Backward Difference Method Research Articles

Two fast numerical methods for a generalized Oldroyd-B fluid model

This work proposes two fast and efficient numerical methods for a generalized Oldroyd-B fluid model with fractional derivatives. The fast evaluations of the Riemann–Liouville fractional derivatives are developed based on convolution quadrature generated by the backward Euler and the second-order backward difference methods. By using these fast algorithms in time and employing the finite element method in space, two fully discrete schemes are established for the considered problem. Further, we discuss the error estimates of these numerical schemes based on the property of the initial value and the right hand function instead of the regularity assumption of the exact solution. Finally, some numerical tests are presented to verify the effectiveness of the numerical schemes and confirm the theoretical results.

Extended FDTD Method for Coanalysis of Superconducting Passive Transmission Lines and Josephson Junction Drivers and Receivers

In rapid single-flux quantum circuits, the signal integrity problem of superconducting passive transmission lines (PTLs) becomes critical. In this article, an extended 3-D finite-difference time-domain (FDTD) method is proposed for coanalysis of superconducting PTLs and Josephson junction (JJ) drivers and receivers. The central difference FDTD scheme is used for PTLs, and the backward difference method is adopted for JJs, which are introduced to FDTD cells as lumped elements. Due to the nonlinearity of JJs, the Newton–Raphson method is introduced to solve the boundary condition of the FDTD method. Compared with the 1-D FDTD method of the transmission line model of PTLs, the proposed method performs full-wave analysis of the physical structure of PTLs. Therefore, the proposed 3-D FDTD method is theoretically more accurate than the conventional method by avoiding parameter extraction and circuit analysis.

A fully explicit staggered algorithm for near‐field wave propagation of fluid‐saturated porous media based on u‐p dynamic formulation

AbstractNear‐field wave propagation of the fluid‐saturated porous media (FSPM) is an important issue in soil dynamics and geotechnical earthquake engineering. In the current study, a fully explicit staggered algorithm for the near‐field wave propagation analysis of the FSPM in the time domain is developed based on the u‐p dynamic formulation. The decoupling of dynamic formulation is implemented by the diagonalization of the mass matrix and pore fluid compressibility matrix. The central difference method and Newmark constant average acceleration method are used for the solution of the displacement and velocity of solid skeleton, and the backward difference method is used for the solution of pore fluid pressure in the time domain. The effectiveness and feasibility of the matrix diagonalization are verified by the comparison with the classical numerical algorithm. The proposed algorithm is also validated by the comparison of the numerical results with the corresponding analytical solution results. Combing with the transmitting artificial boundary condition, the proposed algorithm is then applied to investigate the seismic response of a FSPM free field, and its applicability to the typical near‐field wave propagation problems of the FSPM is indicated. The stability analysis of the proposed algorithm is finally conducted, and the corresponding stability criterion is presented. For the proposed algorithm, all the dynamic variables of the FSPM are solved in an iterative way and the coupling dynamic equations need not to be solved at each time step. The computational effort can be reduced considerably, and the computational efficiency can be improved remarkably.

Radial basis function partition of unity procedure combined with the reduced-order method for solving Zakharov–Rubenchik equations

A meshless radial basis function based on partition of unity (RBF-PU) method is proposed to solve Zakharov–Rubenchik equations. In this local method, the domain is split into overlapping patches forming a covering of it and also, it provides accurate results for PDEs. Time discretization is performed using a second-order implicit explicit backward difference method (IMEX-BDF2). Although the proper orthogonal decomposition (POD) is applied to reduce the dimension of the governing model, the computational complexity of the reduced model for nonlinear terms still depends on the number of variables of the full model. To overcome this subject, we employ the discrete empirical interpolation method (DEIM). Two problems with different situations are solved by the proposed method and the comparison of numerical findings with the conservative compact difference scheme and RBF-FD method shows that the presented method provides accurate results at a low computing cost.

