# Ernst HairerUniversity of Geneva | UNIGE · Section of Mathematics

Ernst Hairer

Dr. Drhc.

## About

269

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Introduction

Additional affiliations

February 2015 - March 2015

December 2009 - February 2010

February 2007 - July 2007

## Publications

Publications (269)

The orders of PDE-convergence in the Euclidean norm of s-stage AMF-W-methods for two-dimensional parabolic problems on rectangular domains are considered for the case of Dirichlet boundary conditions and an initial condition. The classical algebraic conditions for order p with p≤3 are shown to be sufficient for PDE-convergence of order p (independe...

The Boris algorithm, a closely related variational integrator and a newly proposed filtered variational integrator are studied when they are used to numerically integrate the equations of motion of a charged particle in a mildly non-uniform strong magnetic field, taking step sizes that are much larger than the period of the Larmor rotations. For th...

This article considers the numerical treatment of piecewise-smooth dynamical systems. Classical solutions as well as sliding modes up to codimension-2 are treated. An algorithm is presented that, in the case of non-uniqueness, selects a solution that is the formal limit solution of a regularized problem. The numerical solution of a regularized diff...

The study of convergence of time integrators, applied to linear discretized PDEs, relies on the power boundedness of the stability matrix R. The present work investigates power boundedness in the maximum norm for ADI-type integrators in arbitrary space dimension m. Examples are the Douglas scheme, the Craig–Sneyd scheme, and W-methods with a low st...

The Boris algorithm, a closely related variational integrator and a newly proposed filtered variational integrator are studied when they are used to numerically integrate the equations of motion of a charged particle in a non-uniform strong magnetic field, taking step sizes that are much larger than the period of the Larmor rotations. For the Boris...

The study of convergence of time integrators, applied to linear dis-cretized PDEs, relies on the power boundedness of the stability matrix R. The present work investigates power boundedness in the maximum norm for ADI-type integrators in arbitrary space dimension m. Examples are the Douglas scheme, the Craig-Sneyd scheme, and W-methods with a low s...

A modification of the standard Boris algorithm, called filtered Boris algorithm, is proposed for the numerical integration of the equations of motion of charged particles in a strong non-uniform magnetic field in the asymptotic scaling known as maximal ordering. With an appropriate choice of filters, second-order error bounds in the position and in...

The differential equations of motion of a charged particle in a strong non-uniform magnetic field have the magnetic moment as an adiabatic invariant. This quantity is nearly conserved over long time scales covering arbitrary negative powers of the small parameter, which is inversely proportional to the strength of the magnetic field. The numerical...

For the numerical solution of parabolic problems with a linear diffusion term, linearly implicit time integrators are considered. To reduce the cost on the linear algebra level, an alternating direction implicit approach is applied (so-called AMF-W-methods). The present work proves optimal bounds of the global error for two classes of 1-stage metho...

A modification of the standard Boris algorithm, called filtered Boris algorithm, is proposed for the numerical integration of the equations of motion of charged particles in a strong non-uniform magnetic field in the asymptotic scaling known as maximal ordering. With an appropriate choice of filters, second-order error bounds in the position and in...

The conjectures in the title deal with the zeros xj, j=1,2,…,n, of an orthogonal polynomial of degree n>1 relative to a nonnegative weight function w on an interval [a,b] and with the respective elementary Lagrange interpolation polynomials ℓk(n) of degree n−1 taking on the value 1 at the zero xk and the value 0 at all the other zeros xj. They invo...

For the numerical solution of parabolic problems with linear diffusion term, linearly implicit time integrators are considered. To reduce the cost on the linear algebra level an alternating direction implicit (ADI) approach is applied (so-called AMF-W-methods). The present work proves optimal bounds of the global error for two classes of 1-stage me...

The present work considers the numerical solution of differential equations that are obtained by space discretization (method of lines) of parabolic evolution equations. Main emphasis is put on the presence of mixed derivatives in the elliptic operator. An extension of the alternating-direction-implicit (ADI) approach to this situation is presented...

