Archive ouverte UNIGE | last documentshttps://archive-ouverte.unige.ch/Latest objects deposited in the Archive ouverte UNIGEengSymmetric multistep methods for constrained Hamiltonian systemshttps://archive-ouverte.unige.ch/unige:114859https://archive-ouverte.unige.ch/unige:114859A method of choice for the long-time integration of constrained Hamiltonian systems is the Rattle algorithm. It is symmetric, symplectic, and nearly preserves the Hamiltonian, but it is only of order two and thus not efficient for high accuracy requirements. In this article we prove that certain symmetric linear multistep methods have the same qualitative behavior and can achieve an arbitrarily high order with a computational cost comparable to that of the Rattle algorithm.Wed, 06 Mar 2019 15:37:07 +0100Convergence results for multistep Runge-Kutta methods on stiff mechanical systemshttps://archive-ouverte.unige.ch/unige:76258https://archive-ouverte.unige.ch/unige:76258Recently Ch. Lubich proved convergence results for Runge-Kutta meth-ods applied to stiff mechanical systems. The present paper discusses the new ideas necessary to extend these results to general linear methods, in particular BDF and multistep Runge-Kutta methods.Mon, 19 Oct 2015 12:22:56 +0200A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problemshttps://archive-ouverte.unige.ch/unige:41958https://archive-ouverte.unige.ch/unige:41958The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L^2 and the H^1 norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.Tue, 18 Nov 2014 14:28:06 +0100Multi-revolution composition methods for highly oscillatory differential equationshttps://archive-ouverte.unige.ch/unige:41957https://archive-ouverte.unige.ch/unige:41957We introduce a new class of multi-revolution composition methods (MRCM) for the approximation of the Nth-iterate of a given near-identity map. When applied to the numerical integration of highly oscillatory systems of differential equations, the technique benefits from the properties of standard composition methods: it is intrinsically geometric and well-suited for Hamiltonian or divergence-free equations for instance. We prove error estimates with error constants that are independent of the oscillatory frequency. Numerical experiments, in particular for the nonlinear Schr¨odinger equation, illustrate the theoretical results, as well as the efficiency and versatility of the methods.Tue, 18 Nov 2014 14:25:10 +0100Convergence results for multistep Runge-Kutta methods on stiff mechanical systemshttps://archive-ouverte.unige.ch/unige:12757https://archive-ouverte.unige.ch/unige:12757Recently Ch. Lubich proved convergence results for Runge-Kutta methods applied to stiff mechanical systems. The present paper discusses the new ideas necessary to extend these results to general linear methods, in particular BDF and multistep Runge-Kutta methods.Thu, 02 Dec 2010 14:46:05 +0100Order stars and stability for delay differential equationshttps://archive-ouverte.unige.ch/unige:12571https://archive-ouverte.unige.ch/unige:12571abstract not availableMon, 22 Nov 2010 14:28:38 +0100Backward error analysis for multistep methodshttps://archive-ouverte.unige.ch/unige:12568https://archive-ouverte.unige.ch/unige:12568abstract not availableMon, 22 Nov 2010 10:48:01 +0100Unconditionally stable explicit methods for parabolic equationshttps://archive-ouverte.unige.ch/unige:12542https://archive-ouverte.unige.ch/unige:12542This paper discusses rational 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 on n-dimensional linear equations) is given. A second order A_0-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.Fri, 19 Nov 2010 09:10:58 +0100Unconditionally stable methods for second order differential equationshttps://archive-ouverte.unige.ch/unige:12540https://archive-ouverte.unige.ch/unige:12540We use the concept of order stars (see [1]) to prove and generalize a recent result of Dahlquist [2] on unconditionally stable linear multistep methods for second order differential equations. Furthermore a result of Lambert-Watson [3] is generalized to the multistage case. Finally we present unconditionally stable Nyström methods of order 2s (s=1,2,...) and an unconditionally stable modification of Numerov's method. The starting point of this paper was a discussion with G. Wanner and S.P. Nørsett. The author is very grateful to them.Fri, 19 Nov 2010 09:08:23 +0100On the order of iterated defect correction. An algebraic proofhttps://archive-ouverte.unige.ch/unige:12537https://archive-ouverte.unige.ch/unige:12537In a recent article [2] Frank and Ueberhuber define and motivate the method of iterated defect correction for Runge-Kutta methods. They prove a theorem on the order of that method using the theory of asymptotic expansions. In this paper we give similar results using the theory of Butcher series (see [4]). Our proofs are purely algebraic. We don't restrict our considerations to Runge-Kutta methods, but we admit arbitrary linear one-step methods. At the same time we consider more general defect functions as in [2].Fri, 19 Nov 2010 09:04:18 +0100Méthodes de Nyström pour l'équation différentielle y''=f(x,y)https://archive-ouverte.unige.ch/unige:12535https://archive-ouverte.unige.ch/unige:12535In a recent paper (Hairer-Wanner [1]) we have given a theory with which it is easy to calculate the order conditions for Nyström methods. Here we show how it is possible to solve this system of non-linear algebraic equations. Moreover we present all methods of orders for s=2, 3, 4 (s-1 indicates the number of function evaluations per step); methods with one parameter of orders for s=5, 6 and some special methods of order s-1 for s=8, 9.Fri, 19 Nov 2010 08:57:03 +0100A theory for Nyström methodshttps://archive-ouverte.unige.ch/unige:12534https://archive-ouverte.unige.ch/unige:12534For the numerical solution of differential equations of the second order there are two possibilities: 1. To transform it into a system of the first order (of doubled dimension) and to integrate by a standard routine. 