Stiff Initial Value Problems Research Articles

Construction of Five-step Continuous Block General Method for the Solution of Ordinary Differential Equations

In this paper, a self starting five step Continuous Block Hybrid Adams Moulton Method (CBHAM) with three off-grid points is developed using collocation and interpolation procedures. The predictor schemes are then expanded using Taylor’s series expansion. Multiple numerical integrators were produced and arrived at some discrete schemes. The discrete schemes are of uniform order and are assembled into a single block matrix equation. These equations are simultaneously applied to provide the approximate solution for stiff initial value problems for ordinary differential equations. The order of accuracy and stability of the block method is discussed and its accuracy is established numerically.

Derivation of Diagonally Implicit Block Backward Differentiation Formulas for Solving Stiff Initial Value Problems

The diagonally implicit 2-point block backward differentiation formulas (DI2BBDF) of order two, order three, and order four are derived for solving stiff initial value problems (IVPs). The stability properties of the derived methods are investigated. The implementation of the method using Newton iteration is also discussed. The performance of the proposed methods in terms of maximum error and computational time is compared with the fully implicit block backward differentiation formulas (FIBBDF) and fully implicit block extended backward differentiation formulas (FIBEBDF). The numerical results show that the proposed method outperformed both existing methods.

High-Order Implicit Time-Marching Methods Based on Generalized Summation-By-Parts Operators

This article extends the theory of classical finite-difference summation-by-parts (FD-SBP) time-marching methods to the generalized summation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods are shown to retain: A and L-stability, as well as superconvergence of integral functionals when integrated with the quadrature associated with the discretization. This also implies that the solution approximated at the end of each time step is superconvergent. In addition GSBP time-marching methods constructed with a diagonal norm are BN-stable. This article also formalizes the connection between FD-SBP/GSBP time-marching methods and implicit Runge-Kutta methods. Through this connection, the minimum accuracy of the solution approximated at the end of a time step is extended for nonlinear problems. It is also exploited to derive conditions under which nonlinearly stable GSBP time-marching methods can be constructed. The GSBP approach to time marching can simplify the construction of high-order fully-implicit Runge-Kutta methods with a particular set of properties favourable for stiff initial value problems, such as L-stability. It can facilitate the analysis of fully discrete approximations to PDEs and is amenable to to multi-dimensional spcae-time discretizations, in which case the explicit connection to Runge-Kutta methods is often lost. A few examples of known and novel Runge-Kutta methods associated with GSBP operators are presented. The novel methods, all of which are L-stable and BN-stable, include a four-stage seventh-order fully-implicit method, a three-stage third-order diagonally-implicit method, and a fourth-order four-stage diagonally-implicit method. The relative efficiency of the schemes is investigated and compared with a few popular non-GSBP Runge-Kutta methods.

Rational interpolation method for solving initial value problems (IVPs) in ordinary differential equations

In this paper we designed Rational Interpolation Method for solving Ordinary Differential Equations (ODES) and Stiff initial value problems (IVPs).This was achieved by considering the Rational Interpolation Formula. y(x)=U(x)=P0+P1x+P2x2+P3x3+p4x4+P5x51+q1x+q2x2+q3x3+q4x4+q5x5+q6x6, satisfying U(Xn+i)=yn+i,i=0,1,2,3,4,5,6.We also implemented k=6 in Aashikpelokhai (1991) class of rational integration formulas given by yn+1=∑i=05piXn+1i1+∑i=16qiXn+1i where, [Display omitted] The results as analyzed with the computer show that the rational interpolation method copes favorably well with ordinary differential equations and stiff initial value problems.

A-Stable High Order Hybrid Linear Multistep Methods for Stiff Problems

This paper considers a new class of high order hybrid linear multistep methods for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The numerical experiments shows the application of the methods on stiff problems.

