A-stable Methods Research Articles

Convergence results for general linear methods on singular perturbation problems

Many numerical methods used to solve Ordinary Differential Equations, or Differential Algebraic Equations can be written as general linear methods. The B-convergence results for general linear methods are for algebraically stable methods, and therefore useless for nearly A-stable methods. The purpose of this paper is to show convergence for singular perturbation problems for the class of general linear methods without assuming A-stability.

Construction of higher-order algebraic one-step schemes in stiff BVPs

We present a generalization of the SQRT scheme introduced by Schmitt (1990) in the form of implicit Runge-Kutta methods with algebraic matrix coefficients. After setting up a characterization of A-stable methods we discuss the singularly perturbed limit of two-stage schemes, which reveals some possible instabilities for nonautonomous linear problems and explains the different performance of some special schemes in a numerical example.

An asymptotic expansion of the global discretization error of difference schemes for numerically solving a quasilinear parabolic system of differential equations

In this paper a class of implicit, A-stable one-step difference methods for quasilinear strongly coupled parabolic systems is considered. For the global discretization error of this class of finite difference approximations, an asymptotic expansion in power of the step size with respect to the space and the time coordinates is proved. This result allows to obtain more accurate solutions by the principles of local and global Richardson extrapolation or the method of correction by higher order differences without loss of A-stability.

A-stable parallel block methods for ordinary and integro-differential equations

In this paper we study the stability of a class of block methods which are suitable for integrating ordinary and integro-differential equations on parallel computers. A-stable methods of orders 3 and 4 and A(α)-stable methods with α > 89.9° of order 5 are constructed. On multiprocessor computers these methods are of the same computational complexity as implicit linear multistep methods on one-processor computers.

Extended a-stable two-step methods for the numerical solution of ordinary differential equations

A class of two-step finite difference formulae for the numerical solution of first order differential equations is investigated. A particular one parameter family of methods is derived which is shown to be fourth order convergent and A-stable. Numerical results are presented for several test problems.

One-step collocation methods for differential-algebraic systems of index 1

In this paper we propose one-step collocation methods for linear differential-algebraic equation (DAE) systems of index 1. The proposed methods give a continuous approximate solution on the integration interval [ t 0, t N ]. We study the uniform convergence properties of the proposed collocation methods. Some of the methods show an order reduction phenomenon, at the nodes, similar to that observed for Runge-Kutta methods (Petzold, 1986). Since DAE systems may be considered as a kind of very stiff differential systems, the collocation methods for DAE systems have to verify strong stability properties. For example, collocation methods at Radau points are particularly suitable for solving DAE systems. However the numerical results will also show that Gauss-Legendre collocation methods may be applied successfully to DAE systems also if these are only A-stable methods.

Algebraic characterization of A-stable Runge-Kutta methods

Important stability concepts for Runge-Kutta methods are A-stability and B-stability. Whereas A-stability is described by an analytic property of the stability function, B-stability can be characterized by algebraic conditions related to the generating matrix of the method. In this paper, we deduce an algebraic characterization of A-stable Runge-Kutta methods similar to that of B-stable methods. This result gives us information about the gap between the set of A- and B-stable Runge-Kutta methods. Further results, concerning the W-transformation, provide the possibility to derive A-stable methods. To a giver positive quadrature formula we construct all corresponding A-stable Runge-Kutta methods of a certain type.

On the Numerical Integration of Nonlinear Two-Point Boundary Value Problems Using Iterated Deferred Corrections. Part 2: The Development and Analysis of Highly Stable Deferred Correction Formulae

Iterated deferred correction methods have been very widely used for the numerical solution of general nonlinear two-point boundary value problems in ordinary differential equations. However, there may be loss of stability when this procedure is applied, since the process of deferred correction is normally explicit. This phenomenon is well known in the theory of initial value problems, where a classic stability theorem of Dahlquist is central in the design of A-stable methods. As a direct result of the deterioration in stability, deferred correction procedures are sometimes ineffective for problems with “rough” solutions, such as singular perturbation or turning point problems. (Such problems are often referred to as “stiff.”) In this paper we consider the idea of using implicit deferred corrections, which allows the derivation of high-order methods that maintain the good stability properties of the underlying low-order formula. Using the ideas developed in this paper, a highly stable, variable-order deferred correction scheme is derived, and the performance of its implementation is compared with that of COLSYS and D02GAF on some linear test problems.

An algorithm of variable order and step based on methods of the Rozenbrok type

Imbedded A-stable methods of the Rozenbrok type from the first to the third orders are constructed to solve stiff autonomous sets of ordinary differential equations. A new error-estimation method is suggested which enables one to construct a simple and effective algorithm of variable order and step.

An Adaptive Boundary Value Runge–Kutta Solver for First Order Boundary Value Problems

The use of finite difference methods which combine features of both Runge–Kutta processes and Gap schemes is proposed for developing adaptive codes for the solution of first order differential equations with two-point boundary conditions. The order conditions for the coefficients of these processes are given. A set of conditions, by which these order conditions can be reduced in number, are developed. An eighth order A-stable method which has second, fourth and sixth order A-stable methods embedded in it is given as an example. A variable order, variable step finite difference solver using embedded methods of this kind is described.

