Design and analysis of integrators for stiff and Hamiltonian problems

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Thesis

Reference

Design and analysis of integrators for stiff and Hamiltonian problems

ZBINDEN, Christophe

Abstract

This thesis consists of three parts. Part I: Theoretical study on conjugate symplecticity of B-series integrators. Algebraic criteria for conjugate symplecticity up to a certain order are presented in terms of the coefficients of the B-series. These criteria are then applied to characterize the conjugate symplecticity of implicit Runge–Kutta methods and of energy-preserving collocation methods. Part II: Partitioned Runge-Kutta-Chebyshev methods for diffusion-advection-reaction problems. We discuss an integration method based on Runge–Kutta–Chebyshev methods that is designed to treat moderately stiff and non-stiff terms separately. The method, called PRKC, is a one-step, explicit partitioned Runge–Kutta method of second-order with extended real stability interval. Part III: Characterization of Poisson integrators. Series expansions like B-series play a central role in the numerical analysis of ODEs. Part III introduces a new extension of B-series, called P-series, dedicated to integrators for a generalization of Hamiltonian systems, called Poisson systems.

ZBINDEN, Christophe. Design and analysis of integrators for stiff and Hamiltonian problems. Thèse de doctorat : Univ. Genève, 2013, no. Sc. 4615

URN : urn:nbn:ch:unige-323310

DOI : 10.13097/archive-ouverte/unige:32331

Available at:

http://archive-ouverte.unige.ch/unige:32331

Disclaimer: layout of this document may differ from the published version.

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Section de mathématiques Professeur Ernst HAIRER

Design and analysis of integrators for stiff and Hamiltonian problems

THÈSE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention mathématiques.

par

Christophe Jérôme ZBINDEN du

Grand-Saconnex (Genève)

Thèse n^o4615

Genève

Atelier d’impression ReproMail 2013

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Remerciements

En premier lieu, je souhaite adresser mes plus sincères remerciements à mon directeur de thèse Ernst Hairer. Les années passées à travailler sous sa direction ont été très enrichissantes, tant sur le plan scientifique que sur le plan humain. Je tiens, par conséquent, à lui exprimer ma plus profonde gratitude pour m’avoir choisi comme doctorant.

Ma reconnaissance va également à l’ensemble des membres de mon jury : Assyr Abdulle, Philippe Chartier et Martin Gander.

Je tiens aussi à remercier les membres du séminaire d’analyse numérique : Heiko Bernin- ger, Paola Console, Pierre-Henri Cocquet, Yves Courvoisier, Soheil Hajian, Bankim Chandra Mandal, Jérôme Michaud, Erwin Veneros, Hui Zhang, Gerhard Wanner, et les nombreux in- vités : David Cohen, Victorita Dolean, Stefan Güttel, Sébastien Loisel, Christian Lubich, Kevin Santugini, Gilles Vilmart.

Je souhaite exprimer ma gratitude à Felix Kwok pour sa générosité, sa bonne humeur et l’intérêt qu’il a manifesté à l’égard de ma recherche tout au long de ces années.

Mes remerciements vont également aux personnes que j’ai côtoyées au cours de ces an- nées, les membres et ex-membres du bureau 616 : Goulnara Arzhantseva, Loren Coquille, Paul-Henry Leemann, Stefan Sakalos, Kristin Shaw, et également les non-membres : Yves Barmaz, Pierre-Alain Cherix, Caterina Campagnolo, Caroline Giacobino, Justine Louis, Maté Lehel Juhasz, Mucyo Karemera, Xavier Morvan, Aglaia Myropolska, Sylvain Sardy, Sebas- tien Stevan, Shaula Fiorelli Vilmart, Guilherme Vorpe, Rimer Zurita, ainsi que l’ensemble du personnel administratif et des collaborateurs de la Section de Mathématiques.

J’adresse toute mon affection à ma famille : mes parents, mon frère, sa femme et mon neveu, et évidemment à mon épouse.

Merci enfin à tous les lecteurs de ce document.

Christophe J. Zbinden

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Résumé

Cette thèse est composée de trois principales parties :

Première partie : Étude théorique sur la conjugué-symplecticité des intégrateurs dévelop- pables en B-séries.

L’intégration à long terme d’équations différentielles Hamiltoniennes requiert des mé- thodes numériques particulières. Les intégrateurs symplectiques sont un excellent choix, mais il existe des situations (par exemple pour les schémas multi-pas ou pour les méthodes qui pré- servent l’énergie) où la symplecticité n’est pas possible. Il est alors intéressant d’étudier si les méthodes sont conjuguées-symplectiques, et donc ont le même comportement à long terme que les méthodes symplectiques.

