Efficient Finite Difference Method Research Articles

An Efficient Finite Difference Method for Stochastic Linear Second-Order Boundary-Value Problems Driven by Additive White Noises

An Efficient Finite Difference Method for Stochastic Linear Second-Order Boundary-Value Problems Driven by Additive White Noises

Numerical simulations of third-grade fluid flow with exothermic reactions in porous media-saturated vertical microchannels

The aim of this article is to analyze the electro-osmotic flow behavior of a third-grade (TG) incompressible fluid flowing in a vertical microchannel saturated with porous material. The electro-osmotic flow is regarded as an essential part of many scientific and engineering applications used in microfluidic devices. The flow is subjected to joule heating, exothermic reactions, and asymmetrical convective cooling. The viscosity which is assumed temperature dependent via a Naham-type law and the convective heat transfer rates governing the asymmetric convective cooling at the microchannel walls are modeled via Newton’s law of cooling. The governing system of nonlinear partial differential equations is solved computationally using an efficient and reliable finite difference method. The spatial and temporal convergence of the numerical scheme is showcased graphically. Our study findings reveal that the combination of exothermic reactions and Joule heating significantly influences the electroosmotic flow. Higher values of variable viscosity parameter (α), Brinkman number (Br), and Buoyancy parameter (Gr) result an enhancement in the flow variables. Conversely, a reverse trend is observed when augmenting the Prandtl number (Pr), non-Newtonian parameter (γ), and porous media parameters (Ω2). Numerical results for temperature and velocity are also discussed qualitatively and validated with the data previously published.

Explicit Topology Optimization of Voronoi Foams.

Topology optimization can maximally leverage the high DOFs and mechanical potentiality of porous foams but faces challenges in adapting to free-form outer shapes, maintaining full connectivity between adjacent foam cells, and achieving high simulation accuracy. Utilizing the concept of Voronoi tessellation may help overcome the challenges owing to its distinguished properties on highly flexible topology, natural edge connectivity, and easy shape conforming. However, a variational optimization of the so-called Voronoi foams has not yet been fully explored. In addressing the issue, a concept of explicit topology optimization of open-cell Voronoi foams is proposed that can efficiently and reliably guide the foam's topology and geometry variations under critical physical and geometric requirements. Taking the site (or seed) positions and beam radii as the DOFs, we explore the differentiability of the open-cell Voronoi foams w.r.t. its seed locations, and propose a highly efficient local finite difference method to estimate the derivatives. During the gradient-based optimization, the foam topology can change freely, and some seeds may even be pushed out of shape, which greatly alleviates the challenges of prescribing a fixed underlying grid. The foam's mechanical property is also computed with a much-improved efficiency by an order of magnitude, in comparison with benchmark FEM, via a new material-aware numerical coarsening method on its highly heterogeneous density field counterpart. We show the improved performance of our Voronoi foam in comparison with classical topology optimization approaches and demonstrate its advantages in various settings.

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.

Analysis of a non-integer order mathematical model for double strains of dengue and COVID-19 co-circulation using an efficient finite-difference method

An efficient finite difference approach is adopted to analyze the solution of a novel fractional-order mathematical model to control the co-circulation of double strains of dengue and COVID-19. The model is primarily built on a non-integer Caputo fractional derivative. The famous fixed-point theorem developed by Banach is employed to ensure that the solution of the formulated model exists and is ultimately unique. The model is examined for stability around the infection-free equilibrium point analysis, and it was observed that it is stable (asymptotically) when the maximum reproduction number is strictly below unity. Furthermore, global stability analysis of the disease-present equilibrium is conducted via the direct Lyapunov method. The non-standard finite difference (NSFD) approach is adopted to solve the formulated model. Furthermore, numerical experiments on the model reveal that the trajectories of the infected compartments converge to the disease-present equilibrium when the basic reproduction number ({mathbb {R}}_0) is greater than one and disease-free equilibrium when the basic reproduction number is less than one respectively. This convergence is independent of the fractional orders and assumed initial conditions. The paper equally emphasized the outcome of altering the fractional orders, infection and recovery rates on the disease patterns. Similarly, we also remarked the importance of some key control measures to curtail the co-spread of double strains of dengue and COVID-19.