Sliding mode control of discrete-time interval type-2 fuzzy Markov jump systems with the preview target signal

This paper studies the problem of output feedback sliding mode control for uncertain discrete-time interval type-2 (IT2) fuzzy Markov jump systems (MJSs) with the preview target signal. An augmented error system (AES) including the known preview target signal is derived based on the backward difference method. A sliding mode output controller is designed for the AES, which integrates the known future information. The designed controller guarantees the discrete-time reach condition and maintains the states in a neighborhood region of the predesigned sliding surface. The stability criteria of the sliding motion on a linear sliding surface are given for the considered systems. Finally, two practical examples show that the output of the original system can track the target signal better under the designed preview controller.

Error Bounds of a Finite Difference/Spectral Method for the Generalized Time Fractional Cable Equation

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is unconditionally stable. The scheme is proved to have min{2−α,2−β}-order convergence in time and spectral accuracy in space for smooth solutions, where α,β are two exponents of fractional derivatives. We report numerical results to confirm our error bounds and demonstrate the effectiveness of the proposed method. This method can be applied to model diffusion and viscoelastic non-Newtonian fluid flow.

On the application of higher-order Backward Difference (BDF) methods for computing turbulent flows

The Backward Differentiation Formulae (BDF) of orders one to six are implemented in an in-house three-dimensional finite-volume (FVM) compressible flow code that operates on unstructured meshes. The lower-upper symmetric Gauss-Seidel (LUSGS) implicit relaxation technique based on the splitting of convective Jacobians is used to obtain a variable-order stable implementation in the solver. The temporal order of accuracy of the implemented schemes is verified using the Method of Manufactured Solutions (MoMS). Practical engineering flow problems are simulated to investigate the operational stability of BDF methods so that these can be used as a cheaper alternative to multi-stage methods in RANS, hybrid RANS-LES, and LES implementations. Steady-state flow simulations of supersonic flow over a flat plate and high Reynolds number flow over a NACA-0012 airfoil show that algorithm with BDF schemes of orders 1-5 is stable at high CFL numbers (700-1000) and produces a converged solution. Stokes' second problem and a high-resolution delayed detached eddy simulation (DDES) of flow over a 3D circular cylinder using the BDF1-BDF6 methods demonstrate the use of these higher-order temporal schemes in unsteady laminar and turbulent flows. It is concluded that BDF methods of orders 3-5 can be practically employed to achieve higher levels of temporal accuracy in flow simulations when the level of spatial accuracy is also high (ENO/WENO schemes, spectral methods, etc.) in hybrid RANS-LES, LES or DNS. Even without higher-order spatial accuracy, the same BDF schemes can be used in Adaptive Time-stepping (ATS) methods to obtain prescribed temporal accuracy efficiently by algorithmically switching between the different schemes.

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.

Finite difference based iterative learning control with initial state learning for a class of fractional order two‐dimensional continuous‐discrete linear systems

SummaryThis article presents a PI‐type iterative learning control (ILC) law with initial state learning for a class of () fractional order two‐dimensional (2D) linear systems. First, by using backward difference method, the finite difference approximation of the fractional order derivative is obtained, which leads to globally order accuracy. Then, a PI‐ILC law is constructed at the nodes and the convergence analysis of the iterative scheme is proved. A linear matrix inequality‐based sufficient condition is derived to guarantee that the tracking error is asymptotically convergent. The obtained convergence condition is fractional order dependent. Most of the classical ILC conditions for fractional order one‐dimensional linear systems fall into the special cases of this article. Finally, the simulation results show the effectiveness of the proposed control method.

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.