English

English

L(α)-stable variable-order implicit second derivative Runge-Kutta methods

This paper considers the extension of the popular Runge-Kutta methods (RKM) to second derivative Runge-Kutta methods (SDRKMs) for the direct solution of stiff initial value problems (IVPs) of ordinary differential equations (ODEs). The methods are based on using collocation and interpolation technique. The last stage of the input approximation is identical to the output method. The SDRKMs are L(α)-stable for the methods examined. Numerical experiments are given comparing one of these methods with a two derivative Runge Kutta method (TDRKM) and a second derivative linear multistep method (SDLMM).

Two classes of implicit–explicit multistep methods for nonlinear stiff initial-value problems

The initial value problems of nonlinear ordinary differential equations which contain stiff and nonstiff terms often arise from many applications. In order to reduce the computation cost, implicit–explicit (IMEX) methods are often applied to these problems, i.e. the stiff and non-stiff terms are discretized by using implicit and explicit methods, respectively. In this paper, we mainly consider the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems, and present two classes of the IMEX multistep methods by combining implicit one-leg methods with explicit linear multistep methods and explicit one-leg methods, respectively. The order conditions and the convergence results of these methods are obtained. Some efficient methods are constructed. Some numerical examples are given to verify the validity of the obtained theoretical results.

Exponentially fitted third derivative three-step methods for numerical integration of stiff initial value problems

In this paper, we consider the construction of exponentially fitted third derivative three-step method of order six for the numerical integration of stiff initial-value problems. The numerical analysis of the stability of the new method was discussed and some numerical experiments confirming theoretical expectation are provided. Finally, the numerical results show that the new method is A-stable and compete favorably with the existing methods in terms of efficiency and accuracy.

A family of highly stable second derivative block methods for stiff IVPs in ODEs

This paper considers a class of highly stable block methods for numerically solving initial value problems (IVPs) in ordinary differential equations (ODEs). The boundary locus of the proposed parallel one-block, r-output point algorithms shows that the new schemes are A-stable for output points r = 2(2)8 and A(α)-stable for output points r = 10(2)20, where r is the number of processors in a particular block method in the family. Numerical results of the block methods are compared with the second derivative linear multistep method in [8].

Second derivative general linear methods

A class of second derivative Runge Kutta methods (SD-RKM) which has a simple transformation to the general linear methods (GLM) are employed for the numerical integration of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The new GLM which incorporates second derivative (SD) terms have the advantage of higher order L-stable methods for a given number of stages compared with the classical GLM.

A Modified Block Adam Moulton (MOBAM) Method for the Solution of Stiff Initial Value Problems of Ordinary Differential Equations

Stiff ordinary differential equations pose computational difficulties as they present severe step size restrictions on the numerical methods to be used. Construction of numerical methods that possess suitable stability properties for the solution of such systems has been the target of many researchers. Development of methods suitable for these systems of equations has been either through the use of derivative of the solution or by introducing off-step points, additional stages or super future points. These processes have been exploited in Runge-Kutta methods or linear multistep methods. In this study, an improved class of linear multistep block method has been constructed based on Adams Moulton block methods. The improved methods are shown to be A-stable, a property desirable to handle stiff ODEs. Methods of uniform orders 10 and 11 have been constructed. The efficiency of the new methods tested on stiff systems of ODEs and the results reveal that the MOBAM methods compare favourably with results obtained using the state of the art Matlab Ode23 solver.

A NON-LINEAR ABSOLUTELY-STABLE EXPLICIT NUMERICAL INTEGRATION ALGORITHM FOR STIFF INITIAL-VALUE PROBLEMS

The time-step in integration process has two restri ctions. The first one is the time step restriction due to accuracy requirement τac and the second one is the time-step restriction du e to stability requirement τst . The most of explicit methods have small stability regio ns and consequently small τst . It obliges us to solve stiff problems with small step size τst << τac . The implicit methods work well with stiff problem s but these methods require more work per step than the explici t methods. In this study, a non-linear absolutly st able explicit one step numerical integration algorithm i s proposed for solving non linear stiff initial-val ue problems in ordinary differential equations. The al gorithm is based on deriving a non-linear relation between the dependent variable and its derivatives from the well known Taylor expansion. The accuracy of the method depends on some unknown parameter inserted in Taylor expansion and determined from the error analysis. The accuracy and stability properties of the method are investigated and shown to yield at least third order and A-stable. The results obtained in the numerical exp eriments show the efficiency of the present method in solving stiff initial value problems.