The Boris algorithm is a widely used numerical integrator for the motion of particles in a magnetic field. This article proves near-conservation of energy over very long times in the special cases where the magnetic field is constant or the electric potential is quadratic. When none of these assumptions is satisfied, it is illustrated by numerical...

The time integration of differential equations obtained by the space discretization via Finite Differences of evolution parabolic PDEs with mixed derivatives in the elliptic operator is considered (MOL approach). W-methods (Rosenbrock-type methods) are combined with the Approximate Matrix Factorization technique (AMF), which is applied in alternati...

The time integration of differential equations obtained by the space discretization via finite differences of evolution parabolic PDEs with mixed derivatives in the elliptic operator is considered (the method of lines approach). W-methods (Rosenbrock-type methods) are combined with the approximate matrix factorization (AMF) technique, which is appl...

Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical low-rank approximation of...

This talk discusses the construction of skew-symmetric differentiation matrices of high order. Such matrices play an important role in the space discretization of partial differential equations. Recent results of [1] are presented, which permit to obtain differentiation matrices up to order 6 that are banded, stable, and skew symmetric.

Piece-wise smooth differential equations (their regularization, numerical integration, and classification of solutions) is the topic of the present work. The behaviour close to one discontinuity surface and also the entering into the intersection of two discontinuity surfaces is well understood. Here, we study the solutions that exit a codimension-...

Differentiation matrices play an important role in the space discretization of first-order partial differential equations. This work considers grids on a finite interval and treats homogeneous Dirichlet boundary conditions. Differentiation matrices of orders up to $6$ are derived that are banded, stable and skew symmetric. To achieve these desirabl...

This talk discusses the concept of solutions of piecewise smooth, discontinuous differential equations. Recent results of [1] are presented, which give new insight into the solution close to the intersection of two discontinuity hyper-surfaces. The study is connected to space regularizations where, close to the discontinuities, the vector field is...

A semilinear wave equation with slowly varying wave speed is considered in one to three space dimensions on a bounded interval, a rectangle or a box, respectively. It is shown that the action, which is the harmonic energy divided by the wave speed and multiplied with the diameter of the spatial domain, is an adiabatic invariant: it remains nearly c...

For differential equations with discontinuous right-hand side and, in particular, for neutral delay equations it may happen that classical solutions do no exist beyond a certain time instant. In this situation, it is common to consider weak solutions of Utkin (Filippov) type. This article extends the concept of weak solutions and proposes a new reg...

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential
equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods.

The long-time behaviour of the Störmer–Verlet–leapfrog method is studied when this method is applied to highly oscillatory Hamiltonian systems with a slowly varying, solution-dependent high frequency. Using the technique of modulated Fourier expansions with state-dependent frequencies, which is newly developed here, the following results are proved...

The main concern of this paper is with the stable discretisation of linear partial differential equations of evolution with time-varying coefficients. We commence by demonstrating that an approximation of the first derivative by a skew-symmetric matrix is fundamental in ensuring stability for many differential equations of evolution. This motivates...

This talk is concerned with the long-time energy preservation of numerical integrators applied to Hamiltonian differential equations. A well established result (backward error analysis) tells us that symplectic integrators nearly preserve the total energy over very long time intervals provided that the step size is sufficiently small. For the situa...

Ordinary differential equations with discontinuous right-hand side, where the discontinuity of the vector field arises on smooth surfaces of the phase space, are the topic of this work. The main emphasis is the study of solutions close to the intersection of two discontinuity surfaces. There, the so-called hidden dynamics describe the smooth transi...

Much effort is put into the construction of general linear methods with the aim of achieving an excellent long-time behavior for the integration of Hamiltonian systems. In this article, a backward error analysis is presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. For careful...

