Implicit Linear Multistep Methods Research Articles

Analysis of Fractional Linear Multi-Step Methods of Order Four from Super-Convergence

We analyze two implicit fractional linear multi-step methods of order four for solving fractional initial value problems. The methods are derived from the Grunwald-Letnikov approximation of fractional derivative at a non-integer shift point with super-convergence. The weight coefficients of the methods are computed from fundamental G unwald weights, making them computationally efficient when compared with other known methods of order four. We also show that the stability regions are larger than that of the fractional Adams-Moulton and fractional backward difference formula methods. We present numerical results and illustrations to verify that the theoretical results obtained are indeed satisfied.

A New Hybrid Block Method via Combined Hermite Polynomials and Exponential Functions as Basis Function

This paper presents the derivation and implementation of a hybrid block method for solving stiff and oscillatory first-order initial value problems of ordinary differential equations (ODEs). The hybrid block method was derived by continuous collocation and interpolation using combined Hermite polynomials and exponential functions as the basis function to produce a continuous implicit Linear Multistep Method (LMM) of order nine and implement in block form. The basic properties of the derived method were studied, and the hybrid block integrator was demonstrated to be zero-stable, convergent, consistent, and to have an A-stable region of absolute stability, which made it suitable for stiff and oscillatory ordinary differential equations. The use of a combined basis in the generation of LMMs is worthy of universal acceptance. The technique indicates that, utilizing an interpolation and collocation approach, continuous LMMs can be derived from combinations of any polynomials and exponential functions. On two sampled stiff and oscillatory problems, the new integrator was tested. The numerical results indicate that our new hybrid block integrator is computationally efficient and outperforms existing methods in terms of accuracy

A Filtered Milne-Simpson ODE Method with Enhanced Stability Properties

The Milne-Simpson method is a two-step implicit linear multistep method for the numerical solution of ODEs that obtains the theoretically highest order of convergence for such a method. The stability region of the method is only an interval on the imaginary axis and the method is classified as weakly stable which causes non-physical oscillations to appear in numerical solutions. For this reason, the method is seldom used in applications. This work examines filtering techniques that improve the stability properties of the Milne-Simpson method while retaining its fourth-order convergence rate. The resulting filtered Milne-Simpson method is attractive as a method of lines integrator of linear time-dependent partial differential equations.

Analysis of New Type of Second-order Fractional Linear Multi-step Method with Improved Stability

We present and investigate a new type of implicit fractional linear multi-step method of order two for fractional initial value problems. The method is obtained from the second-order superconvergence of the Grünwald-Letnikov approximation of the fractional derivative at a non-integer shift point. The method coincides with the backward difference method of order two for the classical initial value problem when the order of the derivative is one. The weight coefficients of the proposed method are obtained from the Grünwald weights and are hence computationally efficient compared with that of the fractional backward difference formula of order two. The stability properties are analyzed and it is shown that the stability region of the method is larger than that of the fractional Adams-Moulton method of order two and the fractional trapezoidal method. Numerical results and illustrations are presented to justify the analytical theories.

The global stability investigation of the mathematical design of a fractional-order HBV infection.

This work presents approximate solutions of a fractional-order design for hepatitis B virus infection. The numerical solution of the system is given by using an implicit fractional linear multi-step method of the second order. Here, Caputo fractional derivative is considered for fractional derivative. Basic theoretical properties are discussed. We prove the global stability analysis of the fractional-order model. Numerical simulations are demonstrated to display our theoretical results. This current study is to reveal that the order of the fractional derivative beta does not affect the regular state’s stability concerning both theoretical and numerical results. Besides, if the fractional-order beta increases, the solutions converge more rapidly to the regular states. Finally, we note that this study can provide beneficial outcomes for understanding and estimating the dissipation of distinct epidemics.

BLOCK PROCEDURE WITH IMPLICIT SIXTH ORDER LINEAR MULTISTEP METHOD USING LEGENDRE POLYNOMIALS FOR SOLVING STIFF INITIAL VALUE PROBLEMS

In this paper, a discrete implicit linear multistep method in block form of uniform step size for the solution of first-order ordinary differential equations is presented using the power series as a basis function. To improve the accuracy of the method, a perturbation term is added to the approximated solution. The method is based on collocation of the differential equation and interpolation of the approximate solution using power series at the grid points. The procedure yields four linear multistep schemes which are combined as simultaneous numerical integrators to form block method. The method is found to be consistent and zero-stable, and hence convergent. The accuracy of the method is tested with some standard stiff first order initial value problems. The results are compared with the two point implicit block backward difference formula and two point implicit block extended backward difference formula. All numerical examples show that our proposed method has a better accuracy than the existing literature.