2. To apply a direct method as those invented by Nyström. The benefit of these direct methods is not generally accepted, a historical reason for them was surely the fact that at that time the theories did not consider systems, but single equations only. In any case the second approach is more general, since the class of methods defined in this paper contains the first approach as a special case. So there is more freedom for extending stability or accuracy. This paper begins with the development of a theory, which extends our theory for first order equations [1] to equations of the second order, and which is applicable to the study of possibly all numerical methods for problems of this type. As an application, we obtain Butcher-type results for Nyström-methods, we characterize numerical methods as applications of a certain set of trees, give formulas for a group-structure (expressing the composition of methods) etc. Recently in [2] the equations of conditions for Nyström methods have been tabulated up to order 7 (containing errors). Our approach yields not only the correct equations of conditions in a straight-forward way, but also an insight in the structure of methods that is useful for example in choosing good formulas.Fri, 19 Nov 2010 08:54:20 +0100Dense output for extrapolation methodshttps://archive-ouverte.unige.ch/unige:12476https://archive-ouverte.unige.ch/unige:12476This paper is concerned with dense output formulas for extrapolation methods for ordinary differential equations. In particular, the extrapolated explicit Euler method, the GBS method (for non-stiff equations) and the extrapolated linearly implicit Euler method (for stiff and differential-algebraic equations) are considered. Existence and uniqueness questions for dense output formulas are discussed and an algorithmic description for their construction is given. Several numerical experiments illustrate the theoretical results.Tue, 16 Nov 2010 09:13:39 +0100Equilibria of Runge-Kutta methodshttps://archive-ouverte.unige.ch/unige:12475https://archive-ouverte.unige.ch/unige:12475It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4.Tue, 16 Nov 2010 09:12:30 +0100Extrapolation at stiff differential equationshttps://archive-ouverte.unige.ch/unige:12472https://archive-ouverte.unige.ch/unige:12472Asymptotic expansions of the global error of numerical methods are well-understood, if the differential equation is non-stiff. This paper is concerned with such expansions for the implicit Euler method, the linearly implicit Euler method and the linearly implicit mid-point rule, when they are applied tostiff differential equations. In this case perturbation terms are present, whose dominant one is given explicitly. This permits us to better understand the behaviour of extrapolation methods at stiff differential equations. Numerical examples, supporting the theoretical results, are included.Tue, 16 Nov 2010 08:49:33 +0100One-step and extrapolation methods for differential-algebraic systemshttps://archive-ouverte.unige.ch/unige:12470https://archive-ouverte.unige.ch/unige:12470The paper analyzes one-step methods for differential-algebraic equations (DAE) in terms of convergence order. In view of extrapolation methods, certain perturbed asymptotic expansions are shown to hold. For the special DAE extrapolation solver based on the semi-implicit Euler discretization, the perturbed order pattern of the extrapolation tableau is derived in detail. The theoretical results lead to modifications of the known code. The efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion.Tue, 16 Nov 2010 08:43:16 +0100A- and B-stability for Runge-Kutta methods-characterizations and equivalencehttps://archive-ouverte.unige.ch/unige:12468https://archive-ouverte.unige.ch/unige:12468Using a special representation of Runge-Kutta methods (W-transformation), simple characterizations of A-stability and B-stability have been obtained in [9, 8, 7]. In this article we will make this representation and their conclusions more transparent by considering the "exact Runge-Kutta method". Finally we demonstrate by a numerical example that for difficult problems B-stable methods are superior to methods which are only A-stable.Mon, 15 Nov 2010 18:50:45 +0100Asymptotic expansions of the global error of fixed-stepsize methodshttps://archive-ouverte.unige.ch/unige:12464https://archive-ouverte.unige.ch/unige:12464In his fundamental paper on general fixed-stepsize methods, Skeel [6] studied convergence properties, but left the existence of asymptotic expansions as an open