Certain symmetric linear multistep methods have an excellent long-time behavior when applied to second order Hamiltonian systems with or without constraints. For high accuracy computations round-off can be the dominating source of errors. This article shows how symmetric multistep methods should be implemented such that round-off errors are minimiz...

For gradient systems in Euclidean space or on a Riemannian manifold the energy decreases monotonically along solutions. Algebraically stable Runge-Kutta methods are shown to also reduce the energy in each step under a mild step-size restriction. In particular, Radau IIA methods can combine energy monotonicity and damping in stiff gradient systems....

Long-time integrations are an important issue in the numerical solution of Hamiltonian systems. They are time consuming and it is natural to consider the use of parallel architectures for reasons of efficiency. In this context the parareal algorithm has been proposed by several authors. The present work is a theoretical study of the parareal algori...

For trigonometric and modified trigonometric integrators applied to
oscillatory Hamiltonian differential equations with one or several constant
high frequencies, near-conservation of the total and oscillatory energies are
shown over time scales that cover arbitrary negative powers of the step size.
This requires non-resonance conditions between the...

This article presents two regularization techniques for systems of state-dependent neutral delay differential equations which have a discontinuity in the derivative of the solution at the initial point. Such problems have a rich dynamics and besides classical solutions can have weak solutions in the sense of Utkin. Both of the presented techniques...

Long-time integration of Hamiltonian systems is an important issue in many applications – for example the planetary motion in astronomy or simulations in molecular dynamics. Symplectic and symmetric one-step methods are known to have favorable numerical features like near energy preservation over long times and at most linear error growth for nearl...

Geometric Numerical Integration is a subfield of the numerical treatment of differential equations. It deals with the design and analysis of algorithms that preserve the structure of the analytic flow. The present review discusses numerical integrators, which nearly preserve the energy of Hamiltonian systems over long times. Backward error analysis...

This note proves that the underlying one-step method of a G-symplectic general linear method is conjugate to a symplectic method. Parasitic solution components are not considered.

A 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 qual...

We consider multiscale Hamiltonian systems in which harmonic oscillators with
several high frequencies are coupled to a slow system. It is shown that the
oscillatory energy is nearly preserved over long times eps^{-N} for arbitrary
N>1, where eps^{-1} is the size of the smallest high frequency. The result is
uniform in the frequencies and does not...

For neutral delay differential equations the right-hand side can be multi-valued, when one or several delayed arguments cross a breaking point. This article studies a regularization via a singularly perturbed problem, which smooths the vector field and removes the discontinuities in the derivative of the solution. A low-dimensional dynamical system...

Ordinary differential equations arise everywhere in science – Newton’s law in physics, N-body problems in astronomy and in molecular dynamics, engineering problems in robotics, population models in biology, and many more. Since their analytic solution can be obtained only in exceptional situations, one is usually restricted to numerical simulations...

Section de mathématiques, Université deGe eve, 2-4 rue duLì evre, CH-1211Ge eve 4, Switzerland The long-time integration of Hamiltonian differential equations requires special numerical methods. Symplectic integrators are an excellent choice, but there are situations (e.g., multistep schemes or energy-preserving methods), where symplecticity is not...

For FPU chains with large particle numbers, the formation of a packet of modes with geometrically decaying harmonic energies from an ini-tially excited single low-frequency mode and the metastability of this packet over longer time scales are rigorously studied in this paper. The analysis uses mod-ulated Fourier expansions in time of solutions to t...

For the long-time integration of Hamiltonian differential equations the
use of symplectic methods is recommended. In practice it is often
sufficient to apply a method that is conjugate (up to a sufficiently
high order) to a symplectic integrator. This article gives a criterion
on the conjugate symplecticity of methods that can be represented as a
B...

We consider the problem of the long-time stability of plane waves under nonlinear perturbations of linear Klein-Gordon equations. This problem reduces to studying the distribution of the mode energies along solutions of one-dimensional semilinear Klein–Gordon equations with periodic boundary conditions when the initial data are small and concentrat...

