Rational-fractional methods for solving stiff systems of differential equations

This paper proposes new numerical methods for solving stiff systems of first-order differential equations not resolved with respect to the derivative. These methods are based on rational-fractional approximations of the vector-valued function of solution of the system considered. The authors study the stability of the constructed methods of arbitrary finite order of accuracy. Analysis of the results of experimental studies of these methods by test examples of various types confirms their efficiency. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 203–208, 2006.     

Nonlinear iterative operator splitting methods and applications for nonlinear differential equations

In this paper we discuss decomposition methods that are used for solving nonlinear differential equations. The motivation arose from the need to decouple nonlinear operator equations into simpler, computable operator equations [1]; [2]. We consider iterative operator-splitting methods for the decoupling of the differential equations and we apply iterative steps to achieve linearisation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient higher order implicit one-step methods for integration of stiff differential equations

A new implicit integration method is presented which can efficiently be applied in the solution of (stiff) differential equations. The given formulas are of a modified implicit Runge-Kutta type and areA-stable. They may containA-stable embedded methods for error estimation and step-size control.     

Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

Summary GeneralizedA()-stable Runge-Kutta methods of order four with stepsize control are studied. The equations of condition for this class of semiimplicit methods are solved taking the truncation error into consideration. For application anA-stable and anA(89.3°)-stable method with small truncation error are proposed and test results for 25 stiff initial value problems for different tolerances are discussed.     

High order stiffly stable composite multistep methods for numerical integration of stiff differential equations

Let \(\dot y\) =f(y,t) withy(t ₀)=y ₀ possess a solutiony(t) fort≧t ₀. Sett _n=t ₀+nh, n=1, 2,.... Lety ₀ denote the approximate solution ofy(t _n) defined by the composite multistep method with \(\dot y_n \) =f(y _n ,t _n ) andN=1, 2,.... It is conjectured that the method is stiffly stable with orderp=l for alll≧1 and shown to be so forl=1,..., 25. The method is intrinsically efficient in thatl future approximate solution values are established simultaneously in an iterative solution process with only one function evaluation per iteration for each of thel future time points. Step and order control are easily implemented, in that the approximate solution at only one past point appears in each component multistep formula of the method and in that the local truncation error for the first component multistep formula of the method is easily evaluated as $$T^{[l]} = \frac{h}{{t_{Nl} - t_{(N - 1)l - 1} }}\{ y_{Nl}^{PRED} - y_{Nl} \} ,$$ wherey _Nl ^PRED denotes the value att _Nl of the Lagrange interpolating polynomial passing through the pointsy _(N?1)l+j att _(N?1)l+j withj=?1, 0,...,l ? 1.     

On one class of numerical methods for solving stiff systems of differential equations

The paper is concerned with a modification of a Cowell-type method: at each step the solution is evaluated at several points and only some first of these values are retained. Stability of these methods is examined. In particular, among the methods of this group we single out one method that has the fourth order of accuracy and is stable for stiff systems with any choice of integration step.     

A class of one-step one-stage methods for stiff systems of ordinary differential equations

A new class of one-step one-stage methods (ABC-schemes) designed for the numerical solution of stiff initial value problems for ordinary differential equations is proposed and studied. The Jacobian matrix of the underlying differential equation is used in ABC-schemes. They do not require iteration: a system of linear algebraic equations is once solved at each integration step. ABC-schemes are A- and L-stable methods of the second order, but there are ABC-schemes that have the fourth order for linear differential equations. Some aspects of the implementation of ABC-schemes are discussed. Numerical results are presented, and the schemes are compared with other numerical methods.     

Heterogeneous multiscale methods for stiff ordinary differential equations

The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are based on higher-order estimates of the effective force by kernels satisfying certain moment conditions and regularity properties. These new methods have superior computational complexity compared to traditional methods for stiff problems with oscillatory solutions.

     

A-stable one-step methods with step-size control for stiff systems of ordinary differential equations

Two efficient third-and fourth-order processes for solving the initial value problem for ordinary differential equations are studied. Both are A-stable and so recommended for stiff systems. An economic and efficient way of step-size control is given for each of them. Numerical examples are considered.     

Contractive methods for stiff differential equations part I

An integration method for ordinary differential equations is said to be contractive if all numerical solutions of the test equationx=x generated by that method are not only bounded (as required for stability) but non-increasing. We develop a theory of contractivity for methods applied to stiff and non-stiff, linear and nonlinear problems. This theory leads to the design of a collection of specific contractive Adams-type methods of different orders of accuracy which are optimal with respect to certain measures of accuracy and/or contractivity. Theoretical and numerical results indicate that some of these novel methods are more efficient for solving problems with a lack of smoothness than are the familiar backward differentiation methods. This lack of smoothness may be either inherent in the problem itself, or due to the use of strongly varying integration steps. In solving smooth problems, the efficiency of the low-order contractive methods we propose is approximately the same as that of the corresponding backward differentiation methods.This work was done during the first author's stay at the IBM Thomas J. Watson Research Center under his appointment as Senior Researcher of the Academy of Finland. It was sponsored in addition by the IBM Corporation and by the AirForce Office of Scientific Research (AFSC), United States Air Force, under contracts No. F44620-75-C-0058 and F49620-77-C-0088. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.     

Relaxed Newton-like methods for stiff differential systems

Newton-like methods are commonly used to solve the nonlinear equations arising in the numerical solution of stiff differential equations. We show that easily calculable relaxation factors may be used to improve the convergence properties of such methods. The technique is also applicable when partitioning methods are used.     

Spectral methods for some singularly perturbed third order ordinary differential equations

Spectral methods with interface point are presented to deal with some singularly perturbed third order boundary value problems of reaction-diffusion and convection-diffusion types. First, linear equations are considered and then non-linear equations. To solve non-linear equations, Newton’s method of quasi-linearization is applied. The problem is reduced to two systems of ordinary differential equations. And, then, each system is solved using spectral collocation methods. Our numerical experiments show that the proposed methods are produce highly accurate solutions in little computer time when compared with the other methods available in the literature.      

Overrelaxation methods for hybrid systems of first order partial differential equations

The classical overrelaxation method, applicable to second order elliptic partial differential equations, is extended to hybrid systems of first order equations. It is shown both by theory and by an example that the method has first order convergence rate.     

Efficient parallel multivalue hybrid methods for stiff differential equations

A class of efficient parallel multivalue hybrid methods for stiff differential equations are presented, which are all extremely stable at infinity,A-stable for orders 1–3 and A(α)-stable for orders 4–8. Each method of the class can be performed parallelly using two processors with each processor having almost the same computational amount per integration step as a backward differentiation formula (BDF) of the same order with the same stepsize performed in serial, whereas the former has not only much better stability properties but also a computational accuracy higher than the corresponding BDF. Theoretical analysis and numerical experiments show that the methods constructed in this paper are superior in many respects not only to BDFs but also to some other commonly used methods.