Runge-Kutta schemes that maintain invariant solutions for differential algebraic systems

Implicit Runge-Kutta (IRK) methods and projected IRK methods for the solution of semiexplicit index-2 systems of differential algebraic systems (DAEs) have been proposed by several authors. In this paper we prove that if a method satisfiesBA+A ^t B–bb ^t =0, it conserves quadratic invariants of DAEs.     

Geometric Properties of Runge-Kutta Discretizations for Nonautonomous Index 2 Differential Algebraic Systems

We analyze Runge-Kutta discretizations applied to nonautonomous index 2 differential algebraic equations (DAEs) in semi-explicit form. It is shown that for half-explicit and projected Runge-Kutta methods there is an attractive invariant manifold for the discrete system which is close to the invariant manifold of the DAE. The proof combines reduction techniques to autonomou index 2 differential algebraic equations with some invariant manifold results of Schropp [9]. The results support the favourable behavior of these Runge-Kutta methods applied to index 2 DAEs for t = 0.     

Local error estimates for moderately smooth problems: Part I – ODEs and DAEs

The paper consists of two parts. In the first part, we propose a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs). Based on the idea of defect correction we develop local error estimates for the case when the problem data is only moderately smooth. Numerical experiments illustrate the performance of the mesh adaptation based on the error estimation developed in this paper. In the second part of the paper, we will consider the estimation of local errors in context of stochastic differential equations with small noise. AMS subject classification (2000)  65L06, 65L80, 65L50, 65L05     

Numerical Treatment of Second Order Differential-Algebraic Systems

We consider the numerical treatment of systems of second order differential-algebraic equations (DAEs). The classical approach of transforming a second order system to first order by introducing new variables can lead to difficulties such as an increase in the index or the loss of structure. We show how we can compute an equivalent strangeness-free second order system using the derivative array approach and we present Runge-Kutta methods for the direct numerical solution of second order DAEs. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Approximate Runge-Kutta Computational Process

Implicit Runge-Kutta methods are efficient for solving stiff ODEs and DAEs. To bridge the gap between their theoretical analysis and practical implementation, we introduce the notion of the (, K)-approximate Runge- Kutta process to account for inevitable iteration errors. We prove iteration error bounds uniform with respect to stiffness, and investigate stage derivative reuse for methods having a first explicit stage. The latter technique may result in significant performance gains, also when such methods are used as error estimators. Previous computational heuristics can therefore be replaced by a consistent approach supported by theoretical analysis. The approximate but well-defined computational process is evaluated using approved test problems.     

Efficient Runge-Kutta integrators for index-2 differential algebraic equations

In seeking suitable Runge-Kutta methods for differential algebraic equations, we consider singly-implicit methods to which are appended diagonally-implicit stages. Methods of this type are either similar to those of Butcher and Cash or else allow for the importation of a final derivative from a previous step. For these two classes, with up to three additional diagonally-implicit stages, we derive methods that satisfy appropriate order conditions for index-2 DAEs.

     

Global error estimation for index-1 and -2 DAEs

In this paper, we consider the extension of three classical ODE estimation techniques (Richardson extrapolation, Zadunaisky's technique and solving for the correction) to DAEs. Their convergence analysis is carried out for semi-explicit index-1 DAEs solved by a wide set of Runge-Kutta methods. Experimentation of the estimation techniques with RADAU5 is also presented: their behaviour for index-1 and -2 problems, and for variable step size integration is investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Global modified Hamiltonian for constrained symplectic integrators

Summary. We prove that the numerical solution of partitioned Runge-Kutta methods applied to constrained Hamiltonian systems (e.g., the Rattle algorithm or the Lobatto IIIA–IIIB pair) is formally equal to the exact solution of a constrained Hamiltonian system with a globally defined modified Hamiltonian. This property is essential for a better understanding of their longtime behaviour. As an illustration, the equations of motion of an unsymmetric top are solved using a parameterization with Euler parameters. Mathematics Subject Classification (2000):65L06, 65L80, 65P10     

New Rosenbrock W-Methods of Order 3 for Partial Differential Algebraic Equations of Index 1

In this note new Rosenbrock methods for ODEs, DAEs, PDEs and PDAEs of index 1 are presented. These solvers are of order 3, have 3 or 4 internal stages, and fulfil certain order conditions to obtain a better convergence if inexact Jacobians and approximations of are used. A comparison with other Rosenbrock solvers shows the advantages of the new methods. AMS subject classification (2000) 34A09, 65L80     

Half-explicit Runge-Kutta methods for semi-explicit differential-algebraic equations of index 1

Summary For the numerical solution of non-stiff semi-explicit differentialalgebraic equations (DAEs) of index 1 half-explicit Runge-Kutta methods (HERK) are considered that combine an explicit Runge-Kutta method for the differential part with a simplified Newton method for the (approximate) solution of the algebraic part of the DAE. Two principles for the choice of the initial guesses and the number of Newton steps at each stage are given that allow to construct HERK of the same order as the underlying explicit Runge-Kutta method. Numerical tests illustrate the efficiency of these methods.     

Specialized Runge-Kutta methods for index differential-algebraic equations

We consider the numerical solution of systems of semi-explicit index differential-algebraic equations (DAEs) by methods based on Runge-Kutta (RK) coefficients. For nonstiffly accurate RK coefficients, such as Gauss and Radau IA coefficients, the standard application of implicit RK methods is generally not superconvergent. To reestablish superconvergence projected RK methods and partitioned RK methods have been proposed. In this paper we propose a simple alternative which does not require any extra projection step and does not use any additional internal stage. Moreover, symmetry of Gauss methods is preserved. The main idea is to replace the satisfaction of the constraints at the internal stages in the standard definition by enforcing specific linear combinations of the constraints at the numerical solution and at the internal stages to vanish. We call these methods specialized Runge-Kutta methods for index DAEs (SRK-DAE).