Direct Integrators for the General Third-Order Ordinary Differential Equations with an Application to the Korteweg–de Vries Equation

We construct a new class of implicit continuous linear multistep methods (LMMs) which are used as boundary value methods for the numerical integration of the general third order initial and boundary value problems in ordinary differential equations, including the Korteweg–de Vries equation. The boundary value methods obtained from these continuous LMMs are weighted the same and are used to simultaneously generate approximate solutions to the exact solutions in the entire interval of integration. We established the convergence analysis of the methods and several numerical examples are given to show the performance of the methods.

Delay-dependent stability of linear multistep methods for DAEs with multiple delays

This paper aims to investigate the asymptotic stability of linear multistep (LM) methods for linear differential-algebraic equations (DAEs) with multiple delays. Based on the argument principle, we first establish the delay-dependent stability criteria of analytic solutions; then, we propose some practically checkable conditions for weak delay-dependent stability of numerical solutions derived by implicit LM methods. Lagrange interpolations are used to compute the delayed terms. Several numerical examples are given to illustrate the theoretical results.

The solution of fractional order epidemic model by implicit Adams methods

We consider the numerical solution of the fractional order epidemic model on long time intervals of a non-fatal disease in a population. Under real-life initial conditions the problem needs to be treated by means of an implicit numerical scheme. Here we consider the use of implicit fractional linear multistep methods of Adams type. Numerical results are presented.

Numerical Solution of Stiff and Oscillatory Differential Equations Using a Block Integrator

This paper presents the derivation and implementation of a block integrator for the solution of stiff and oscillatory first-order initial value problems of Ordinary Differential Equations (ODEs). The integrator was derived by collocation and interpolation of the combination of power series and exponential function to generate a continuous implicit Linear Multistep Method (LMM). The basic properties of the derived integrator were investigated and the integrator was implemented on some sampled stiff and oscillatory problems. From the results obtained, it is obvious that the block integrator gives better approximation than some existing ones.

Some one-step embedded hybrid schemes of maximal orders

Hybrid schemes are known to produce lower error constants, and hence more accurate than their standard counterparts. In this research work, some explicit and implicit Linear Multi-step Methods (LMMs) of maximal orders defined on the interval [xn,xn+k] are embedded into one-step hybrid schemes defined on the interval [xn,xn+1]. The results show that the hybrid schemes have the same orders as the standard methods for each step number. However, the hybrid schemes have much lower error constants than the corresponding standard schemes. More importantly, the hybrid schemes are consistent and zero-stable, hence convergent, unlike the standard methods which are known to be unstable.

Four Steps Continuous Method for the Solution of <i>y″</i>= <i>f</i> (<i>x, y, y′</i>)

This paper proposes a continuous block method for the solution of second order ordinary differential equation. Collocation and interpolation of the power series approximate solution are adopted to derive a continuous implicit linear multistep method. Continuous block method is used to derive the independent solution which is evaluated at selected grid points to generate the discrete block method. The order, consistency, zero stability and stability region are investigated. The new method was found to compare favourably with the existing methods in term of accuracy.

Stability properties of a predictor-corrector implementation of an implicit linear multistep method

We examine the stability properties of a predictor-corrector implementation of a class of implicit linear multistep methods. The method has recently been described in the literature as suitable for the efficient integration of stiff systems and as having stability regions similar to well known implicit methods. A more detailed analysis reveals that this is not the case.

Approximate factorization for a viscous wave equation

A general procedure to construct ADI methods for multidimensional problems was originated by Beam and Warming using the method of approximate factorization. In this paper, we extend the method of approximate factorization to solve a viscous wave equation. The method can be combined with any implicit linear multistep method for the time integration of the wave equation. The stability of the factored schemes and their underlying schemes is analyzed based on a discrete Fourier analysis and the energy method. Convergence proofs are presented and numerical results supporting our analysis are provided.

