Backward Finite Difference Method Research Articles

Graded mesh modified backward finite difference method for two parameters singularly perturbed second-order boundary value problems

The existence of boundary layers in the solutions of two-parameter singularly perturbed boundary value problems makes classical numerical methods insufficient in providing accurate approximations. Consequently, the development of layer-adapted mesh methods that achieve parameter uniform convergence and are specifically designed to accurately handle these layers becomes highly significant. The aim of this paper is to construct and examine a numerical approach for obtaining approximate solutions to a specific class of two-parameter singularly perturbed second order boundary value problems whose solutions exhibit boundary layers at both ends of the domain. The problem is discretized by employing a modified backward finite difference method on a mesh that is graded. To validate the theoretical findings, well known test problems from the existing literature are utilized. Moreover, the efficiency of the proposed method is demonstrated by comparing it to other existing methods in the literature. The stability and uniform convergence with respect to the parameters of the proposed method have been verified, revealing that it attains second-order convergence in the maximum norm. The numerical outcomes illustrate that the proposed method offers remarkably accurate approximations of the solution. Theoretical findings are in agreement with the experimental results.

On a moving boundary problem associated with the swelling drug release platforms

This paper studies a parabolic-type system of time-dependent partial differential equations coupled with Robin boundary conditions to model the solvent penetration and subsequently a controlled drug release from a swellable polymeric spherical platform. The model's key feature is that the solvent penetration and the polymer swelling result in moving boundaries. From an analytical perspective, some properties of the solutions, such as the positivity and the uniqueness of the solutions, are established. A second-order iterative procedure based on the backward finite difference method is carried out to solve the problem numerically. Finally, the numerical results and the asymptotic solutions are compared, intending to illustrate the validity and applicability of the numerical scheme.

Numerical Solution of Linear Fractional Differential Equation with Delay Through Finite Difference Method

This article addresses a new numerical method to find a numerical solution of the linear delay differential equation of fractional order , the fractional derivatives described in the Caputo sense. The new approach is to approximating second and third derivatives. A backward finite difference method is used. Besides, the composite Trapezoidal rule is used in the Caputo definition to match the integral term. The accuracy and convergence of the prescribed technique are explained. The results are shown through numerical examples.

An estimation of water levels associated with a storm along the coast of Bangladesh using a non-central difference method of lines

In this paper, depth-averaged shallow water equations were solved using a non-central difference method of lines together with the classical RK(4,4) method to estimate water levels associated with a cyclone along the coast of Bangladesh. A 4-point backward finite difference method was adopted in discretizing spatial derivatives retaining time derivatives as continuous. Thus, the shallow water equations together with the boundary conditions were transformed into a system of initial value problems, which was solved using the RK(4,4) method. A nesting approach was employed to incorporate the land-sea interface and bottom topography effectively with economic cost. A stable tidal oscillation was generated in the domain applying the M2 tidal constituent along the southern boundary of the parent model and the surge model was then run to get water levels owing to dynamic tide-surge interaction. Numerical experiments were performed to simulate water levels due to tide, surge, and nonlinear tide-surge interactions associated with the cyclonic storm, AILA. The obtained results were in good agreement with those attained in some previous studies and observations and the WLs due to the nonlinear tide-surge interaction were in a reasonable agreement with respect to the root mean square error values and predictive skills.

Large eddy simulation of the wind flow in a realistic full-scale urban community with a scalable parallel algorithm

Turbulence and large computational domains make urban wind flow simulation a computationally challenging problem. To reduce the total simulation time and efficiently use today's new generation of supercomputers with massive processors, scalable parallel solvers are needed. In this work, we present a scalable domain decomposition method based 3D incompressible Navier-Stokes solver for the simulation of unsteady, complex wind flows around urban communities. A large eddy simulation with Smagorinsky modeling is employed for the simulation of the turbulent wind flows. On a fine unstructured grid, we discretize the spatial and temporal dimensions using a stabilized P1−P1 finite element method and an implicit backward Euler finite difference method, respectively. The resulting large-scale nonlinear algebraic system at each timestep is solved by a Newton-Krylov-Schwarz algorithm in parallel. The solver is first validated by a benchmark case where the numerical results are compared with measured data from a wind tunnel. Then, the wind flow field of a realistic actual-scale urban community with a group of buildings in the downtown area of Shenzhen, China, is studied. The numerical results show that the flow field matches with the experimental data for the benchmark problem, and some reasonable, detailed, and complex flow structures are obtained for the urban community simulation case. The scalability of the solver is studied on the TianHe-2A and Sunway TaihuLight supercomputers, and the solver scales up to 16,384 processor cores for a grid with over 20 million elements, which implies that the solver has the potential to perform fast and high-fidelity simulations of large-scale wind flows in complicated computational domains.