Fast and efficient numerical method for solving the Allen–Cahn equation on the cubic surface

In this study, we present a fast and efficient finite difference method (FDM) for solving the Allen–Cahn (AC) equation on the cubic surface. The proposed method applies appropriate boundary conditions in the two-dimensional (2D) space to calculate numerical solutions on cubic surfaces, which is relatively simpler than a direct computation in the three-dimensional (3D) space. To numerically solve the AC equation on the cubic surface, we first unfold the cubic surface domain in the 3D space into the 2D space, and then apply the FDM on the six planar sub-domains with appropriate boundary conditions. The proposed method solves the AC equation using an operator splitting method that splits the AC equation into the linear and nonlinear terms. To demonstrate that the proposed algorithm satisfies the properties of the AC equation on the cubic surface, we perform the numerical experiments such as convergence test, total energy decrease, and maximum principle.

Efficient pure qP-wave simulation and reverse time migration imaging for vertical transverse isotropic (VTI) media

ABSTRACT Anisotropic pseudo-acoustic forward modeling and migration imaging are critical for high-precision seismic exploration. However, the wavefields simulated by the traditional coupled anisotropic acoustic wave equation have the problems of shear wave noise and numerical simulation instability for epsilon that is less than delta. Furthermore, although the pure anisotropic acoustic wave equation expressed by the differential operators can solve the aforementioned noise interference and numerical simulation instability issues, its numerical simulation calculation is large, particularly in 3D industrial applications, because it has to be solved by the spectral-based method. In this paper, a pure anisotropic acoustic wave equation in a vertical transverse isotropic (VTI) medium that can be numerically computed using the efficient finite difference method is derived. This equation not only eliminates noise interference and numerical simulation instabilities, but also allows for an efficient wavefield simulation. We also implement reverse time migration (RTM) using the proposed VTI pure anisotropic acoustic wave equation. Two synthetic tests and one field data test are performed to evaluate the accuracy and robustness of the developed VTI RTM. The imaging results show that the proposed VTI RTM can correct the anisotropy effect on seismic wave propagation and improve migration imaging precision.

Local-basis difference potentials method for elliptic PDEs in complex geometry

We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the difference potentials framework. The main novelty of the developed schemes is the use of local basis functions defined at near-boundary grid points. The use of local basis functions allows unified numerical treatment of (i) explicitly and implicitly defined geometry; (ii) geometry of more complicated shapes, such as those with corners, multi-connected domain, etc; and (iii) different types of boundary conditions. This geometrically flexible approach is complementary to the classical difference potentials method using global basis functions, especially in the case where a large number of global basis functions are needed to resolve the boundary, or where the optimal global basis functions are difficult to obtain. Fast Poisson solvers based on FFT are employed for standard, centered finite difference stencils regardless of the designed order of accuracy. Proofs of convergence of difference potentials in maximum norm are outlined both theoretically and numerically.

A fast and accurate elastic reverse-time migration method based on decoupled elastic wave equations

The extrapolation and decomposition of different mode waves are very important to efficiently obtain multi-wave depth images for elastic reverse-time migration (ERTM). To achieve this goal successfully, we firstly develop an efficient and accurate staggered-grid finite-difference method (SFDM) based on decoupled elastic wave equations. This developed SFDM uses two sets of difference operator lengths for decoupled elastic wave equations to respectively perform the extrapolation of decoupled P- and S-wavefields and employs the least squares (LS) to minimize the objective functions respectively constructed from the time-space dispersion equations of P- and S-waves to generate their corresponding difference coefficients. This developed method can theoretically be regarded as the optimal SFDM based on least squares (LSSFDM), and it can provide the high-precision separated wavefields of different mode waves simultaneously and reduce the computing time effectively for elastic wave modeling under the given accuracy requirements. The modeling results of different models illustrate that our developed LSSFDM based on decoupled elastic wave equations can generate the decomposed P- and S-wavefields and suppress their numerical dispersions effectively. Secondly, we present a fast and accurate ERTM method based on this developed LSSFDM to successfully achieve the multi-wave imaging results, including PP, PS, SP and SS images. The ERTM examples of different models demonstrate that our ERTM based on this developed LSSFDM can accurately obtain the high-precision depth images of multimode waves and effectively reduce the computing time of wavefield extrapolation.

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.