problem. In this paper we give a complete answer to this question. For the special cases of one-step and linear multistep methods our proof is shorter than the published ones. Asymptotic expansions are the theoretical base for extrapolation methods.Mon, 15 Nov 2010 17:30:24 +0100Constructive characterization of A-stable approximations to exp(z) and its connection with algebraically stable Runge-Kutta methodshttps://archive-ouverte.unige.ch/unige:12457https://archive-ouverte.unige.ch/unige:12457All rational approximations to exp(z) of order >=2m-beta (m denotes the maximal degree of nominator and denominator) are given by a closed formula involving beta real parameters. Using the theory of order stars [9], necessary and sufficient conditions for A-stability (respectively I-stability) are given. On the basis of this characterization relations between the concepts of A-stability and algebraic stability (for implicit Runge-Kutta methods) are investigated. In particular we can partly prove the conjecture that to any irreducible A-stable R(z) of oder >=0 there exist algebraically stable Runge-Kutta methods of the same order with R(z) as stability function.Mon, 15 Nov 2010 17:09:42 +0100Order conditions for numerical methods for partitioned ordinary differential equationshttps://archive-ouverte.unige.ch/unige:12454https://archive-ouverte.unige.ch/unige:12454Motivated by the consideration of Runge-Kutta formulas for partitioned systems, the theory of P-series is studied. This theory yields the general structure of the order conditions for numerical methods for partitioned systems, and in addition for Nyström methods for y'=f(y,y'), for Rosenbrock-type methods with inexact Jacobian (W-methods). It is a direct generalization of the theory of Butcher series [7, 8]. In a later publication, the theory of P-series will be used for the derivation of order conditions for Runge-Kutta-type methods for Volterra integral equations [1].Mon, 15 Nov 2010 17:04:27 +0100The life-span of backward error analysis for numerical integratorshttps://archive-ouverte.unige.ch/unige:12431https://archive-ouverte.unige.ch/unige:12431Backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations. The numerical solution is formally interpreted as the exact solution of a perturbed differential equation, given as a formal and usually divergent series in powers of the step size. For a rigorous analysis, this series has to be truncated. In this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation. Results on the long-time behaviour of numerical solutions are obtained in this way. We present applications to the numerical phase portrait near hyperbolic equilibrium points, to asymptotically stable periodic orbits and Hopf bifurcation, and to energy conservation and approximation of invariant tori in Hamiltonian systems.Fri, 12 Nov 2010 14:49:05 +0100Second order Chebyshev methods based on orthogonal polynomialshttps://archive-ouverte.unige.ch/unige:12357https://archive-ouverte.unige.ch/unige:12357Stabilized methods (also called Chebyshev methods) are explicit Runge-Kutta methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. The aim of this paper is to show that with the use of orthogonal polynomials, we can construct nearly optimal stability polynomials of second order with a three-term recurrence relation. These polynomials can be used to construct a new numerical method, which is implemented in a code called ROCK2. This new numerical method can be seen as a combination of van der Houwen-Sommeijer-type methods and Lebedev-type methods.Mon, 08 Nov 2010 13:53:37 +0100Global modified Hamiltonian for constrained symplectic integratorshttps://archive-ouverte.unige.ch/unige:12278https://archive-ouverte.unige.ch/unige:12278We 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.Mon, 01 Nov 2010 17:57:15 +0100Collocation methods for differential-algebraic equations of index 3https://archive-ouverte.unige.ch/unige:12143https://archive-ouverte.unige.ch/unige:12143This article gives sharp convergence results for stiffly accurate collocation methods as applied to differential-algebraic equations (DAE's) of index 3 in Hessenberg form, proving partially a conjecture of Hairer, Lubich, and Roche.Tue, 19 Oct 2010 10:39:33 +0200Symmetric multistep methods over long timeshttps://archive-ouverte.unige.ch/unige:12123https://archive-ouverte.unige.ch/unige:12123For computations of planetary motions with special linear multistep methods an excellent long-time behaviour is reported in the literature, without a theoretical explanation. Neither the total energy nor the angular momentum exhibit secular error terms. In this paper we completely explain this behaviour by studying the modified equation of these methods and by analyzing the remarkably stable propagation of parasitic solution components.Mon, 18 Oct 2010 09:26:27 +0200Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equationshttps://archive-ouverte.unige.ch/unige:5202https://archive-ouverte.unige.ch/unige:5202For classes of symplectic and symmetric time-stepping methods - trigonometric integrators and the Stšrmer-Verlet or leapfrog method - applied to spectral semi-discretizations of semilinear wave equations in a weakly nonlinear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.Tue, 16 Feb 2010 13:25:51 +0100