Backward Differentiation Formula Method Research Articles

MODELING STOCHASTIC ADVERSE EFFECTS OF CBN 2023 REDESIGNED NAIRA NOTES POLICY ON RURAL FARMERS IN NIGERIA

The recent Central Bank of Nigeria (CBN) 2023 redesigned naira notes is of good benefits to strengthen the economy of the country by checking counterfeiting and hoarding of large volume of banknotes by the public. Despite all the efforts made by the CBN for citizens to enjoy the benefits of this implementation, most rural farmers were faced with adverse effects of uncertainties in the production and marketing of their agricultural produce due to lack of redesigned new naira notes in circulation. The adverse effects of these uncertainties are modeled as Advanced Stochastic Time-Delay Differential Equation (ASTDDE). The modeled equation is solved using Extended Second Derivative Block Backward Differentiation Formulae Method (ESDBBDFM) without the use of interpolation techniques in the evaluations of the delay term and noise term. In comparing the numerical results of this method with other existing methods in literature, the newly developed mathematical expressions for the evaluations of the delay term and the noise term in solving ASDDEs with the discrete schemes of ESDBBDFM gives better results for step number than step numbers and 3 by producing Least Minimum Absolute Random Error (LMARE) in a Lower Computational Processing Unit Time (LCPUT) faster than other existing methods that applied interpolation techniques in evaluations of the delay term and the noise term.

The Second-Order Numerical Approximation for a Modified Ericksen–Leslie Model

In this study, two numerical schemes with second-order accuracy in time for a modified Ericksen–Leslie model are constructed. The highlight is based on a novel convex splitting method for dealing with the nonlinear potentials, which is integrated with the second-order backward differentiation formula (BDF2) and leap frog method for temporal discretization and the finite element method for spatial discretization. The unconditional energy stability of both schemes is further demonstrated. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed schemes.

A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers’ equation

This study explores a third-order backward differentiation formula (BDF3) and nonlinear fourth-order difference method (FODM) for solving the generalized viscous Burgers’ equation (GVBE). The BDF3 method is employed for discretizing the time derivative, while the nonlinear term uλux is handled using a newly constructed nonlinear fourth-order difference operator. The spatial second derivative is discretized using a linear fourth-order difference formula. We establish the convergence of the proposed BDF3-FODM by introducing a cut-off function (COF). The main contributions include the construction of a novel nonlinear fourth-order difference operator, addressing the limitation observed in Zhang et al. (2021), Guo et al. (2023), where the order of spatial convergence was restricted to second order. Finally, a numerical test is presented to validate our theoretical analysis.

NUMERICAL SOLUTION OF NON-SIMILAR BOUNDARY-LAYER FLOW OVER A CYLINDER

This paper presents а numerical solution of a non-similar two-dimensional boundary-layer flow over a cylinder. The assumptions and related two-dimensional flow equations are presented. The fourth-order Runge-Kutta method with the backward differentiation formula (BDF) method to separation point is implemented in the numerical solution using MATLAB software. Numerical solution results are compared with well-known analytic solutions. Shear stress diagram, friction ratio based on θ and x, and output results are illustrated. The results of numerical solution demonstrate good consistency with analytic solutions.

Dynamic Analysis of Elastically Supported Functionally Graded Sandwich Beams Resting on Elastic Foundations Under Moving Loads

Vibration behaviors of elastically supported functionally graded (FG) sandwich beams resting on elastic foundations under moving loads are investigated. The transformed-section method is first applied to establish the bending vibration equations of FG sandwich beams, then the Chebyshev collocation method is used to study free and forced vibrations. Two types of sandwich beams with FG faces-isotropic core and isotropic faces-FG core are considered. The material properties of FG materials are assumed to vary across the beam thickness according to a simple power function. Regarding the free vibration analysis, bending vibration frequencies are calculated numerically by forming a matrix eigenvalue equation. As for the forced vibration analysis, the backward differentiation formula method is employed to solve the time-dependent ordinary differential equations to obtain the dynamic deformations of the beam. To ensure the accuracy of the proposed model, some calculated results are compared with those in the published literature. Parametric studies are then performed to demonstrate the effects of material gradient indexes, stack types, layer thickness ratios, slenderness ratios, excitation frequency and speed of moving loads, and foundation and support stiffness parameters on the dynamic characteristics of FG sandwich beams.