Numerical approximations for space–time fractional Burgers’ equations via a new semi-analytical method

The objectives of this research describes a novel meshless technique for solving space–time fractional Burgers’ equation by using concept of weighted and Caputo fractional derivative type. An efficient and accurate meshfree method based on backward substitution method and finite difference method is applied for problem on the various computational domain with scattering nodes. According to proposal method, Crank–Nicolson techniques is exploited to find the approximation in time level and the backward substitution method is utilized to compute node on the boundary. Subsequently, we obtain the corresponding weighted parameters the governing equations, we implemented collocation approach based on radial basis function. To eliminate nonlinearity, quasilinearization technique is applied to transform nonlinear source term into a linear source term. For investigation accuracy and reliably, this paper is compared with other previous researches. It is found that proposed scheme outperforms compared with existing numerical method.

Semi-Analytical Solutions for Barrier and American Options Written on a Time-Dependent Ornstein–Uhlenbeck Process

In this article, we develop semi-analytical solutions for the barrier (perhaps, time-dependent) and American options written on the underlying stock that follows a time-dependent Ornstein–Uhlenbeck process with a lognormal drift. Semi-analytical means that given the time-dependent interest rate, continuous dividend and volatility functions, one need to solve a linear (for the barrier option) or nonlinear (for the American option) Volterra equation of the second kind (or a Fredholm equation of the first kind). After that, the option prices in all cases are presented as one-dimensional integrals of combination of the preceding solutions and Jacobi theta functions. We also demonstrate that computationally our method is more efficient than the backward finite difference method traditionally used for solving these problems, and can be as efficient as the forward finite difference solver while providing better accuracy and stability. TOPICS:Derivatives, options, statistical methods Key Findings ▪ For the first time the method of generalized integral transform, invented in physics for solving an initial-boundary value parabolic problem at [0, y(t)] with a moving boundary [y(t)], is applied to finance. ▪ Using this method, pricing of barrier and American options, where the underlying follows a time-dependent OU process (the Bachelier model with drift) are solved in a semi-analytical form. ▪ It is demonstrated that computationally this method is more efficient than the backward and even forward finite difference method traditionally used for solving these problems whereas providing better accuracy and stability.

Design of Atangana–Baleanu–Caputo fractional-order digital filter

Atangana–Baleanu–Caputo (ABC) fractional differential operator based upon Mittag-Leffler kernel exhibits all the advantages of conventional Riemann–Liouville and Caputo fractional differential operators; in addition, the kernel associated is non-singular. Therefore, this paper puts forward a closed-form analytical formulation for the design of an ABC-based fractional-order FIR filter for various signal processing and filtering applications. The closed-form expression is derived by utilizing backward finite difference method and fractional sample delay interpolation techniques. Furthermore, several design examples are considered to illustrate the effectiveness of the proposed method. From the analytical and simulation studies done, it is observed that the proposed design efficiently approximates the ideal frequency response of ABC-fractional differential operator. Finally, one-dimensional and two-dimensional applications of the proposed method are validated and compared against state-of-the-art methods for electrocardiogram (ECG) R-peak detection as well as for digital image sharpening.

Using a shifted Grünwald‐Letnikov scheme for the Caputo derivative to study anomalous solute transport in porous medium

SummaryWe propose an extension of the shifted Grünwald‐Letnikov method to solve fractional partial differential equations in the Caputo sense with arbitrary fractional order derivative α and with an advective term. The method uses the relation between Caputo and Riemann‐Liouville definitions, the shifted Grünwald‐Letnikov, and the traditional backward and forward finite difference method. The stability of the method is investigated for the implicit and explicit scheme with homogeneous boundary conditions, and a stability criterion is found for the advective‐dispersive equation. An application of the method is used to solve contaminant diffusion and advective‐dispersive problems. The numerical solution for the fractional diffusion and fractional advection‐dispersion is compared with their respective analytical solutions for different time and space grid refinements. The diffusion simulation exhibited a good fit between the analytical and numerical solutions, with the explicit scheme going from stable to unstable as the time and space refinement changes. The fractional advection‐dispersion application produced small deviations from the analytical solution. These deviations, however, are analogous to the numerical dispersions encountered in conventional finite difference solutions of the advection‐dispersion equation. The new method is also compared with the traditional L2 method. Notably, an example that involves asymmetrical fractional conditions, a fractional diffusivity that depends on time, and a source term show how the methods compare. Overall, this study assesses the quality and easiness of use of the numerical method.