Singularly perturbed delay differential equations arising from the regularization of state dependent neutral delay equations are considered. Much insight into the solution of the regularized problem and into its limit for ε → 0 + is obtained by the study of asymptotic expansions. Due to discontinuities in the derivative of the solution of the neutr...

Dieses Kapitel erläutert den Ursprung der grundlegendenFunktionen und den Einfluss von Descartes’ Géométrieauf ihre Berechnung. Das Interpolationspolynom führt zu Newtons binomischem Lehrsatz und zu den Reihenentwicklungen der Exponential-
und Logarithmusfunktion und der trigonometrischen Funktionen.

For Hamiltonian systems with non-canonical structure matrix a new class of numerical integrators is proposed. The methods
exactly preserve energy, are invariant with respect to linear transformations, and have arbitrarily high order. Those of optimal
order also preserve quadratic Casimir functions. The discussion of the order is based on an interpr...

Funktionen mehrerer Variablen haben ihren Ursprung in der Geometrie (z.B. in Form von Kurven, die von Parametern abhängen
(Leibniz 1694a)) und in der Physik. Ein im ganzen 17ten Jahrhundert berühmtes Problem bestand in der Berechnung der Bewegung
einer schwingenden Saite (d’Alembert 1748, Abb. 0.1). Der Ort einer Saite u(x, t) ist tatsäachlich sowo...

Auf Eulers Tod im Jahr 1783 folgte eine Periode des Stillstands in der Mathematik. Er hatte virtuell jedes Problem gelöst:
zwei unübertroffene Abhandlungen über die Analysis des Unendlichen und der Differentiale (Euler 1748, 1755), lösbare Integrale
gelöst, lösbare Differentialgleichungen gelöst (Euler 1768, 1755), 186 III. Grundlagen der klassisch...

Dieses Kapitel stellt die Differential- und Integralrechnung vor, die größte Errungenschaftder Mathematik überhaupt.Wir erklären
die Ideen von Leibniz, der Bernoullis und von Euler. Eine strenge Behandlung im Sinne des 19ten Jahrhunderts wird in den Abschnitten
III.5 und III.6 gegeben. Die Entwicklung dieser zwei Kalküle wirft auch Licht auf die of...

Diese Einfuhrung in die Analysis orientiert sich in ihrem Aufbau an der zeitlichen Entwicklung der Themen. Die ersten zwei Kapitel schlagen den Bogen von historischen Berechnungsmethoden praktischer Problemen hin zu unendlichen Reihen, Differential- und Integralrechnung und zu Differentialgleichungen. Das Etablieren einer mathematisch stringenten D...

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods.

A class of partitioned methods, combining collocation with averaged vector fields, is presented. The methods exactly preserve energy for general Hamiltonian systems, they are invariant with respect to linear transformations, and they can be of arbitrarily high order.

B-series are a fundamental tool in practical and theoretical aspects of numerical integrators for ordinary differential equations. A composition law for B-series permits an elegant derivation of order conditions, and a substitution law gives much insight into modified differential equations of backward error analysis. These two laws give rise to al...

http://www.scholarpedia.org/article/Linear_multistep_method

In Sect. V.6 we have seen a convergence result for one-leg methods (Theorem 6.10) applied to nonlinear problems satisfying a one-sided Lipschitz condition. An extension to linear multistep methods has been given in Theorem 6.11. A different and direct proof of this result will be the first goal of this section. Unfortunately, such a result is valid...

Extrapolation of explicit methods is an interesting approach to solving nonstiff differential equations (see Sect. II.9). Here we show to what extent the idea of extrapolation can also be used for stiff problems. We shall use the results of Sect. I1.8 for the existence of asymptotic expansions and apply them to the study of those implicit and linea...