Cette question est abordée dans le chapitre 1 (publié dans [25], à l’exception des annexes D et E) pour la classe des intégrateurs développables en B-séries. Des critères algébriques pour la conjugué-symplecticité jusqu’à un certain ordre sont présentés en termes de coefficients du développement en B-séries. L’effet des hypothèses simplificatrices (en anglais simplifying assumptions) est étudié. Ces critères sont ensuite appliqués pour caractériser la conjugué-symplecticité des méthodes de Runge-Kutta implicites (Lobatto IIIA et Lobatto IIIB) et des méthodes de collocation qui préservent l’énergie (en anglais energy-preserving collocation methods).

Deuxième partie : Méthodes de Runge–Kutta–Chebyshev partitionnées pour des problèmes de diffusion-advection-réaction.

Dans le chapitre 2 (publié dans [48], à l’exception des annexes A et B), une méthode d’in- tégration basée sur les méthodes de Runge–Kutta–Chebyshev (RKC) et conçue pour traiter les termes modérément raides et non raides séparément est présentée. La méthode, appelée Partitioned Runge–Kutta–Chebyshev (PRKC), est une méthode de Runge–Kutta partition- née d’ordre 2 à un pas. Elle appartient à la classe des méthodes stabilisées, c’est-à-dire des méthodes de Runge–Kutta explicites qui possèdent des intervalles de stabilité réelle étendus.

L’objectif de la méthode PRKC est de réduire le nombre d’évaluations de la fonction corres- pondant aux termes non raides et d’obtenir un intervalle de stabilité imaginaire à l’origine non réduit à zéro.

Troisième partie : Caractérisation des intégrateurs de Poisson.

Les développements en série jouent un rôle essentiel dans l’analyse numérique des équa- tions différentielles ordinaires. Parmi ces développements en série, le développement en B-

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séries est sans aucun doute le plus couramment utilisé pour étudier les méthodes de Runge–

Kutta (RK). Au fil du temps, diverses extensions du développement en B-séries ont été éla- borées en fonction de la famille des méthodes à analyser. Par exemple, le développement en P-séries pour les méthodes de RK partitionnées [18], le développement en NB-séries pour les méthodes de RK additives [27] et le développement en B_∞-séries pour la composition des méthodes de RK [35].

Le chapitre 3 introduit une nouvelle extension du développement en B-séries, appelé P-séries, consacré aux intégrateurs de Poisson (sorte de généralisation des systèmes Hamil- tonien où la matrice de structure est désormais non constante).

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Summary

This thesis consists of three main parts:

Part I: Theoretical study on conjugate symplecticity of B-series integrators.

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 possible. It is then of interest to study if the methods are conjugate symplectic and thus have the same long-time behaviour as symplectic methods.

This question is addressed in the Chapter 1 (published in [25], except for appendices D and E) for the class of B-series integrators. Algebraic criteria for conjugate symplecticity up to a certain order are presented in terms of the coefficients of the B-series. The effect of simplifying assumptions is investigated. These criteria are then applied to characterize the conjugate symplecticity of implicit Runge–Kutta methods (Lobatto IIIA and Lobatto IIIB) and of energy-preserving collocation methods.

Part II: Partitioned Runge-Kutta-Chebyshev methods for diffusion-advection-reaction prob- lems.

In the Chapter 2 (published in [48], except for appendices A and B), we discuss an integration method based on Runge–Kutta–Chebyshev (RKC) methods that is designed to treat moderately stiff and non-stiff terms separately. The method, called Partitioned Runge–

Kutta–Chebyshev (PRKC), is a one-step, partitioned Runge–Kutta method of second-order.

It belongs to the class of stabilized methods, viz. explicit Runge–Kutta methods possessing extended real stability intervals. The aim of the PRKC method is to reduce the number of function evaluations of the non-stiff terms and to get a non-zero imaginary stability boundary.

Part III: Characterization of Poisson integrators.

Series expansions play a central role in the numerical analysis of ordinary differential equations. Among them B-series [23] are the most commonly used to study Runge–Kutta (RK) methods. Over time, various extensions of B-series have been developed depending on the family of methods that need be analyzed, such as P-series for partitioned RK methods [18], NB-series for additive RK methods [27] and B_∞-series for composition of RK methods [35].