     

MULTISTEP DISCRETIZATION OF INDEX 3 DAES

1. IntroductionIn this paper, we will consider the multistep discrezations of the differential--algebraicequations (DAEs) in Hessenberg formwhere F e AN M M R", K E AN M L - AM, G E AN - RL, the initial value(yo, ic, no) at xo are assumed to be consistent, i.e., they satisfyWe supposes, F, G and K are sufficiently differentiable, and thatin a neighbourhood of the solution. Such problems often appear in the simulation ofmechanical systems with constraints and the singularly perturbed…     

Stability Analysis of Runge-Kutta Methods for Non-Linear Delay Differential Equations

This paper is concerned with the numerical solution of delay differential equations(DDEs). We focus on the stability behaviour of Runge-Kutta methods for nonlinear DDEs. The new concepts of GR(l)-stability, GAR(l)-stability and weak GAR(l)-stability are further introduced. We investigate these stability properties for (k, l)-algebraically stable Runge-Kutta methods with a piecewise constant or linear interpolation procedure.     

Higher order fully discrete scheme combined withH 1-Galerkin mixed finite element method for semilinear reaction-diffusion equations

We first apply a first order splitting to a semilinear reaction-diffusion equation and then discretize the resulting system by anH ¹-Galerkin mixed finite element method in space. This semidiscrete method yields a system of differential algebraic equations (DAEs) ofindex one. Apriori error estimates for semidiscrete scheme are derived for both differential as well as algebraic components. For fully discretization, an implicit Runge-Kutta (IRK) methods is applied to the temporal direction and the error estimates are discussed for both components. Finally, we conclude the paper with a numerical example.     

On the construction of stiffly accurate and B-stable Runge-Kutta methods

In the solution of stiff ODEs and especially DAEs it is desirable that the method used is stiffly accurate and B-stable. In this paper guidelines for the construction of Runge-Kutta methods with these properties are presented.     

Approximate compositions of a near identity map by multi-revolution Runge-Kutta methods

Summary. The so-called multi-revolution methods were introduced in celestial mechanics as an efficient tool for the long-term numerical integration of nearly periodic orbits of artificial satellites around the Earth. A multi-revolution method is an algorithm that approximates the map _T^N of N near-periods T in terms of the one near-period map _T evaluated at few s << N selected points. More generally, multi-revolution methods aim at approximating the composition ^N of a near identity map . In this paper we give a general presentation and analysis of multi-revolution Runge-Kutta (MRRK) methods similar to the one developed by Butcher for standard Runge-Kutta methods applied to ordinary differential equations. Order conditions, simplifying assumptions, and order estimates of MRRK methods are given. MRRK methods preserving constant Poisson/symplectic structures and reversibility properties are characterized. The construction of high order MRRK methods is described based on some families of orthogonal polynomials.Mathematics Subject Classification (1991): 65L05, 65L06This material is based upon work supported by the National Science Foundation Grant No. 9983708 and by the DGI Grant BFM2001–2562     

Applications of Rosenbrock-type methods to the p-version of finite elements

In quasistatic solid mechanics the spatial as well as the temporal domain need to be discetized. For the spatial discretization usually elements with linear shape functions are used even though it has been shown that generally the p-version of the finite elemente method yields more effective discretizations, see e.g. [1], [2]. For the temporal discretization diagonal-implicit, see e.g. [4], and especially linear-implicit Runge-Kutta schemes, see e.g. [5], [6], have for smooth problems proven to be superior to the frequently applied Backward-Euler scheme (BE). Thus an approach combining the p-version of the finite element method with linear-implicit Runge-Kutta schemes, so-called Rosenbrock-type methods, is presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs

A computationally efficient a posteriori error estimator is introduced and analyzed for collocation solutions to linear index-1 DAEs (differential-algebraic equations) with properly stated leading term exhibiting a singularity of the first kind. The procedure is based on a modified defect correction principle, extending an established technique from the context of ordinary differential equations to the differential-algebraic case. Using recent convergence results for stiffly accurate collocation methods, we prove that the resulting error estimate is asymptotically correct. Numerical examples demonstrate the performance of this approach. To keep the presentation reasonably self-contained, some arguments from the literature on DAEs concerning the decoupling of the problem and its discretization, which is essential for our analysis, are also briefly reviewed. The appendix contains a remark about the interrelation between collocation and implicit Runge-Kutta methods for the DAE case.     

Distributional solution theory for linear DAEs

A solution theory for switched linear differential–algebraic equations (DAEs) is developed. To allow for non–smooth coordinate transformation, the coefficients matrices may have distributional entries. Since also distributional solutions are considered it is necessary to define a suitable multiplication for distribution. This is achieved by restricting the space of distributions to the smaller space of piecewise–smooth distributions. Solution formulae for two special DAEs, distributional ordinary differential equations (ODEs) and pure distributional DAEs, are given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence Aspects of Step-Parallel Iteration of Runge-Kutta Methods for Delay Differential Equations

Implicit Runge-Kutta methods are known as highly accurate and stable methods for solving differential equations. However, the iteration technique used to solve implicit Runge-Kutta methods requires a lot of computational efforts. To lessen the computational effort, one can iterate simultaneously at a number of points along the t-axis. In this paper, we extend the PDIRK (Parallel Diagonal Iterated Runge-Kutta) methods to delay differential equations (DDEs). We give the region of convergence and analyze the speed of convergence in three parts for the P-stability region of the Runge-Kutta corrector. It is proved that PDIRK methods to DDEs are efficient, and the diagonal matrix D of the PDIRK methods for DDES can be selected in the same way as for ordinary differential equations (ODEs).