We propose a modification of collocation methods extending the 'averaged vector field method' to high order. These new integrators exactly preserve energy for Hamiltonian systems, are of arbitrarily high order, and fall into the class of B-series integrators. We discuss their symmetry and conjugate-symplecticity, and we compare them to energy- pres...

For the long‐time integration of Hamiltonian systems (e.g., planetary motion, molecular dynamics simulation) much insight can be gained with “backward error analysis.” For example, it explains why symplectic integrators nearly conserve the energy, and why they have at most a linear error growth for integrable systems. This theory breaks down in the...

Ordinary differential equations are coupled with mixed constrained optimization problems when modeling the thermodynamic equilibrium of a system evolving with time. A particular application arises in the modeling of atmospheric particles. Discontinuity points are created by the activation/deactivation of inequality constraints. A numerical method f...

For general optimal control problems, Pontryagin's maximum principle gives necessary optimality conditions, which are in the form of a Hamiltonian differential equation. For its numerical integration, symplectic methods are a natural choice. This article investigates to which extent the excellent performance of symplectic integrators for long-time...

In long-time numerical integration of Hamiltonian systems, and especially in molecular dynamics simulation, it is important that the energy is well conserved. For symplectic integrators applied with sufficiently small step size, this is guaranteed by the existence of a modified Hamiltonian that is exactly conserved up to exponentially small terms....

The first book to approach high oscillation as a subject of its own, Highly Oscillatory Problems begins a new dialog and lays the groundwork for future research. It ensues from the six-month program held at the Newton Institute of Mathematical Sciences, which was the first time that different specialists in highly oscillatory research, from diverse...

For the numerical treatment of Hamiltonian differential equations, symplectic integra-tors are the most suitable choice, and methods that are conjugate to a symplectic integrator share the same good long-time behavior. This note characterizes linear multistep methods whose underlying one-step method is conjugate to a symplectic integrator. The boun...

The long-time behaviour of spectral semi-discretisations of weakly non-linear wave equations is analysed. It is shown that
the harmonic actions are approximately conserved for the semi-discretised system as well. This permits to prove that the energy
of the wave equation along the interpolated semi-discrete solution remains well conserved over long...

A modulated Fourier expansion in time is used to show long-time near- conservation of the harmonic actions associated with
spatial Fourier modes along the solutions of nonlinear wave equations with small initial data. The result implies the long-time
near-preservation of the Sobolev-type norm that specifies the smallness condition on the initial da...

Systems of implicit delay differential equations, including state-dependent problems, neutral and differential-algebraic equations,
singularly perturbed problems, and small or vanishing delays are considered. The numerical integration of such problems is
very sensitive to jump discontinuities in the solution or in its derivatives (so-called breakin...

For classes of symplectic and symmetric time-stepping methods— trigonometric integrators and the Störmer–Verlet or leapfrog
method—applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that
energy, momentum, and all harmonic actions are approximately preserved over long times. For the case...

In high accuracy long-time integration of differential equations, round-off errors may dominate truncation errors. This article
studies the influence of round-off on the conservation of first integrals such as the total energy in Hamiltonian systems.
For implicit Runge–Kutta methods, a standard implementation shows an unexpected propagation. We pro...

"For the last 20 years, one has tried to speed up numerical computa- tion mainly by providing ever faster computers. Today, as it appears that one is getting closer to the maximal speed of electronic com- ponents, emphasis is put on allowing operations to be performed in parallel. In the near future, much of numerical analysis will have to be recas...

Inspired by the theory of modified equations (backward error analysis),
a new approach to high-order, structure-preserving numerical integrators
for ordinary differential equations is developed. This approach is
illustrated with the implicit midpoint rule applied to the full dynamics
of the free rigid body. Special attention is paid to methods repr...

Motivated by the theory of modified differential equations (backward error analysis) an approach for the construction of high order numerical integrators that preserve geometric properties of the exact flow is developed. This summarises a talk presented in honour of Michel Crouzeix. Motivé par la théorie des équations modifiées (analyse rétrograde...