Multistep methods using higher derivatives and damping at infinity

Linear multistep methods using higher derivatives are discussed. The order of damping at infinity which measures the stability behavior of a k-step method for large h is introduced, A-stable methods with positive damping order are most suitable for stiff problems. A method for computing the damping order is given. Necessary and sufficient conditions for A-stability, A A ( α ) A(\alpha ) -stability and stiff stability are presented. A new A-stable two-step method of order 4 with damping order 1 is found and numerical results are given.

Multistep Methods Using Higher Derivatives and Damping at Infinity

Linear multistep methods using higher derivatives are discussed. The order of damping at infinity which measures the stability behavior of a k-step method for large h is introduced, A-stable methods with positive damping order are most suitable for stiff problems. A method for computing the damping order is given. Necessary and sufficient conditions for A-stability, A $A(\alpha )$-stability and stiff stability are presented. A new A-stable two-step method of order 4 with damping order 1 is found and numerical results are given.

Hermite interpolation and A-stable methods for stiff ordinary differential equations

Hermite interpolation in the form of Newton's divided difference expressions is employed to give a generating function for A-stable difference methods of order 2 n. These methods can be used to solve the initial value ordinary differential equation y′= g( y,t), y( a)=η. The extension to higher dimensions is considered, and practical suggestions are given for step size changes and order changes.

Solving Nonlinear Second Kind Volterra Equations by Modified Increment Methods

One step procedures for solving Volterra integral equations have appeared in the literature but have generally been ignored because of their complicated nature as well as their inefficiency. In this paper, a general class of methods are modified to give increased efficiency. Error bounds are provided, a class of A-stable methods are presented and numerical examples follow.

High-order methods for parabolic problems

Error estimates valid for all t ⩾ 0 for the semi-discrete Galerkin approximation of a parabolic mixed boundary-initial value problem are presented. The solution of the resulting system of ordinary differential equations by implicit Runge-Kutta formulae for arbitrarily high order of accuracy, are discussed. Strongly A-stable methods are found to be advantageous. Theoretical and experimental results for the solution of the resulting system of algebraic equations using a preconditioned outer iteration scheme are discussed. Even the inner linear algebraic equations are preferably solved by iteration.

On the efficiency of a class of a-stable methods

The numerical solution of systems of differential equations of the formB dx/dt=σ(t)Ax(t)+f(t),x(0) given, whereB andA (withB and —(A+AT) positive definite) are supposed to be large sparse matrices, is considered.A-stable methods like the Implicit Runge-Kutta methods based on Radau quadrature are combined with iterative methods for the solution of the algebraic systems of equations.

A nonexistence theorem for explicit 𝐴-stable methods

It is proved that there are no A-stable explicit methods in a general class of "linear" methods. The class contains, for example, Runge-Kutta methods, linear multistep methods, predictor-corrector formulas, cyclic multistep methods and linear multistep methods with higher derivatives.

A -Stable Composite Multistep Methods

Consider the set of multistep formulas ∑ l -1 j mn - k α ij x mn + j - h ∑ l -1 j mn - k β ij x mn + j = 0, i = 1, ···, l , where x mn + j = y mn + j for j = - k , ···, -1 and x n = ƒ n = ƒ( x n , t n ). These formulas are solved simultaneously for the x mn + j with j = 0, ···, l - 1 in terms of the x mn + j with j = - k , ··· , - 1, which are assumed to be known. Then y mn + j is defined to be x mn + j for j = 0, ··· , m - 1. For j = m , ··· , l - 1, x mn + j is discarded. The set of y 's generated in this manner for successive values of n provide an approximate solution of the initial value problem: y = ƒ( y, t ), y ( t 0 ) = y 0 . It is conjectured that if the method, which is referred to as the composite multistep method, is A -stable, then its maximum order is 2 l . In addition to noting that the conjecture conforms to Dahlquist's bound of 2 for l = 1, the conjecture is verified for k = 1. A third-order A -stable method with m = l = 2 is given as an example, and numerical results established in applying a fourth-order A -stable method with m = 1 and l = 2 are described. A -stable methods with m = l offer the promise of high order and a minimum of function evaluations—evaluation of ƒ( y, t ) at solution points. Furthermore, the prospect that such methods might exist with k = 1—only one past point—means that step-size control can be easily implemented

Some 𝐴-stable methods for stiff ordinary differential equations

This paper gives some A A -stable methods of order 2 n 2n , with variable coefficients, based on Hermite interpolation polynomials, for the stiff system of ordinary differential equations, making use of n n starting values. The method is exact if the problem is of the form y ′ ( t ) = P y ( t ) + Q ( t ) y’(t) = Py(t) + Q(t) , where P P is a constant and Q ( t ) Q(t) is a polynomial of degree 2 n 2n .

Some A-Stable Methods for Stiff Ordinary Differential Equations

Some A-Stable Methods for Stiff Ordinary Differential Equations