Newton Backward Difference Application for Automating Fertilizer Distribution in Agriculture using Wireless Sensor Networks

Background: The Indian economy and population have been largely influenced by the green revolution which has caused a big change in India. Yet farmers are not able to make full-blown usage of technology to improve crop cultivation. As Indian economy and per capita income, in trade terms, CAPEX (Capital Expenses); is based on agro products, to rejuvenate the demand-supply chain of the food sector, farmers are using fertilizers to increase yield. Due to the excessive use of these puissance fertilizers, soil fertility is adversely impacted. It is hazardous for direct consumption by people. The current study was aimed to optimize the use of fertilizer as per crop requirement using Newton backward difference (NBD) method. Methods: The experiment was conducted for the last three years (2017-2020) in the Agrotech farm of MIT Academy of Engineering, Pune. The tomato crop was used for taking the readings. Using the NBD, we compared the values between standard, conventional and actual which gives the optimized value and saves the wastage of fertilizer and farmers’ burden economically. These NBD can lead to a reduction in the number of sensor nodes in the farm field. The reduction in sensor nodes saves the overall working cost. Result: The implemented system saves the quantity of nitrogen (N), phosphorus (P) and potassium (K) only by providing the required amount of fertilizer to the crop in comparison with a conventional approach.

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.

A backward difference formulation for analyzing the dynamics of capital stocks

The current study provides a numerical method that is derived in a backward difference formulation for ordinary differential equations. The proposed method employs a constant step size algorithm of order 12. The backward difference formulation serves as a competitive algorithm for solving ordinary differential equations. In the current study, the backward difference method is used to analyze the dynamics of capital stocks in terms of depreciation rate for the capital–labor ratio. Results provided in this study will validate the accuracy of the backward difference algorithm hence proving it as a viable alternative for analyzing economic problems in the form of ordinary differential equations.

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.

A Finite Difference-Spectral Method for Solving the European Call Option Black–Scholes Equation

In this paper, we present a novel technique based on backward-difference method and Galerkin spectral method for solving Black–Scholes equation. The main propose of this method is to reduce the solution of this problem to the solution of a system of algebraic equations. The convergence order of the proposed method is investigated. Also, we provide numerical experiment to show the validity of proposed 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.

Numerical schemes for the time-fractional mobile/immobile transport equation based on convolution quadrature

In this work, the numerical approximation of the time-fractional mobile/immobile transport equation is considered. We investigate the solution regularity for two types of the initial data regularities. By applying the continuous piecewise linear finite elements in space, we obtain the spatial semidiscrete Galerkin scheme and derive its error estimates. We then propose two finite element schemes for the equation by employing convolution quadrature based on the backward Euler and the second-order backward difference methods. The corresponding error estimates for the two schemes are also given. Numerical examples of the two-dimensional problems are shown to confirm the convergence theory results.

Uniform [formula omitted] behavior of the first-order interpolant quadrature scheme for some partial integro-differential equations

We study the time discretization for the solution of a Volterra equation with weakly singular kernel. The time discrete scheme is based on the first-order backward difference method. The memory term is approximated by the interpolating quadrature. We use Laplace transform technique to show that this interpolating quadrature scheme has a convergence rate of O(△t) where △t denotes the time step, for smooth and non-smooth initial data in homogeneous case. Special attention is that the uniform l1 convergence property of the discretization in time is given.

Fast algorithm based on TT-M FE method for modified Cahn–Hilliard equation

We consider numerical methods for solving the modified Cahn–Hilliard equation involving strong nonlinearities. A fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme is proposed to overcome the time-consuming computation of the nonlinear terms. The TT-M FE algorithm includes three main steps: Firstly, a nonlinear FE scheme is solved on a coarse time-mesh τc. Here, the FE method is used for spatial discretization and the implicit second-order 𝜃 scheme (containing both implicit Crank–Nicolson scheme and second-order backward difference method) is used for temporal discretization. Secondly, the Lagrange’s interpolation is used to obtain the interpolation result on the fine time-mesh. Finally, a linearized FE system is solved on a fine time-mesh τ (τ<τc). The stability analysis and priori error estimates are provided in detail. Numerical examples are given to demonstrate the validity of the proposed scheme. The TT-M FE method is compared with the traditional Galerkin FE method and it is evident that the TT-M FE method can save the calculation time.