Ernst Hairer

Solving Ordinary Differential Equations II

Physics (nature?) provides a plethora of ordinary differential equations (relations involving a function of a single variable, its derivatives, and some external function of time, say) – Newton’s second law, the time-independent Schrödinger equation, and the equations governing the generation of spherically symmetric electromagnetic and gravitational fields, are a few examples. While there are a variety of techniques for solving specific problems, and an interesting class of techniques for characterizing the solutions to problems in the absence of an explicit form, it always surprises (slash embarrasses) me that a large number of physically relevant, interesting, and well-posed problems have little in the way of complete solution.

Geometric numerical integration illustrated by the Störmer–Verlet method

The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved long-time behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Störmer–Verlet method. It thus presents a cross-section of the recent monograph by the authors, enriched by some additional material. After an introduction to the Newton–Störmer–Verlet–leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent long-time behaviour of the method: long-time energy conservation, linear error growth and preservation of invariant tori in near-integrable systems, a discrete virial theorem, and preservation of adiabatic invariants.

Order stars and stability theorems

AbstractThis paper clears up to the following three conjectures:1.The conjecture of Ehle [1] on theA-acceptability of Padé approximations toez, which is true;2.The conjecture of Nørsett [5] on the zeros of the “E-polynomial”, which is false;3.The conjecture of Daniel and Moore [2] on the highest attainable order of certainA-stable multistep methods, which is true, generalizing the well-known Theorem of Dahlquist. We further give necessary as well as sufficient conditions forA-stable (acceptable) rational approximations, bounds for the highest order of “restricted” Padé approximations and prove the non-existence ofA-acceptable restricted Padé approximations of order greater than 6.The method of proof, just looking at “order stars” and counting their “fingers”, is very natural and geometric and never uses very complicated formulas.

On the Butcher group and general multi-value methods

This paper proves a theorem (“Theorem 6”) on the composition of, what we call, Butcher series. This Theorem is shown to be fundamental for the theory of Runge-Kutta methods: the formulas for the Taylor expansion of RK-methods and multiderivative RK-methods as well as formulas for the operation of the “Butcher group” (which describes the composition of RK-methods) are easy consequences. We do not attempt to realize the series as (generalized) Runge-Kutta methods, so we are not forced to restrict ourselves to the finite dimensional case. The theory extends to the multiderivative case as well, and the formulas remain valid for series which are not realizable as Runge-Kutta methods at all. Finally we extend the multi-value methods of J. Butcher [2] to the multiderivative case, which leads to a big class of integration methods for ordinary differential equations, including the methods of Nordsieck and Gear [3].The defintions and notations of [4] are used throughout this paper, many of the results are proved here again.ZusammenfassungEs wird ein Satz (“Theorem 6”) über die Zusammensetzung von “Butcherreihen” bewiesen. Dieser Satz, so zeigt sich, ist grundlegend für die Theorie der Runge-Kutta Methoden; die Formeln für die Taylorrrihen von RK-Methoden, auch von RK-Methoden mit mehrfachen Knoten, und Formeln für die Operation der Butchergruppe ergeben sich als leichte Folgerungen. Da wir von der Betrachtung der Reihen ausgehen und nicht von den zugehörigen Runge-Kutta Methoden, gelten die hergeleiteten Formeln auch für Reihen, die durch keine Runge-Kutta Methode realisierbar sind.Schließlich erweitern wir die Multi-Value Methoden von J. Butcher [2], welches zu einer weit größeren Klasse von Integrationsmethoden für gewöhnliche Differentialgleichungen führt. Diese enthalten z. B. die Methoden von Nordsieck und Gear [3].In diesem Bericht werden gewisse Definitionen und Bezeichnungen von [4] benützt und viele Ergebnisse neu bewiesen.

Stiff differential equations solved by Radau methods

Radau IIA methods are successful algorithms for the numerical solution of stiff differential equations. This article describes RADAU, a new implementation of these methods with a variable order strategy. The paper starts with a survey on the historical development of the methods and the discoveries of their theoretical properties. Numerical experiments illustrate the behaviour of the code.

Unconditionally stable methods for second order differential equations

We 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 Numerovs method. The starting point of this paper was a discussion with G. Wanner and S.P. Nørsett. The author is very grateful to them.

Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations

We consider second-order differential systems where high-frequency oscillations are generated by a linear part. We present a frequency expansion of the solution, and we discuss two invariants of the system that determine the coefficients of the frequency expansion. These invariants are related to the total energy and the oscillatory harmonic energy of the original system. For the numerical solution we study a class of symmetric methods that discretize the linear part without error. We are interested in the case where the product of the step size with the highest frequency can be large. In the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system. This allows us to prove the near-conservation of the total and the oscillatory energy over very long time intervals.

Fast Numerical Solution of Nonlinear Volterra Convolution Equations

Numerical methods for general Volterra integral equations of the second kind need

A theory for Nyström methods

O(n^2 )

Algebraically Stable and Implementable Runge-Kutta Methods of High Order

kernel evaluations and