Chapter 3 introduces a new extension of B-series, calledP-series, dedicated to integrators for a generalization of Hamiltonian systems, called Poisson systems, where the structure

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matrix is replaced with a non-constant matrix.

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Chapter 0 Overview

This thesis is devoted to the design and analysis of integrators for ordinary differential equations.

Ordinary differential equations (ODEs) arise in many applications in various disciplines such as physics (classical mechanics), chemistry (chemical reaction rates), biology (popula- tion dynamics), engineering (simulation of electrical circuits), meteorology (weather mod- eling), economics (interest rates), etc. In fact ODEs appear in almost all models involving quantities which are defined as the rate of change of other quantities (derivatives).

In general, finding the solutions of an ODE is complicated and requires sophisticated methods. This difficulty stems in part from the fact that it is rarely possible to express the solutions of an ODE by elementary functions. For this reason, one uses numerical methods to generate approximate solutions.

The Runge–Kutta (RK) methods are an important family of numerical methods for the approximation of solutions of ODEs. Historically, the principal goal when constructing explicit or implicit RK methods was to achieve the highest order possible with a given number of stages. However, depending on the type of differential equations, taking the most accurate numerical method may not be the wisest choice. Indeed, on the one hand, an accurate method can be really costly in terms of CPU time; that can be problematic when the system to be solved is very large. On the other hand, the accuracy of a numerical method is based on the analysis of local truncation errors, which do not take into account the geometric properties of the flow of a differential equation.

Chapters 1 and 3 deal with some aspects of geometric numerical integration, in particular conjugate symplecticity and Poisson integrators, while Chapter 2 focuses on the construction of methods that are very attractive for solving large systems arising from spatial discretization of parabolic partial differential equations (PDEs).

0.1 Chapter 1

Consider a Hamiltonian system, viz. a system of differential equations

˙

y=J⁻¹∇H(y), J=

0 I

−I 0

,

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where H :R^2d →Ris sufficiently differentiable. It is well known that the exact flowϕt(y)of a Hamiltonian system has the following properties:

• energy preservation : H ϕt(y)

=Const,

• symplecticity of the flow: ϕ_t^′(y)^TJϕ_t^′(y) =J, and

• volume preservation : detϕt^′(y) =1.

The first question that comes to mind is “Is it possible for an approximate solution obtained from a numerical method to also satisfy all of these properties ?” The answer is no, a numerical method cannot satisfy energy preservation and symplecticity at the same time ([17], see also [13]). However, a symplectic method is formally conjugate to a method that preserves the Hamiltonian exactly [13]. This result was one of the motivations that led us to study the conjugate symplecticity of numerical integrators. Indeed, given an energy-preserving integrator, the question is “How close can it be to a symplectic integrator ?” The study of the conjugate symplecticity of the method gives an answer to this question.

The second motivation for this choice of subject comes from the long-time integration of Hamiltonian systems. The discrete flow y₁=Φh(y0)of a numerical method of order p applied to Hamiltonian systems with constant step-size can be

Non-symplectic :

H(yn)−H(y0) =O(th^p) for t=nh.

There is a linear drift of the numerical Hamiltonian.

Symplectic : Φ^′_h(yn)^T

JΦ^′_h(yn) =J

H(yn)−H(y₀) =O(h^p).

No drift is caused by the numerical method. In fact, symplectic methods preserve exactly a modified Hamiltonian that isO(h^p)close to the original Hamiltonian.

Conjugate symplectic up to order p+r with r≥0 : There exists z= χ(y) =y+O(h^p) such thatΨh=χh◦Φh◦(χh)⁻¹satisfiesΨ^′_h(z)^TJΨ^′_h(z) =J+O(h^p+r+1).

H(yn) =H(y0) +O(h^p) +O(th^p+r) for t=nh.

There is no drift on intervals of lengthO(h⁻^r).

This means that non-symplectic methods which are conjugate symplectic have the same long- time behaviour as symplectic methods.

A classical example is given by the implicit midpoint rule and the trapezoidal rule [40].

These two methods are mutually conjugate, but the implicit midpoint rule is a symplectic method and the trapezoidal rule is not. However they both share excellent long-time behaviour.

