A-stability of Runge-Kutta methods for systems with additive noise

Numerical stability of both explicit and implicit Runge-Kutta methods for solving ordinary differential equations with an additive noise term is studied. The concept of numerical stability of deterministic schemes is extended to the stochastic case, and a stochastic analogue of Dahlquist'sA-stability is proposed. It is shown that the discretization of the drift term alone controls theA-stability of the whole scheme. The quantitative effect of implicitness uponA-stability is also investigated, and stability regions are given for a family of implicit Runge-Kutta methods with optimal order of convergence.This author was partially supported by the Italian Consiglio Nazionale delle Ricerche.     

Über die Effektivität einiger nichtlinearer Verfahren bei der numerischen Behandlung steifer Differentialgleichungssysteme

Summary The stability and accuracy of some explicit nonlinear methods for the numerical integration of stiff systems of ordinary differential equations are investigated. It is shown, that in the general case they can produce the essential error. The special class of stiff systems is singled out, for which these methods are highly efficient. Some numerical results are also presented.

     

Global error estimation and control in linearly-implicit parallel two-step peer W-methods

The class of linearly-implicit parallel two-step peer W-methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations. Those schemes allow for parallelism across the method, that is an important feature for implementation on modern computational devices. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff problems. In this paper, we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user-supplied accuracy requirement in automatic mode. An algorithm of such global error control and other technical particulars are also discussed here. Numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions, including nonstiff, stiff, very stiff and large-scale differential equations. A comparison with the well-known stiff solver RODAS is also shown.     

Partitioned adaptive Runge-Kutta methods and their stability

Summary This paper deals with the solution of partitioned systems of nonlinear stiff differential equations. Given a differential system, the user may specify some equations to be stiff and others to be nonstiff. For the numerical solution of such a system partitioned adaptive Runge-Kutta methods are studied. Nonstiff equations are integrated by an explicit Runge-Kutta method while an adaptive Runge-Kutta method is used for the stiff part of the system.The paper discusses numerical stability and contractivity as well as the implementation and usage of such compound methods. Test results for three partitioned stiff initial value problems for different tolerances are presented.     

Unconditionally stable explicit methods for parabolic equations

Summary This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA ₀-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.     

Methods for parallel integration of stiff systems of odes

This paper presents a class of parallel numerical integration methods for stiff systems of ordinary differential equations which can be partitioned into loosely coupled sub-systems. The formulas are called decoupled backward differentiation formulas, and they are derived from the classical formulas by restricting the implicit part to the diagnonal sub-system. With one or several sub-systems allocated to each processor, information only has to be exchanged after completion of a step but not during the solution of the nonlinear algebraic equations.The main emphasis is on the formula of order 1, the decoupled implicit Euler formula. It is proved that this formula even for a wide range of multirate formulations has an asymptotic global error expansion permitting extrapolation. Besides, sufficient conditions for absolute stability are presented.     

The composite Milstein methods for the numerical solution of Ito stochastic differential equations

In this paper, we present the composite Milstein methods for the strong solution of Ito stochastic differential equations. These methods are a combination of semi-implicit and implicit Milstein methods. We give a criterion for choosing either the implicit or the semi-implicit scheme at each step of our numerical solution. The stability and convergence properties are investigated and discussed for the linear test equation. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. The stability properties of the composite Milstein methods are found to be more superior compared to those of the Milstein, the Euler and even better than the composite Euler method. This superiority in stability makes the methods a better candidate for the solution of stiff SDEs.     

Three-stage stochastic Runge–Kutta methods for stochastic differential equations

In this paper we discuss three-stage stochastic Runge–Kutta (SRK) methods with strong order 1.0 for a strong solution of Stratonovich stochastic differential equations (SDEs). Higher deterministic order is considered. Two methods, a three-stage explicit (E3) method and a three-stage semi-implicit (SI3) method, are constructed in this paper. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of several standard test problems.     