An extension of the theory of Dorsselaer and Spijker

We are concerned with the numerical solution of ini- tial value problems (IVPs) for systems of sti¤ ordi- nary di¤erential equations (ODEs). Our emphasis is on implicit linear multistep methods (LMM), partic-ularly the backward di¤erentiation formulae (BDF). In our quest for saving computations in the time in- tegration of the initial value problems we adopt and extend a theory of Dorsselaer and Spijker The the- ory of Spijker introduces the notion of classifying sti¤ problems into highly nonlinear and mildly nonlinear. In this paper we derive expressions that relate the convergence rate factor for the highly nonlinear and mildly nonlinear problems to variations in the step-size and the Jacobian matrix. The expressions reveal that when the iteration matrix uses the current step- size, hn = h0 then a poor convergence rate is not solely due to the change in the Jacobian (absolute or relative). From these relations it is evident that a vastly changing solution can also lead to poor conver- gence rate. Our expressions further show that for the mildly nonlinear problems it is the absolute change of the Jacobian that can contribute to the convergence rate while for strongly nonlinear problems the relative change in the Jacobian can play a signi cant part in the convergence rate.Key words: Reynolds equation, random roughness, squeeze film, pressure, load carrying capacity

Implicit solution of hyperbolic equations with space-time adaptivity

Adaptivity in space and time is introduced to control the error in the numerical solution of hyperbolic partial differential equations. The equations are discretised by a finite volume method in space and an implicit linear multistep method in time. The computational grid is refined in blocks. At the boundaries of the blocks, there may be jumps in the step size. Special treatment is needed there to ensure second order accuracy and stability. The local truncation error of the discretisation is estimated and is controlled by changing the step size and the time step. The global error is obtained by integration of the error equations. In the implicit scheme, the system of linear equations at each time step is solved iteratively by the GMRES method. Numerical examples executed on a parallel computer illustrate the method.

Regular and singular [formula omitted]-blocking of difference corrected multistep methods for nonstiff index-2 DAEs

There are several approaches to using nonstiff implicit linear multistep methods for solving certain classes of semi-explicit index 2 DAEs. Using β-blocked discretizations (Arévalo et al., 1996) Adams–Moulton methods up to order 4 and difference corrected BDF (Söderlind, 1989) methods up to order 7 can be stabilized. As no extra matrix computations are required, this approach is an alternative to projection methods. Here we examine some variants of β-blocking. We interpret earlier results as regular β-blocking and then develop singular β-blocking. In this nongeneric case the stabilized formula is explicit, although the discretization of the DAE as a whole is implicit. We investigate which methods can be stabilized in a broad class of implicit methods based on the BDF ρ polynomials. The class contains the BDF, Adams–Moulton and difference corrected BDF methods as well as other high order methods with small error constants. The stabilizing difference operator τ is selected by a minimax criterion for the moduli of the zeros of σ+τ. The class of explicit methods suitable as β-blocked methods is investigated. With singular β-blocking, Adams–Moulton methods up to order 7 can be stabilized with the stabilized method corresponding to the Adams–Bashforth methods.

Behind and beyond the Matlab ODE suite

The paper explains the concepts of order and absolute stability of numerical methods for solving systems of first-order ordinary differential equations (ODE) of the form ▪ describes the phenomenon of problem stiffness, and reviews explicit Runge-Kutta methods, and explicit and implicit linear multistep methods. It surveys the five numerical methods contained in the Matlab ODE suite (three for nonstiff problems and two for stiff problems) to solve the above system, lists the available options, and uses the odedemo command to demonstrate the methods. One stiff ode code in Matlab can solve more general equations of the form M( t) y′ = f( t, y) provided the Mass option is on.

The stabilization of linear multistep methods for constrained mechanical systems

Linear multistep methods from ODE theory may be applied straightforwardly to index-2 DAEs in Hessenberg form if they are strictly stable at infinity (Hairer and Wanner, 1996, Theorem VII.3.6). This condition is very restrictive and excludes, e.g., all higher order Adams methods. In the paper we present an alternative way to apply implicit linear multistep methods to index-2 systems. The convergence of these partitioned linear multistep methods is guaranteed whenever the underlying ODE method is convergent with order p ≥ 3. We discuss the new approach in detail for the application to model equations of constrained mechanical systems. The theoretical results are illustrated by a numerical comparison of multistep methods for index-2 DAEs in Hessenberg form.

On Spurious Behavior of Super-Stable Implicit Methods

The objective of this paper is to gain a better understanding of the asymptotic nonlinear behavior of super-stable implicit linear multistep methods for constant time steps. Examples of the nonlinear effect caused by grid adaptation and super-stable implicit total variation diminishing (TVD) schemes on the overall performance of the numerical procedure are given. A method to minimize spurious steady-state numerical solutions is discussed.