Modeling of heating and cooling behaviors of laminated glass facades exposed to fire with three-dimensional flexibilities

To develop a precise and efficient computer model for predicting the heating and cooling behaviors of laminated glass facades exposed to fire, there is an urgent need to reduce the huge computational requirements associated with simulating heat transfer in layered structures that feature a down-flowing water film. We overcome this challenge by proposing an efficient three-dimensional finite difference method (3DFDM) which has high numerical stability when solving heat transfer equations with water film and air convection. To capture the moving particles of the water film, we developed a unique computational algorithm for particle labeling, which has two significant advantages: (1) it eliminates the time-consuming process of searching for neighboring particles in conventional meshfree methods, and (2) it ensures that every main particle interacts only with limited neighboring particles without utilizing any weights, thus significantly reducing the computational effort. We validated our 3DFDM through experiments in heating and cooling scenarios and compared its thermal results with those obtained from commercial packages to demonstrate its high efficiency and accuracy. Furthermore, we examined the feasibility of our model in evaluating the effects of thickness of the interlayer and water film release time on the cooling behavior of laminated glass during a fire.

Computation of Stagnation Point Convection Flow of Carbon Nanotube Nanofluids From a Stretching Sheet With Melting: Dual Solutions

Abstract A theoretical study in stagnation point flow is presented where melting heat transfer effects of carbon nanotube (CNT) from a stretching surface is appeared. Both carbon nanotubes like single-wall CNT (SWCNT) and multiwall CNT (MWCNT) are homogeneously dispersed in the base fluid. As the ordinary (or base) fluids, water and kerosene oil are employed. A set of nonlinear ordinary differential equations with appropriate boundary conditions is formed by transforming the governing equations via similarity transformations. The transformed nonlinear ordinary differential equations are then solved numerically using the bvp4c solver in matlab, an efficient numerical finite difference method. The impact of nanoparticle volume fraction, velocity, melting, stretching parameter, and CNT type on transport characteristics are explored and visualized graphically and in tabular forms. Verification of the matlab computations with available data in certain limiting cases is included showing excellent agreement. Existence of dual (upper and lower branch) solution is shown for a certain range of stretching sheet parameter. The obtained dual solutions are examined for velocity and temperature in detail. A stability analysis demonstrates that the first solution is a stable solution, and the second solution is an unstable solution. Local skin friction and local Nusselt number are also computed in order to determine critical values that can permit dual solutions. It is observed that when a dimensionless melting parameter is greater than 1, SWCNT nanofluids attain greater velocities than MWCNT nanofluids for water as well as kerosene oil base fluids. Moreover, the flow is accelerated for SWCNT compared with MWCNT for both water and kerosene oil. With increasing stretching parameter, the heat transfer rate (Nusselt number) increases, whereas skin friction coefficients decrease. Higher skin friction and Nusselt number are obtained for SWCNTs compared to MWCNTs due to their greater density and thermal conductivity. The study is relevant to phase change manufacturing fluid dynamics of nanomaterials.

Efficient Finite Difference Methods for the Numerical Analysis of One-Dimensional Heat Equation

Efficient Finite Difference Methods for the Numerical Analysis of One-Dimensional Heat Equation

Efficient sixth-order finite difference method for the two-dimensional nonlinear wave equation with variable coefficient

Efficient sixth-order finite difference method for the two-dimensional nonlinear wave equation with variable coefficient

Collision of hybrid nanomaterials in an upper-convected Maxwell nanofluid: A theoretical approach

Many viscoelastic fluid problems are solved using the notion of fractional derivative. However, most researchers paid little attention to the effects of nonlinear convective in fluid flow models with time-fractional derivatives and were mainly interested in solving linear problems. Furthermore, the nonlinear fluid models with a fractional derivative for an unsteady state are rare, and these constraints must be overcome. On the other hand, nanofluids are thought to be trustworthy coolants for enhancing the cooling process in an electrical power system. Therefore, this research has been conducted to analyze the unsteady upper-convected Maxwell (UCM) hybrid nanofluid model with a time-fractional derivative. Incorporating the Cattaneo heat flux into the energy equation has increased the uniqueness of the research. The numerical solutions for the coupled partial differential equations describing velocity and temperature are presented using an efficient finite difference method assisted by the Caputo fractional derivative. Significant changes in heat transfer and fluid flow properties due to governing parameters, including the nanomaterial volume fraction, fractional derivative, relaxation time, and viscous dissipation, are graphically demonstrated. The nanomaterial concentration, the fractional derivative parameter, and the relaxation time parameter must all be substantial to manifest a surface heat increase.