As the trapezoidal rule is equivalent to the Lobatto IIIA method with 2 stages and the midpoint rule to the Lobatto IIIB method with 2 stages, it is interesting to study the conjugate symplecticity of Lobatto IIIA and IIIB methods. The results obtained in [31] imply that

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Lobatto IIIA

Lobatto IIIB Gauss

ErrorinHamiltonian

Time

×10⁻⁴

2 1 0

0

−1

−2

−3

−4

−5

50 100 150 200 250 300 350 400 450 500

Figure 0.1: Error in numerical Hamiltonian of three fourth-order methods (2-stage Gauss method, 3- stage Lobatto IIIA and Lobatto IIIB methods) for the perturbed pendulum H(p,q) = ¹₂p²−cos(q) +

1

5sin(2q)with step size h=0.2 and initial values p(0) =2.5, q(0) =0.

• Lobatto IIIA method with 3 stages of order 4 is conjugate symplectic up to at least order 5,

• and Lobatto IIIB method with 3 stages of order 4 is conjugate symplectic up to order 4, but not higher.

These results raise the following questions: “Is Lobatto IIIA method with 3 stages conjugate symplectic up to 6, or to a higher order ?” “What happens when the number of stages gets larger ?”

The techniques developed in Chapter 1 based on backward error analysis and some well- known simplifying assumptions enable us to answer these open questions. We proved that for s≥3,

• Lobatto IIIA method with s stages of order 2s−2 is conjugate symplectic up to order 2s, but it is not conjugate symplectic up to a higher order.

• Lobatto IIIB method with s stages of order 2s−2 is conjugate symplectic up to order 2s−2, but not higher.

Fig. 0.1 compares the errors in the numerical Hamiltonian of three different fourth-order methods, namely the 2-stage Gauss method (symplectic), and the 3-stage Lobatto IIIB (non- symplectic) and Lobatto IIIA (conjugate symplectic up to order 6) methods. As Lobatto IIIB is a symmetric method, we chose a perturbed pendulum problem [16] instead of the classical pendulum (which is an integrable reversible system) in order to show the energy drift of size O(th⁴)for Lobatto IIIB. Lobatto IIIA seems to have the same behaviour as the Gauss method on Fig. 0.1, because its error isO(h⁴) +O(th⁶)and the linear drift is not visible on this scale.

We also proved that the energy-preserving collocation methods [20] of maximal order 2s, which as their names suggest belong to the class of energy-preserving integrators, are only conjugate symplectic up to order 2s+2.

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s-stage first-order RKC methods s steps of explicit Euler method

s=2 s=2

s=3 s=3

0

0 0

0

0 0

0

−2

2 2

−5

−10

−15

Figure 0.2: The scaled stability domain of first-order s-stage Runge–Kutta–Chebyshev methods with damping and s steps of explicit Euler method for s=2,3.

We also mention that the results obtained in Chapter 1 have recently been used in [15]

to prove that the underlying one-step method of a G-symplectic general linear method is conjugate to a symplectic method.

0.2 Chapter 2

In Chapter 2, we deal with moderately stiff initial value problems arising from the spatial discretization of parabolic partial differential equations (PDEs). Roughly speaking, stiff problems are problems where certain implicit methods perform better than explicit ones.

This occurs, for example, when different components of the system evolve on different time scales (e.g. chemical reactions systems or parabolic PDEs converted by the method of lines (MOL) to a system of ordinary differential equations (ODEs) ).

The use of implicit methods makes stiff problems much harder to solve numerically than non-stiff ones, because each integration step of an implicit method requires the solution of one or more linear or nonlinear algebraic systems. Although explicit methods are generally not advocated for solving stiff problems due to their severe step size restrictions, for many moderately stiff problems of large dimension and with eigenvalues known to lie in a long narrow strip along the negative real axis, a special class of explicit methods called stabilized Runge–Kutta methods is especially efficient.

Stabilized RK methods are explicit methods, usually of low order, with an extended stability domain along the negative real axis [42, 4, 1, 33]. Their real stability interval has a length proportional to the square of the number of stages. This remarkable feature, derived from properties of Chebyshev polynomials, makes stabilized RK methods particularly suitable for the time integration of systems of ODEs

w^′(t) =F t,w(t)

, w(0) =w₀,

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resulting from the spatial discretization (MOL) of parabolic PDEs in multiple space dimensions. The Jacobian matrix∂F(t,w)/∂w of this kind of systems typically has all its eigenval- ues near the negative real axis, and usually has a spectral radius proportional to h⁻², where h denotes the spatial mesh width. Therefore the quadratic dependence of the real stability interval compensates for the quadratic growth of the spectral radius, and thus makes it possible by adding stages (implying a reduction of the step size restriction) to maintain a reasonable workload per step. Fig. 0.2 illustrates the difference between stabilized RK methods and standard explicit methods, for which the stability domain would only grow linearly with the number of function evaluations. Note that the stability domains have been scaled in order to have comparable numerical work for the same advance in step size (see [24]).