Time discretization in the solution of parabolic PDEs with ANNs

We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks. An Artificial Neural Network (ANN) is an interconnected group of nodes -where activation functions act- organized in numbered layers. From a mathematical point of view, an ANN represents a combination of activation functions that is linear if only one hidden layer is admitted. In the present paper, the ELMs setting is considered, and this gives that a single hidden layer is admitted and that the ANN can be trained at a modest computational cost as compared to Deep Learning Neural Networks. Our approach addresses the time evolution by applying classical ODEs techniques and uses ELM-based collocation for solving the resulting stationary elliptic problems. In this framework, the θ-method and Backward Differentiation Formulae (BDF) techniques are investigated on some linear parabolic PDEs that are challenging problems for the stability and accuracy properties of the methods. The results of numerical experiments confirm that ELM-based solution techniques combined with BDF methods can provide high-accuracy solutions of parabolic PDEs.

An improved stiff-ODE solving framework for reacting flow simulations with detailed chemistry in OpenFOAM

The integration of stiff ordinary differential equation (ODE) systems associated with detailed chemical kinetics is computationally demanding in practical combustion simulations. Despite the various approaches in expediting the computational efficiency, it is still necessary to optimise the cell-wise calculation in operator-splitting type simulations of reactive flow. In this work, we proposed an improved stiff-ODE solver framework targeting to speed up the simulation of reactive flow in OpenFOAM. This framework combines the Radau-IIA and backward differentiation formula (BDF) ODE-integration algorithms, the pyJac-based fully analytical Jacobian formulation, and dense-based LAPACK and sparse-based KLU sophisticated linear system solvers. We evaluate the performance of the efficient solver framework on various benchmark combustion problems across a wide range of chemical kinetic complexities. A comprehensive investigation of the key elements of stiff ODE solvers is conducted in the homogeneous reactor, focusing respectively on the influences of error tolerance, integration time interval, Jacobian evaluation methodology, and linear system solver on the accuracy and efficiency trade-off. More realistic simulation results are presented regarding the one-dimensional laminar flame and three-dimensional turbulent flame. The results indicate that the Radau-IIA is more preferable in both efficiency and accuracy compared with the widely used BDF and Seulex methods for large integration interval, whereas the differences between three methods diminish as the integration time interval decreases. In all cases, it is found that the full analytical Jacobian is more advantageous for small mechanisms of species number around 50–100 while the approximated formulation of Jacobian is recommended for larger ones. Furthermore, the more robust linear system solvers provide significant improvement on computational efficiency with the dense-based LAPACK solver being more suitable for small to moderate-scale mechanisms while sparse-based KLU being superior for large-scale mechanisms. The proposed efficient solver framework in its optimal configuration obtains more than 2.6 times speedup in realistic high-fidelity flame simulation with a 57 species combustion mechanism.

Reliable numerical treatment with Adams and BDF methods for plant virus propagation model by vector with impact of time lag and density

Plant disease incidence rate and impacts can be influenced by viral interactions amongst plant hosts. However, very few mathematical models aim to understand the viral dynamics within plants. In this study, we will analyze the dynamics of two models of virus transmission in plants to incorporate either a time lag or an exposed plant density into the system governed by ODEs. Plant virus propagation model by vector (PVPMV) divided the population into four classes: susceptible plants [S(t)], infectious plants [I(t)], susceptible vectors [X(t)], and infectious vectors [Y(t)]. The approximate solutions for classes S(t), I(t), X(t), and Y(t) are determined by the implementation of exhaustive scenarios with variation in the infection ratio of a susceptible plant by an infected vector, infection ratio of vectors by infected plants, plants' natural fatality rate, plants' increased fatality rate owing to illness, vectors' natural fatality rate, vector replenishment rate, and plants' proliferation rate, numerically by exploiting the knacks of the Adams method (ADM) and backward differentiation formula (BDF). Numerical results and graphical interpretations are portrayed for the analysis of the dynamical behavior of disease by means of variation in physical parameters utilized in the plant virus models.

Maximum time step for high order BDF methods applied to gradient flows

For a backward differentiation formula (BDF) applied to the gradient flow of a semiconvex function, quadratic stability implies gradient stability. Namely, it is possible to build a Lyapunov functional for the discrete-in-time dynamical system, with a restriction on the time step. The maximum time step which can be derived from quadratic stability has previously been obtained for the BDF1, BDF2 and BDF3 schemes. Here, we compute it for the BDF4 and BDF5 methods. We also prove that the BDF6 scheme is not quadratically stable. Our results are based on the tools developed by Dahlquist and other authors to show the equivalence of A-stability and G-stability. We give several applications of gradient stability to the asymptotic behaviour of sequences generated by BDF schemes.