Using Micropolar Nanofluid under a Magnetic Field to Enhance Natural Convective Heat Transfer around a Spherical Body

An analysis is discussed of the heat and mass transfer for micropolar nanofluid in presence of natural convection from a spherical body with magneto-hydrodynamic (MHD) effects. The constant wall temperature boundary condition is also studied. By employing proper similarity transformations, the governing equations are converted into a set of partial differential equations (PDEs) with the used boundary conditions, which can then be solved numerically via the efficient Keller-box implicit numerical finite difference method. The numerical results of impacts of the controlling parameters on heat transfer physical quantities have been presented, tabular and graphically, by MATLAB symbolic software. Comparisons of the current study results to previously published results show good agreement, indicating that our numerical computations are legitimate and accurate. Increasing nanoparticle volume fraction is observed to depress local skin friction, Nusselt number, and angular velocity while the reverse effects are observed for velocity and temperature.

Direct simulation Monte Carlo for new regimes in aggregation-fragmentation kinetics

We revisit two basic Direct Simulation Monte Carlo Methods to model aggregation kinetics and extend them for aggregation processes with collisional fragmentation (shattering). We test the performance and accuracy of the extended methods and compare their performance with the efficient deterministic finite-difference method applied to the same model. We validate the stochastic methods on the test problems and apply them to verify the existence of oscillating regimes in the aggregation-fragmentation kinetics recently detected in deterministic simulations. We confirm the emergence of steady oscillations of densities in such systems and prove the stability of the oscillations with respect to fluctuations and noise.

High order compact split step finite difference method for two-dimensional coupled nonlinear Schrödinger system

In this paper, a decoupled and efficient finite difference method is developed for two-dimensional coupled nonlinear Schrödinger (CNLS) system. The proposed method uses split step technique for the temporal discretization and high order compact (HOC) difference approximation for the spatial discretization. The original problem is decomposed into a two-dimensional linear subproblem and a two-dimensional nonlinear subproblem. For the two-dimensional linear subproblem, the Lie-Trotter splitting formula is adopted in time to reduce computational cost. While for the nonlinear subproblem, it can be integrated directly and exactly. By the von Neumann approach, it is showed that the proposed method is unconditionally stable. Numerical examples are conducted to compare it with other scheme and numerical results verified the superiority of the proposed method in terms of accuracy and efficiency. The new method also exhibits good numerical performance in long-time simulation.

A higher-order finite difference method for two-dimensional singularly perturbed reaction-diffusion with source-term-discontinuous problem

This paper considers a two-dimensional singularly perturbed reaction-diffusion equation with a discontinuous source term. Due to this discontinuity, interior, corner, and boundary layers appear in the solution for adequately small values of the perturbation parameter ϵ. To achieve a decent estimate of the solution, we construct a numerical approach adopting an efficient hybrid finite difference method that includes a proper layer adapted piece-wise uniform Shishkin mesh. Further, we prove that the hybrid finite difference method is almost second-order uniformly convergent with respect to the perturbation parameter. We have implemented our method to test examples. Numerical results are verifying the theoretical results.

Modeling of pure visco-qP-wave propagation in attenuating tilted transversely isotropic media based on decoupled fractional Laplacians

The pseudoviscoacoustic anisotropic wave equation is widely used in the oil and gas industry for modeling wavefields in attenuating anisotropic media. Compared to the full viscoelastic anisotropic wave equation, it can greatly reduce the computational cost of wavefield modeling while maintaining the visco-qP-wave kinematics very well. However, even if we place the source in a thin isotropic layer, there will be some unwanted S-wave artifacts in the qP wavefield simulated by the pseudoviscoacoustic anisotropic wave equation due to the stepped approximation of inclined layer interfaces. Furthermore, the wavefield simulated by the pseudoviscoacoustic anisotropic wave equation may suffer from numerical instabilities when the anisotropy parameter epsilon is less than delta. To overcome these problems, we derive a pure-viscoacoustic tilted transversely isotropic (TTI) wave equation in media with anisotropy in velocity and attenuation based on the exact complex-valued phase velocity formula. The pure-viscoacoustic TTI wave equation has decoupled amplitude dissipation and phase dispersion terms, which is suitable for further reverse time migration with Q compensation. For numerical simulations, we adopt the second-order Taylor series expansion to replace the mixed-domain spatially variable fractional Laplacian operator, which guarantees the decoupling of the wavenumber from the space-related fractional order. Then, we use an efficient and stable hybrid finite-difference and pseudospectral method (HFDPSM) to solve the pure-viscoacoustic TTI wave equation. Numerical tests indicate that the simulation results of the newly derived pure-viscoacoustic TTI wave equation are stable, free from S-wave artifacts, and accurate. We further demonstrate that HFDPSM outperforms the pseudospectral method in terms of numerical simulation stability and computing efficiency.