Family of symplectic implicit Runge-Kutta formulae

A family of formulae for the sympletic IRK method is investigated. Specifically, focus is given to general solutions for formula parameters of IRK under the symplectic and the order conditions. Examples of such formulae are constructed for up to three stages.     

High-order adaptive finite element-singly implicit Runge-Kutta methods for parabolic differential equations

We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-90-0194; the U.S. Army Research Office under Contract Number DAAL 03-91-G-0215; by the National Science Foundation under Grant Number CDA-8805910; and by a grant from the Committee on Research, Tulane University.     

Optimal order diagonally implicit Runge-Kutta methods

In this paper, the optimal order of non-confluent Diagonally Implicit Runge-Kutta (DIRK) methods with non-zero weights is examined. It is shown that the order of aq-stage non-confluent DIRK method with non-zero weights cannot exceedq+1. In particular the optimal order of aq stage non-confluent DIRK method with non-zero weights isq+1 for 1q5. DIRK methods of orders five and six in four and five stages respectively are constructed. It is further shown that the optimal order of a non-confluentq stage DIRK method with non-zero weights isq, forq6.     

Stability of implicit Runge-Kutta methods for nonlinear stiff differential equations

Under the assumption that an implicit Runge-Kutta method satisfies a certain stability estimate for linear systems with constant coefficientsl ₂-stability for nonlinear systems is proved. This assumption is weaker than algebraic stability since it is satisfied for many methods which are not evenA-stable. Some local smoothness in the right hand side of the differential equation is needed, but it may have a Jacobian and higher derivatives with large norms. The result is applied to a system derived from a strongly nonlinear parabolic equation by the method of lines.     

Stability andB-convergence of linearly implicit Runge-Kutta methods

Summary In this paper we study stability and convergence properties of linearly implicit Runge-Kutta methods applied to stiff semi-linear systems of differential equations. The stability analysis includes stability with respect to internal perturbations. All results presented in this paper are independent of the stiffness of the system.     

Monotonicity and boundedness in implicit Runge-Kutta methods

Summary This paper concerns the analysis of implicit Runge-Kutta methods for approximating the solutions to stiff initial value problems. The analysis includes the case of (nonlinear) systems of differential equations that are essentially more general than the classical test equationU=U (with a complex constant). The properties of monotonicity and boundedness of a method refer to specific moderate rates of growth of the approximations during the numerical calculations. This paper provides necessary conditions for these properties by using the important concept of algebraic stability (introduced by Burrage, Butcher and by Crouzeix). These properties will also be related to the concept of contractivity (B-stability) and to a weakened version of contractivity.     

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if ans-stage SRK contains Stratonovich integrals up to orderp then the strong order of the SRK cannot exceed min{(p+1)/2, (s−1)/2},p≥2,s≥3 or 1 ifp=1.     

On the D-suitability of implicit Runge-Kutta methods

The concept of suitability means that the nonlinear equations to be solved in an implicit Runga-Kutta method have a unique solution. In this paper, we introduce the concept of D-suitability and show that previous results become special cases of ours. In addition, we also give some examples to illustrate the D-suitability of a matrixA.     

A note on pseudo-symplectic Runge-Kutta methods

Aubry and Chartier introduced (1998) the concept of pseudo-symplecticness in order to construct explicit Runge-Kutta methods, which mimic symplectic ones. Of particular interest are methods of order (p, 2p), i.e., of orderp and pseudo-symplecticness order 2p, for which the growth of the global error remains linear. The aim of this note is to show that the lower bound for the minimal number of stages can be achieved forp=4 andp=5.     

On implicit Runge-Kutta methods with higher derivatives

Two families of implicit Runge-Kutta methods with higher derivatives are (re-)considered generalizing classical Runge-Kutta methods of Butcher type and f Ehle type. For generalized Butcher methods the characteristic functionG() is represented by means of the node polynomial directly, thereby showing that in methods of maximum order,G() is connected withs-orthogonal polynomials in exactly the same way as Padé approximations in the classical case.     

Relationships among some classes of implicit Runge-Kutta methods and their stability functions

In this paper we apply the theory for implicit Runge-Kutta methods presented by Stetter to a number of subclasses of methods that have recently been discussed in the literature. We first show how each of these classes can be expressed within this theoretical framework and from this we are able to establish a number of relationships among these classes. In addition to improving the current state of understanding of these methods, their expression within this theoretical framework makes it possible for us to obtain results giving general forms for their stability functions.This work was supported by the Natural Sciences and Engineering Research Council of Canada.     

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.     

Runge-Kutta methods for quadratic ordinary differential equations

Many systems of ordinary differential equations are quadratic: the derivative can be expressed as a quadratic function of the dependent variable. We demonstrate that this feature can be exploited in the numerical solution by Runge-Kutta methods, since the quadratic structure serves to decrease the number of order conditions. We discuss issues related to construction design and implementation and present a number of new methods of Runge-Kutta and Runge-Kutta-Nyström type that display superior behaviour when applied to quadratic ordinary differential equations.     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

On the relations between B2V Ms and Runge-Kutta collocation methods

The principal aim of this paper is a rigorous analysis of the relations between Block Boundary Value Methods (B₂V Ms) with minimal blocksize defined over a suitable nonuniform finer mesh and well-known Runge-Kutta collocation methods. Moreover, a further aspect that will be briefly investigated is the construction of an extended finer mesh for building B₂V Ms with nonminimal blocksize. Some advantages that may arise from the use of the so-obtained methods will be also discussed.     

Supplement: Efficient weak second order stochastic Runge-Kutta methods for non-commutative Stratonovich stochastic differential equations

This paper gives a modification of a class of stochastic Runge-Kutta methods proposed in a paper by Komori (2007). The slight modification can reduce the computational costs of the methods significantly.     

New embedded explicit pairs of exponentially fitted Runge-Kutta methods

Two new embedded pairs of exponentially fitted explicit Runge-Kutta methods with four and five stages for the numerical integration of initial value problems with oscillatory or periodic solutions are developed. In these methods, for a given fixed ω the coefficients of the formulae of the pair are selected so that they integrate exactly systems with solutions in the linear space generated by {sinh(ωt),cosh(ωt)}, the estimate of the local error behaves as O(h⁴) and the high-order formula has fourth-order accuracy when the stepsize h→0. These new pairs are compared with another one proposed by Franco [J.M. Franco, An embedded pair of exponentially fitted explicit Runge-Kutta methods, J. Comput. Appl. Math. 149 (2002) 407-414] on several problems to test the efficiency of the new methods.     

A criterion forP-stability properties of Runge-Kutta methods

P-stability is an analogous stability property toA-stability with respect to delay differential equations. It is defined by using a scalar test equation similar to the usual test equation ofA-stability. EveryP-stable method isA-stable, but anA-stable method is not necessarilyP-stable. We considerP-stability of Runge-Kutta (RK) methods and its variation which was originally introduced for multistep methods by Bickart, and derive a sufficient condition for an RK method to have the stability properties on the basis of an algebraic characterization ofA-stable RK methods recently obtained by Schere and Müller. By making use of the condition we clarify stability properties of some SIRK and SDIRK methods, which are easier to implement than fully implicit methods, applied to delay differential equations.     

On theA-stability of implicit Runge-Kutta processes

A new technique to calculate the characteristic functions and to examine theA-stability of implicit Runge-Kutta processes is presented. This technique is based on a direct algebraic approach and an application of theC-polynomial theory of Nørsett. New processes are suggested. These processes can be exponentially fitted in anA-stable manner.