Accuracy Improvement of Block Backward Differentiation Formulas for Solving Stiff Ordinary Differential Equations Using Modified Versions of Euler's Method

In this study, the fully implicit 2-point block backward differentiation formulas (BBDF) method has been successfully utilized for solving stiff ordinary differential equations (ODEs) by taking into account the uses of new starting methods namely, modified Euler's method (MEM), improved modified Euler's method (IMEM), and new Euler's method (NEM). The reason of proposing the BBDF is that the method has been proven useful for stiff ODEs due to its A-stable properties. Furthermore, the method is able to approximate the solutions at two points simultaneously at each step. The proposed method is also implemented through Newton's iteration procedure, which involves the calculation of the Jacobian matrix. Accuracy of the method is evaluated based on its performance in solving linear and non-linear initial value problems (IVPs) of first order stiff ODEs with transient and steady-state solutions. Some comparisons are made with the conventional BBDF approach for indicating the reliability of the proposed method. Numerical results indicate that not only classical Euler's method provides accurate solutions for BBDF, but also the numerous modified versions of Euler's methods improve the accuracy of BBDF, in terms of absolute error at certain step size and stage of iteration.

On Some Numerical Methods for Solving Large Differential Nonsymmetric Stein Matrix Equations

In this paper, we propose a new numerical method based on the extended block Arnoldi algorithm for solving large-scale differential nonsymmetric Stein matrix equations with low-rank right-hand sides. This algorithm is based on projecting the initial problem on the extended block Krylov subspace to obtain a low-dimensional differential Stein matrix equation. The obtained reduced-order problem is solved by the backward differentiation formula (BDF) method or the Rosenbrock (Ros) method, the obtained solution is used to build the low-rank approximate solution of the original problem. We give some theoretical results and report some numerical experiments.

Stability of Variable-Step BDF2 and BDF3 Methods

We prove that the two-step backward differentiation formula (BDF) method is stable on arbitrary time grids; while the variable-step three-step backward differentiation formula scheme is stable if almost all adjacent step ratios are less than 2.553. These results relax the severe mesh restrictions in the literature and provide a new understanding of variable-step BDF methods. Our main tools include the discrete orthogonal convolution kernels and an elliptic type matrix norm.

A variational splitting of high-order linear multistep methods for heat transfer and advection–diffusion parabolic problems

The main result of this paper is the generalization of the concept of variational splitting proposed by Behnoudfar et al. (2018) into higher-order time-stepping schemes. The key novelty of the technique proposed here is the following: when using the isogeometric finite elements, one can factorize the matrix of the linear system and obtain the linear complexity. This matrix has the form M+ηK, where M is the mass matrix, K is the stiffness matrix. Following the technique of Behnoudfar et al. (2018), and factorizing M+ηK one would get the additional error of order η2. We propose the technique to couple the possibility of factorization of M+ηK with the higher-order time-stepping schemes and preserve this higher order. Thus, we introduce a variational splitting for implicit, linear multistep methods of order s for parabolic partial differential equations. Our technique exploits the tensor-product structure of the discretization in space (mesh and basis functions) to derive an update strategy with a linear cost regarding the total number of degrees of freedom for multi-dimensional problems. As an example, we consider two popular methods: Adams–Moulton (AM) and backward differentiation formulae (BDF). We derive the corresponding coefficients for AM and BDF methods. We present a C++ implementation of the time-integration schemes, targeting shared-memory Linux clusters. We present numerical results for the heat transfer problem and advection–diffusion problem. We also discuss the limitations of the method and possible extensions overcoming this limitation by using our method as a preconditioner for an iterative solver.