Current research on stabilized RK methods [39, 45] focuses on the time integration of advection-diffusion-reaction (ADR) problems, sometimes also with additional stochastic terms [5]. For example, a three-dimensional ADR problem is given by

u_t+∇·(a u) =∇·(D∇u) +r(u), where the scalar function u depends on time and space, and

∇=



∂x

∂y

∂z



, a=



a₁ a₂ a3



, D=



d₁ d₂

d3



, ∇u=



∂xu

∂yu

∂zu



.

The semi-discrete system arising from space discretization of the problem is first written in an additive fashion by combining or splitting diffusion, advection and reaction terms into one, two or more terms. For instance, the problem could be split into three parts as follows

w^′(t) =F t,w(t)

=F_D t,w(t)

+F_A t,w(t)

+F_R t,w(t) ,

where F_D, F_A and F_R represent diffusion, advection and reaction terms respectively. In N space dimensions, the size of the semi-discrete system is O(h⁻^N), with h representing the spatial mesh width. The eigenvalues of the Jacobian of F_D are typically on the negative real axis, and those of the Jacobian of F_A on the imaginary axis. The spectral radius of

∂^FD(t,w)/∂w is generally proportional to h⁻², while that of ∂^FA(t,w)/∂^{w is to h}⁻¹^{. Re-} garding the reaction terms, the eigenvalues of∂^FR(t,w)/∂w are usually not related to h, but could have a very large stiffness ratio maxj

ℜλj

/minj

ℑλj

.

The difficulty now lies in the design of a single algorithm capable of integrating each term in the most suitable way. An idea that has been developed over the years is to couple stabilized RK methods with other explicit or implicit RK methods. Nowadays there are several implementations of such numerical integrators that couple many individual methods;

some of these implementations, such as IRKC [36] and PIROCK[5], are available on the Internet.

In this spirit, Chapter 2 discusses a one-step, stabilized RK method of order two based on the Runge–Kutta–Chebyshev (RKC) method [42]. The method, called Partitioned Runge–

Kutta–Chebyshev (PRKC), treats moderately stiff and non-stiff terms separately and explic- itly. PRKC method is designed to reduce the number of function evaluations of the non-stiff terms and to improve imaginary stability close to the origin. Numerical examples obtained with our FORTRAN95 codePRKCare also presented.

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0.3 Chapter 3

A Poisson system is a system of differential equations of the form

˙

y=Λ(y)∇H(y),

where H :Rⁿ→Ris sufficiently differentiable andΛ(y)is a skew-symmetric matrix which satisfies the Jacobi identity

∑

n l=1

∂Λi j(y)

∂^yl

Λlk(y) +∂Λjk(y)

∂^yl

Λli(y) +∂Λki(y)

∂^yl

Λl j(y)

=0 for all i,j,k.

An important motivation for studying Poisson systems is given by the following example (see for instance [21]). For a constrained mechanical system, the equations of motion are given by

˙

q=∇pH(p,q)

˙

p=−∇qH(p,q)−G(q)^Tλ 0=g(q),

where G(q) =g^′(q). Assuming G(q)∇²_ppH(p,q)G(q)^T invertible, it is possible to express λ in terms of canonical coordinates (p,q), and then to formulate this differential-algebraic equation as a differential equation

˙ x=J⁻¹

∇H(x) +

∑

^m

i=1

λi(x)∇gi(x)

on the manifoldM =

x∈R^2d ; c(x) =0 with c(x) = g(q),G(q)∇pH(p,q)^T

and x= (p,q)^T. Using local coordinates y ∈R^2(d−m) of the manifold M via the transformation x=χ(y), the Hamiltonian system on the manifold becomes

X(y)y˙=J⁻¹

∇H χ^(y)⁺

∑

^m

i=1

λi χ^(y)∇gi χ^(y)

where X(y) =χ^′(y). Multiplying this equation from the left with X(y)^TJ yields X(y)^TJ X(y)y˙=X(y)^T∇H χ(y)

.