KINETICS OF HYDROGEN SEPARATION FROM REFINERY OFF-GAS

Adsorption and desorption kinetic curves for equimolar hydrogen – methane mixture on molecular sieve type 5A were experimentally obtained for pressure range 0.122 – 3.546 MPa.The linear driving force rate expression model was used to simulate the dynamic of adsorption and desorption in adiabatic fixed bed adsorber. The model takes into account the interference effects for non-linear isotherms and non-isothermal system. The equations were solved by backward finite difference method with a fixed gridding technique. The individual mass transfer parameters were obtained by matching the theoretical with the experimental data and found to be equal to 8.510 s-1 and 0.783 s-1 for hydrogen and methane, respectively.The predicted effluent histories were shown to be in close agreement with the experimental data for the system. The lowest relative capacity of the bed for methane was almost approximately 95% of that predicted equilibrium capacity. The predicted temperature profiles tracked the experimental temperature data points, but with higher values. Furthermore, the maximum temperature increasing was observed for the adsorption of methane onto 5A molecular sieve at 35 atmospheres and was recorded as 44 K.

Full-Discrete Weak Galerkin Finite Element Method for Solving Diffusion-Convection Problem.

This paper applied and analyzes full discrete weak Galerkin (WG) finite element method for non steady two dimensional convection-diffusion problem on conforming polygon. We approximate the time derivative by backward finite difference method and the elliptic form by WG finite element method. The main idea of WG finite element methods is the use of weak functions and their corresponding discrete weak derivatives in standard weak form of the model problem. The theoretical evidence proved that the error estimate in norm, the properties of the bilinear form, (v-elliptic and continuity), stability, and the energy conservation law.

Experimental study on the stability of an impedance-type force-feedback architecture based on an augmented-state observer for a haptic system under time delay using a LMI approach

One of the most important performance measures of a haptic interface may be characterized by its behaviour in the implementation of a virtual wall in terms of the biggest value of the virtual stiffness that still leaves the device stable. This value depends on many factors including time delays and the virtual world implementation. In this paper we propose a new architecture for this latter factor based on the use of an augmented-state observer which contributes to improving, without additional costs, the performance of a haptic device with respect to the standard implementation based on the use of the backward finite-difference method. In order to compute the stability boundaries of the device in terms of virtual wall parameters, some conditions expressed in terms of LMI (linear matrix inequality) for the stability of discrete-time haptic systems under constant or variable time delay are given. An implementation of a force-feedback haptic test bench is presented. The obtained experimental results confirm the efficiency of the proposed force-feedback architecture with respect to the traditional backward finite-difference method in enlarging the choice for the parameters of the virtual environment for which the haptic system is stable.

An Implicit Harmonic Balance Method in Graphics Processing Units for Oscillating Blades

An implicit harmonic balance (HB) method for modeling the unsteady nonlinear periodic flow about vibrating airfoils in turbomachinery is presented. An implicit edge-based three-dimensional Reynolds-averaged Navier–Stokes equations (RANS) solver for unstructured grids, which runs both on central processing units (CPUs) and graphics processing units (GPUs), is used. The HB method performs a spectral discretization of the time derivatives and marches in pseudotime, a new system of equations where the unknowns are the variables at different time samples. The application of the method to vibrating airfoils is discussed. It is shown that a time-spectral scheme may achieve the same temporal accuracy at a much lower computational cost than a backward finite-difference method at the expense of using more memory. The performance of the implicit solver has been assessed with several application examples. A speed-up factor of 10 is obtained between the spectral and finite-difference version of the code, whereas an additional speed-up factor of 10 is obtained when the code is ported to GPUs, totalizing a speed factor of 100. The performance of the solver in GPUs has been assessed using the tenth standard aeroelastic configuration and a transonic compressor.

Augmented observer approach for high-impedance haptic system with time delay

Among others measures of performance, a haptic device may be characterized by its behavior in the implementation of a virtual wall in terms of the biggest value of the virtual stiffness leaving stable the device. This value depends on many factors including time delays and the virtual world implementation. The paper proposes a new architecture for this latter one based on the use of an augmented state observer which contributes to improve without additional costs the performance of a haptic device with respect to the standard implementation based on the use of the backward finite difference method. In order to compute the stability boundaries of the device in terms of virtual wall parameters, a new LMI criterion for the stability of discrete-time systems with time-varying delay is given. Some numerical simulation results are included to prove the effectiveness of the proposed approach.

A 2N order compact finite difference method for solving the generalized regularized long wave (GRLW) equation

The generalized regularized long wave (GRLW) equation is solved by fully different numerical scheme. The equation is discretized in space by 2N order compact finite difference method and in time by a backward finite difference method. At the inner and the boundary nodes, the first and the second order derivatives with 2N order of accuracy are obtained. To determine the conservation properties of the GRLW equation three invariants of motion are evaluated. The single solitary wave and the interaction of two and three solitary waves are presented to validate the efficiency and the accuracy of the proposed scheme.