Application of MGA and EGA algorithms on large-scale linear systems of ordinary differential equations

In this paper, we propose two new approaches to solve the large linear system of ordinary differential equations (LSODE), which are based on the projection technique on the extended global Krylov (EGKS) or global Krylov (GKS) subspaces if A is not invertible. The first approach (EGA-expm or MGA-expm if A is not invertible) is based on the projection of matrix exponential function (expm), the second approach (EGA-ROS2, EGA-BDF2 or MGA-ROS2, MGA-BDF2 if A is not invertible) is based on a projection of the initial problem onto an EGKS (or GKS if A is not invertible) to get a new LSODE of small-size then solve it with the 2-stage Rosenbrock method (ROS2) or the backward differentiation formula (BDF2) method and the 2-stage obtained solution is used to have a low-rank (LR) approximate solution of the initial problem. Moreover, we will present the expression of the residual and the error norms. Finally, some attractive and interesting examples of specific problems are presented to confirm the strength of the new approaches.

Optimal a posteriori estimators for the variable step-size BDF2 method for linear parabolic equations

Optimal a posteriori error estimates for time discretizations of linear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. Based on second-order BDF reconstructions of the piecewise linear approximate solutions, the optimality of residual-based a posteriori error estimators is proved by using a novel stability inequality when the starting value is computed by the trapezoidal method. With a reasonable choice for the starting step-size, the optimality of the estimators when the starting value is computed by the backward Euler scheme can be also ensured. The effectiveness of the a posteriori error estimators is illustrated by a numerical example.

The Computational Solution of First Order Delay Differential Equations Using Second Derivative Block Backward Differentiation Formulae

In this paper, we implemented second derivative block backward differentiation formulae methods in solving first order delay differential equations without the application of interpolation methods in investigating the delay argument. The delay argument was evaluated using a suitable idea of sequence which we incorporated into some first order delay differential equations before its numerical evaluations. The construction of the continuous expressions of these of block methods was executed through the use of second derivative backward differentiation formulae method on the bases of linear multistep collocation approach using matrix inversion method to derive the discrete schemes. After the numerical experiments, the new proposed method was observed to be convergent, stable and less time consuming. From the numerical solutions obtained, the scheme for step number k = 4 performed better in terms of accuracy than that of the schemes for step numbers k = 3 and 2 when compared with other existing methods.

Comparison of implicit time-discretization schemes for hybridized discontinuous Galerkin methods

The present study is focused on the application of two families of implicit time-integration schemes for general time-dependent balance laws of convection-diffusion-reaction type discretized by a hybridized discontinuous Galerkin method in space, namely backward differentiation formulas (BDF) and diagonally implicit Runge-Kutta (DIRK) methods. Special attention is devoted to embedded DIRK methods, which allow the incorporation of time step size adaptation algorithms in order to keep the computational effort as low as possible. The properties of the numerical solution, such as its order of convergence, are investigated by means of suitably chosen test cases for a linear convection-diffusion-reaction equation and the nonlinear system of Navier-Stokes equations. For problems considered in this work, the DIRK methods prove to be superior to high-order BDF methods in terms of both stability and accuracy.

A class of sixth order hybrid extended block backward differentiation formulae for computational solutions of first order delay differential equations

In this paper, we established and carried-out the computational solution of some first order delay differential equations (DDEs) using hybrid extended backward differentiation formulae method in block forms without the application of interpolation techniques in determining the delay term. The discrete schemes were worked-out through the linear multistep collocation technique by matrix inversion approach from the continuous construction of each step number which clearly demonstrated the order and error constants, consistency, zero stability, convergence and region of absolute stability of this method after investigations. The results obtained after the implementation of this method validate that the lower step number integrated with hybrid extended future points performed better than the higher step numbers integrated with hybrid extended future points when compared with the exact solutions and other existing methods.

TheBDF2 FDMfor the fourth‐order equations with the multi‐termR‐Lfractional integral kernels

AbstractIn this paper, we formulated the finite difference method for the fourth order integro‐differential equation with the Riemann‐Liouville multi‐term fractional integral kernels. The formally two‐step backward differentiation formula method and second‐order convolution quadrature are used in time. Theand‐norm stability and convergence at each time level are given. Numerical results show that our scheme obtains the expected optimal convergence rates.

Stability analysis of general multistep methods for Markovian backward stochastic differential equations

Abstract This paper focuses on the stability analysis of a general class of linear multistep methods for decoupled forward–backward stochastic differential equations (FBSDEs). The general linear multistep methods we consider contain many well-known linear multistep methods from the ordinary differential equation framework, such as Adams, Nyström, Milne--Simpson and backward differentiation formula methods. Under the classical root condition, we prove that general linear multistep methods are mean-square (zero) stable for decoupled FBSDEs with generator function related to both y and z. Based on the stability result, we further establish a fundamental convergence theorem.