Note that the columns of X(y), which are tangent vectors, are orthogonal to the gradients

∇g_iof the constraints. By the assumption that allowed us to expressλ in terms of(p,q), the matrix X(y)^TJ X(y)is invertible. With Λ(y) = X(y)^TJ X(y)₋1

and H(y) =H χ(y) , the last differential equation gives rise to a Poisson system.

As for Hamiltonian systems, the flow ϕt(y) of a Poisson system has many important geometric properties:

• energy preservation: H ϕt(y)

=Const,

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• Poisson map: ϕt^′(y)Λ(y) ϕt^′(y)^T

=Λ ϕt(y)

, which is closely related to symplecticity,

• and conservation of Casimirs ofΛ(y): Ci ϕt(y)

=Const where the function C_iis such that∇C(y)^TΛ(y) =0 for all y.

A numerical integrator that satisfies the last two properties is called a Poisson integrator.

Although Poisson systems are close to Hamiltonian systems, the use of symplectic integrators does not guarantee the preservation of the Poisson structure. For instance, the symplectic Euler method is a Poisson integrator when applied to the Lotka–Volterra problem, but the implicit midpoint rule, which is a symplectic method, turns out not to be a Poisson map [21, VII.4.2]. Also note that unlike a symplectic integrator, a Poisson integrator is related to a class of matrixΛ(y)and it will never be a Poisson integrator for all possibleΛ(y).

These observations led us to develop new tools, called P-series, for studying Poisson integrators. These tools, which are introduced in Chapter 3, are an extension of B-series and they allow us to exploit the properties of the matrixΛ(y). Our first results on the algebraic characterization of Poisson B-series integrators are given at the end of the Chapter.

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Chapter 1 On conjugate symplecticity of B-series integrators

Pages 13–34 of this chapter are identical to the publication

[25] E. Hairer and C. J. Zbinden. On conjugate symplecticity of B-series integrators. IMA J Numer Anal, 33(1):57–79, 2013.

1.1 Introduction

Consider a Hamiltonian differential equation

˙

y=J⁻¹∇H(y), J=

0 I

−I 0

, (1.1)

where J is the canonical structure matrix and H :R^2d →Ris sufficiently differentiable (d is the number of degrees of freedom). The function H(y)is called the Hamiltonian or energy of the system. A classical result by Poincaré tells us that the exact flow, denoted by ϕt(y), is for every t a symplectic transformation. This means that the derivative with respect to the initial value satisfies

ϕ_t^′(y)^TJϕ_t^′(y) =J. (1.2) For problems with one degree of freedom this property is equivalent to area preservation, and it implies volume preservation of the flow in the general case. Another property of Hamiltonian systems is energy preservation, which means that H(y(t)) is constant along solutions of (1.1).

We are interested in the numerical treatment of Hamiltonian systems. In the spirit of geometric numerical integration, the ideal situation would be to have a numerical integrator y_n+1=Φh(yn)for which the discrete flow mappingΦh(y)is symplectic, and which exactly preserves the energy. Unfortunately this is not possible ([17], see also [13]). One is therefore constrained to consider methods satisfying one of these properties and to study how well the other is verified.

An important tool for studying the long-time behaviour of numerical methods is backward error analysis (see [21]). It tells us that the discrete flow of a numerical integratorΦh(y),

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when applied to ˙y= f(y), can be (formally) interpreted as the exact flow of a modified differential equation, whose vector field is given as a series in powers of the step size h:

˙

y= f(y) +h f₁(y) +h²f₂(y) +h³f₃(y) +··· .

If the method is of order p we have fj(y) =0 for 1≤ j<p, so that the perturbation is of size O(h^p).

For symplectic methods applied to (1.1) the modified differential equation is Hamiltonian,

˙

y=J⁻¹∇H_h(y) with H_h(y) =H(y) +hH₁(y) +h²H₂(y) +h³H₃(y) +··· , with functions H_j(y)that are globally defined for B-series integrators. This shows that the discrete flow of symplectic methods has the same qualitative behaviour as the exact flow.

Formally, it exactly conserves the modified Hamiltonian H_h(y), so that the energy H(y) is nearly conserved with an error bounded by O(h^p) (without any drift). Moreover it can be shown (see e.g., [21]) that symplectic methods exactly preserve quadratic first integrals of the system and, in the case of nearly integrable systems, they nearly conserve all action variables and have at most a linear error growth in the angle variables.