A new variable step size block backward differentiation formula for solving stiff initial value problems

A new block backward differentiation formula of order 4 with variable step size is formulated. By varying a parameter in the formula, different sets of formulae with A-stability property can be generated. At the cost of an additional function evaluation, the accuracy of the method is seen to outperform some existing backward differentiation formula algorithms. The strategy involved in controlling the step size ratio is also described. The problems tested with the method show its efficiency in solving stiff initial value problems.

A class of A(α)-stable numerical methods for stiff problems in ordinary differential equations

The A(α)-stable numerical methods (ANMs) for the number of steps k ≤ 7 for stiff initial value problems (IVPs) in ordinary differential equations (ODEs) are proposed. The discrete schemes proposed from their equivalent continuous schemes are obtained. The scaled time variable t in a continuous method, which determines the discrete coefficients of the discrete method, is chosen in such a way as to ensure that the discrete scheme attains a high order and A(α)-stability. We select the value of α for which the schemes proposed are absolutely stable. The new algorithms are found to have a comparable accuracy with that of the backward differentiation formula (BDF) discussed in [12] which implements the Ode15s in the Matlab suite.

An Implicit Method for Numerical Solution of Singular and Stiff Initial Value Problems

An implicit method has been presented for solving singular initial value problems. The method is simple and gives more accurate solution than the implicit Euler method as well as the second order implicit Runge-Kutta (RK2) (i.e., implicit midpoint rule) method for some particular singular problems. Diagonally implicit Runge-Kutta (DIRK) method is suitable for solving stiff problems. But, the derivation as well as utilization of this method is laborious. Sometimes it gives almost similar solution to the two-stage third order diagonally implicit Runge-Kutta (DIRK3) method and the five-stage fifth order diagonally implicit Runge-Kutta (DIRK5) method. The advantage of the present method is that it is used with less computational effort.

An Accurate Block Solver for Stiff Initial Value Problems

New implicit block formulae that compute solution of stiff initial value problems at two points simultaneously are derived and implemented in a variable step size mode. The strategy for changing the step size for optimum performance involves halving, increasing by a multiple of 1.7, or maintaining the current step size. The stability analysis of the methods indicates their suitability for solving stiff problems. Numerical results are given and compared with some existing backward differentiation formula algorithms. The results indicate an improvement in terms of accuracy.

The Convergence and Order of the 2–point Improved Block Backward Differentiation Formula

This paper studied the 2-point improved block backward differentiation formula for solving stiff initial value problems proposed by Musa et al (2013) and further established the necessary conditions for the convergence of the method. It is shown that the method is both zero stable and consistent. The order of the method is also derived.

On the construction of high order A(α)-stable hybrid linear multistep methods for stiff IVPs in ODEs

In this paper, we present a class of A(α)-stable hybrid linear multistep methods for numerical solving stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The method considered uses a second derivative like the Enright’s second derivative linear multistep methods for stiff IVPs in ODEs.

Analytic solution of nonlinear singularly perturbed initial value problems through iteration

This paper is concerned with singularly perturbed initial value problems for systems of ordinary differential equations. Here our emphasis will be on nonlinear phenomena and properties, particularly those with physical relevance. Since very few nonlinear systems can be solved explicitly, one must typically rely on a numerical scheme to accurately approximate the solution. However, numerical schemes do not always give accurate results, and we discuss the class of stiff differential equations, which present a more serious challenge to numerical analysts. In this paper, we derive in closed from, analytic solution of stiff nonlinear initial value problems, through iteration. The obtained sequence of iterates is based on the use of Lagrange multipliers. Moreover, the illustrative examples shows the efficiency of the method.