A Finite Element Method for the Multiterm Time-Space Riesz Fractional Advection-Diffusion Equations in Finite Domain

We present an effective finite element method (FEM) for the multiterm time-space Riesz fractional advection-diffusion equations (MT-TS-RFADEs). We obtain the weak formulation of MT-TS-RFADEs and prove the existence and uniqueness of weak solution by the Lax-Milgram theorem. For multiterm time discretization, we use the Diethelm fractional backward finite difference method based on quadrature. For spatial discretization, we show the details of an FEM for such MT-TS-RFADEs. Then, stability and convergence of such numerical method are proved, and some numerical examples are given to match well with the main conclusions.

Inverse heat conduction problems in three-dimensional anisotropic functionally graded solids

A meshless method based on the local Petrov–Galerkin approach is applied to inverse transient heat conduction problems in three-dimensional solids with continuously inhomogeneous and anisotropic material properties. The Heaviside step function is used as a test function in the local weak form, leading to the derivation of local integral equations. Nodal points are randomly distributed in the domain analyzed, and each node is surrounded by a spherical subdomain in which a local integral equation is applied. A meshless approximation based on the moving least-squares method is employed in the implementation. After performing spatial integrations, we obtain a system of ordinary differential equations for certain nodal unknowns. A backward finite-difference method is used for the approximation of the diffusive term in the heat conduction equation. A truncated singular-value decomposition is used to solve the ill-conditioned linear system of algebraic equations at each time step. The effectiveness of the meshless local Petrov–Galerkin (MLPG) method for this inverse problem is demonstrated by numerical examples.

The Midpoint Upwind Finite Difference Scheme for Time-Dependent Singularly Perturbed Convection-Diffusion Equations on Non-Uniform Mesh

A numerical approach is proposed to examine the singularly perturbed time dependent convection-diffusion equation in one space dimension on a rectangular domain. The solution of considered problem exhibits a boundary layer on the right side of the domain. We semidiscretize the continuous problem by means of backward Euler finite difference method in the temporal direction. The semi-discretization process yields a set of ordinary differential equations at each time level. A resulting set of ordinary differential equations are discretized by using midpoint upwind finite difference scheme on a non-uniform mesh of Shishkin type. The resulting finite difference method is shown to be almost of second order accurate in the coarse mesh and almost of first order accurate in fine mesh in the spatial direction. First order accuracy is achieved in the temporal direction. An extensive amount of analysis has been carried out in order to obtain uniform convergence of the method. Finally, we have found that the method is uniformly convergent with respect to the singular perturbation parameter i.e. ϵ−uniform. Some numerical experiments have been carried out to validate the predicted theory.

Adsorption Kinetics in a Dual-Inlet Channel Flow Cell: I. Cetyl Pyridinium Chloride on Hydrophilic Silica

A dual-inlet channel flow cell has been developed for the study of the adsorption kinetics of surfactants to solid−liquid interfaces under hydrodynamic control. This cell, with ellipsometric detection of the adsorbed surfactant, has been used to study the adsorption kinetics of cetyl pyridinium chloride (CPC) to hydrophilic silica in 0.1 M KCl and in pure water. The methodology provides detailed insight into the kinetic parameters and casts light on the adsorption mechanisms. The convection−diffusion behavior in the cell was calculated numerically using the backward implicit finite difference (BIFD) method: the CPC monomer and micelle populations were modeled with constant diffusion coefficients and assumed to equilibrate quickly on the time scale of the experiment. A Frumkin model was used to describe the adsorption behavior at the silica surface: the fitting parameters were determined from the equilibrium adsorption isotherm and kinetically limited desorption measurements. The adsorption kinetics in t...

Electrochemical Quartz Crystal Microbalance in a Channel Flow Cell: A Study of Copper Dissolution

We describe how a quartz crystal microbalance (QCM) may be incorporated in a channel flow system (CFC). It is demonstrated that although deposition is nonuniform the increase in mass loading of the electrode can be related to the change in the resonance frequency of the system. The QCM−CFC is used to investigate the corrosion of copper in acidic sulfate solutions. It is demonstrated that the rate of corrosion in the system is limited by mass transport of oxygen at low volumetric flow rates and by the oxidation of an adsorbed copper(I) intermediate at high volumetric flow rates. Concentration profiles of reactants and products in the QCM−CFC have been calculated, using the backward implicit finite difference method to solve the convective−diffusion equation and rate constants for the oxidation of the Cu(I) species determined