In the present article we are interested in methods that are not necessarily symplectic, but nevertheless have an excellent long-time behaviour. We call a numerical method of order p conjugate symplectic up to order p+r (with r≥0), if there exists a change of coordinates z=χ^(y)^{that is}^O^(h^p⁾close to the identity, such thatΨh=χ◦Φh◦χ⁻¹^satisfies

Ψ^′_h(z)^TJΨ^′_h(z) =J+O(h^p+r+1). (1.3) The method Ψh has the same order as Φh, and the coefficient functions f_j(z) of the corre- sponding modified differential equation are Hamiltonian for j< p+r. Consequently, the error in the energy H(zn)is bounded by O(h^p) +O(th^p+r), so that no drift can be seen on intervals of lengthO(h⁻^r). The same is true for the near preservation of quadratic first integrals and for the action variables in nearly integrable Hamiltonian systems. Since for a method that is conjugate symplectic up to order p+r we have y_n−z_n=O(h^p), the same statements remain true for the numerical approximation{y_n}.

In Section 1.2 we start with recalling the definition of B-series, we present the composition law, and we give explicit formulas for the B-series representing the modified equation.

We also recall algebraic conditions on the coefficients of a B-series that guarantee its symplecticity, and we discuss the B-series that is obtained after conjugation. Section 1.3 is then devoted to criteria for conjugate symplecticity in terms of the coefficients of the modified differential equation. A recurrence relation counting the number of necessary conditions is given. Analogous criteria in terms of the coefficients of the B-series integrator are then proved in Section 1.4. For high order the number of order conditions is very high, and they can be handled only with the use of simplifying assumptions. In Section 1.5 we recall a coordinate-free definition of simplifying assumptions C(η) and D(ζ), and we discuss the simplification of the algebraic criteria for conjugate symplecticity under these simplifying assumptions. Applications of the criteria are the subject of the final Section 1.6. We discuss the conjugate symplecticity of Lobatto IIIA and Lobatto IIIB Runge–Kutta methods, and we prove that the energy-preserving collocation methods of maximal order 2s are conjugate symplectic up to order 2s+2, but not up to a higher order.

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1.2 B-series theory

Based on the seminal publication of [7], the concept of B-series was introduced in the article of [23]. It is motivated by the fact that the exact solution of ˙y= f(y)as well as the numerical solution of nearly all integrators can be written as B-series. For a modern treatment of the theory of B-series we refer to [34], the monograph of [21] and to the recent article by [14].

In the following we collect the definitions and the results that will be needed in this work.

Let

T =

, , , , , , , , . . .

be the set of rooted trees. If it is convenient to consider also the empty tree, we write T₀= T∪ {/0}. We use the notationτ^{= [}τ1, . . . ,τm]for the tree that is obtained by grafting the roots

ofτ1, . . . ,τm ∈T to a new vertex which becomes the root of τ. We denote the number of

vertices by|τ_|and call it the order ofτ. The symmetry coefficient is defined recursively by σ( ) =1, σ(τ) =σ(τ1)···σ(τm)µ1!µ2!··· , (1.4) where the integersµ1,µ2, . . .count equal trees amongτ1, . . . ,τm. For a differential equation

˙

y= f(y), the corresponding elementary differentials F(τ)are given by

F( )(y) = f(y), F(τ)(y) = f^(m)(y) F(τ1)(y), . . . ,F(τm)(y) .

For given real coefficients a(/0)and a(τ),τ _∈T , a B-series is a formal series of the form B(a,y) =a(/0)y+

∑

τ∈T

h^|τ|

σ(τ)a(τ)F(τ)(y). (1.5) B-series integrators. A discrete flowΦh(y), whose (formal) Taylor series is of the form (1.5) with a(/0) =1 is called a B-series integrator,Φh(y) =B(a,y). It is consistent with ˙y= f(y) if in addition a( ) =1. This is a wide class of numerical methods that comprises all Runge–

Kutta methods, the underlying one-step method of multistep methods, the averaged vector field integrator, energy-preserving collocation methods, and many more.

The exact time-h flow of ˙y= f(y) can be interpreted as a B-series integrator ϕh(y) = B(e,y)with coefficients

e(/0) =e( ) =1, e(τ) = 1

|τ_|^e(τ1)···e(τm) for τ = [τ1, . . . ,τm]. (1.6) A B-series integrator is of order p, if its Taylor series matches that of the exact solution up to an error of sizeO(h^p+1). Algebraically, this can be expressed as a(τ) =e(τ)for all trees with|τ_{| ≤}^p.

