast, the side by side arrangement showed a clear three-dimensional transition in the flow when analyzing the vortex structures. Further investigations carried out for staggered arrangement keeping the center to center distance constant and varying the stagger angle (α) reveal that the three-dimensional behavior starts at α = 30° and keeps progressing with the angle increase.

Convergence of a higher order finite difference scheme for two-phase Stefan problems.

This work establishes a sufficient condition for convergence of a semi-implicit finite difference scheme with variable space grid of fourth order of accuracy in space, previously set by the authors. The scheme is intended to solve one-dimensional two-phase Stefan problems. The proofs of the above-mentioned results are based on estimates which are obtained and subsequently employed in order to find an upper bound of the discretization error both at phases and the interface. The computational behavior of the scheme under the sufficient condition is corroborated by a series of numerical tests.