Construction of a multirate RODAS method for stiff ODEs

Multirate time stepping is a numerical technique for efficiently solving large-scale ordinary differential equations (ODEs) with widely different time scales localized over the components. This technique enables one to use large time steps for slowly varying components, and small steps for rapidly varying ones. Multirate methods found in the literature are normally of low order, one or two. Focusing on stiff ODEs, in this paper we discuss the construction of a multirate method based on the fourth-order RODAS method. Special attention is paid to the treatment of the refinement interfaces with regard to the choice of the interpolant and the occurrence of order reduction. For stiff, linear systems containing a stiff source term, we propose modifications for the treatment of the source term which overcome order reduction originating from such terms and which we can implement in our multirate method.     

Comparing the contractivity properties of semi-implicit methods

Contractivity is a desirable property of numerical integration methods for stiff systems of ordinary differential equations. In this paper, numerical parameters are used to allow a direct and quantitative comparison of the contractivity properties of various methods for non-linear stiff problems. Results are provided for popular Rosenbrock methods and some more recently developed semi-implicit methods.     

High order methods for the numerical integration of ordinary differential equations

Summary High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.     

On Multistep Methods for Differential Equations of Fractional Order

This paper concerns with numerical methods for the treatment of differential equations of fractional order. Our attention is concentrated on fractional multistep methods of both implicit and explicit type, for which order conditions and stability properties are investigated. Dedicated to the memory of Professor Aldo Cossu     

Given a one-step numerical scheme,on which ordinary differential equations is it exact?

A necessary condition for a (non-autonomous) ordinary differential equation to be exactly solved by a one-step, finite difference method is that the principal term of its local truncation error be null. A procedure to determine some ordinary differential equations exactly solved by a given numerical scheme is developed. Examples of differential equations exactly solved by the explicit Euler, implicit Euler, trapezoidal rule, second-order Taylor, third-order Taylor, van Niekerk’s second-order rational, and van Niekerk’s third-order rational methods are presented.     

On the Convergence of LMF-type Methods for SODEs

In a previous paper [3], some numerical methods for stochastic ordinary differential equations (SODEs), based on Linear Multistep Formulae (LMF), were proposed. Nevertheless, a formal proof for the convergence of such methods is still lacking. We here provide such a proof, based on a matrix formulation of the discrete problem, which allows some more insight in the structure of LMF-type methods for SODEs.     

The role of coefficients of a general SPDE on the stability and convergence of a finite difference method

In this paper for the approximate solution of stochastic partial differential equations (SPDEs) of Itô-type, the stability and application of a class of finite difference method with regard to the coefficients in the equations is analyzed. The finite difference methods discussed here will be either explicit or implicit and a comparison between them will be reported. We prove the consistency and stability of these methods and investigate the influence of the multiplier (particularly multiplier of the random noise) in mean square stability. From stochastic version of Lax-Richtmyer the convergence of these methods under some conditions are established. Numerical experiments are included to show the efficiency of the methods.     

A study of Rosenbrock-type methods of high order

Summary This paper deals with the solution of nonlinear stiff ordinary differential equations. The methods derived here are of Rosenbrock-type. This has the advantage that they areA-stable (or stiffly stable) and nevertheless do not require the solution of nonlinear systems of equations. We derive methods of orders 5 and 6 which require one evaluation of the Jacobian and oneLU decomposition per step. We have written programs for these methods which use Richardson extrapolation for the step size control and give numerical results.     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.     

一类带有差分扰动项的显式线性多步法的讨论

一、方法的形成 求解常微分程初值问题 y′=f(x,y),y(a)= y_0,x∈[a b).(1.0)一般形式的显式k阶线性多步法表成(见[3]):     

An improved approximate Newton method for implicit Runge-Kutta formulas

Implicit Runge-Kutta (IRK) methods (such as the s-stage Radau IIA method with s=3,5, or 7) for solving stiff ordinary differential equation systems have excellent stability properties and high solution accuracy orders, but their high computing costs in solving their nonlinear stage equations have seriously limited their applications to large scale problems. To reduce such a cost, several approximate Newton algorithms were developed, including a commonly used one called the simplified Newton method. In this paper, a new approximate Jacobian matrix and two new test rules for controlling the updating of approximate Jacobian matrices are proposed, yielding an improved approximate Newton method. Theoretical and numerical analysis show that the improved approximate Newton method can significantly improve the convergence and performance of the simplified Newton method.