Composition law. Let B(c,y) be a B-series with c(/0) =1, so that it is close to the identity mapping. The expressions F(τ)(B(c,y))can then be expanded into a Taylor series around y, and it turns out (see for example [21, p. 62]) that the composition of B-series satisfies

B b,B(c,y)

=B(c b,y) with (c b)(τ) =

∑

θ∈OST(τ)

b(θ)c(τ_\θ). (1.7)

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Here, OST(τ)denotes the set of ordered subtrees ofτ. The empty tree /0 andτ^{are in OST}(τ), as well as treesθ that are formed by a connected subset of vertices ofτ containing its root.

All vertices ofτ are considered different, so that for example the tree appears twice in the set OST( ). The difference setτ_\θ consists of those trees that remain whenθ and its adjacent branches are removed fromτ, and the expression c(τ_\θ)is defined as the product c(τ_\θ) =∏_δ∈τ\θc(δ).

The set G={a : T₀→R; a(/0) =1}provided with the above composition law is called the Butcher group. Let us also mention that (1.7) defines a co-product which makes the algebra of polynomials with the rooted trees as commuting indeterminates to a Hopf algebra.

B-series vector fields and modified differential equation. A B-series B(α,y)with coefficients satisfyingα(/0) =0 is of the form

B(α,y) =hα( )f(y) +h²α( )f^′(y)f(y) +···

and can be interpreted as a vector field. The modified differential equation of a B-series integrator (in the sense of backward error analysis) is such a vector field. To get a relation between the coefficients a(τ)of the method and the coefficients α(τ) of the vector field it is convenient to work with the Lie derivative∂αc as discussed in [21, p. 370]. If y(t) is a solution of the differential equation h ˙y(t) =B(α^,y(t)), then we have

hd

dtB c,y(t)

=B ∂_αc,y(t)

with (∂_αc)(τ) =

∑

θ∈SP(τ)

c(θ)α(τ_\θ), (1.8) for|τ_{| ≥}^{1, and}(∂αc)(/0) =0. Here, SP(τ) ={θ_∈^OST(τ);τ_\θ consists of only one element} denotes the set of splittings of the tree τ. Higher derivatives can be expressed in terms of iterated application of the Lie derivative. It then follows from Taylor series expansion that h ˙y(t) =B(α^,^y(t))is the modified differential equation of the B-series integrator y_n+1= B(a,yn)if and only if

a(τ) =

∑

|τ|

j=1

1

j! ∂_α^j⁻¹α(τ). (1.9)

This formula yields a bijection between the coefficients a(τ)andα(τ), which can be used to compute the modified differential equation from the coefficients of the integrator.

Criteria for symplecticity. The symplecticity of a mapping y7→B(a,y)can be characterized in terms of algebraic conditions on the coefficients of the B-series. To this end, we need the Butcher product of two trees u,v∈T which is defined by

u◦v= [u1, . . . ,u_m,v] for u= [u1, . . . ,u_m].

The B-series B(a,y)is symplectic for all Hamiltonian systems if and only if

a(u◦v) +a(v◦u) =a(u)a(v) for all u,v∈T. (1.10) The differential equation h ˙y=B(α,y) is Hamiltonian whenever f(y) =J⁻¹∇H(y) if and only if

α(u◦v) +α(v◦u) =0 for all u,v∈T. (1.11)

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If the coefficients a(τ)andα(τ)are related via (1.9), then both conditions, (1.10) and (1.11), are equivalent. These statements are discussed in [21, Sections VI.7 and XI.9].

Conjugation. We consider a B-series integrator y_n+1=Φh(yn)withΦh(y) =B(a,y), and a change of coordinates z=χ(y)that can be written as a B-series

z=B(c,y) with c(/0) =1.

In the new coordinates the method becomes z_n+1=Ψh(zn)withΨh=χ_◦Φh◦χ⁻¹^{. Using} the composition law for B-series, this can be expressed as

z_n+1=B(b,z_n) with b=c⁻¹a c. (1.12) If h ˙y=B(α,y)denotes the modified differential equation corresponding to the method B(a,y), then the modified differential equation for the method B(b,z)is given by

h˙z=B(β^,^z) ^with β ⁼^c⁻¹∂αc. (1.13) This is a consequence of (1.8), because h˙z=B(∂_αc,y) =B ∂_αc,B(c⁻¹,z)

.

1.3 Conjugate symplecticity in terms of the modified equation

We consider a B-series integrator B(a,y) of order p≥1. For the coefficients of the corre- sponding modified differential equations h ˙y=B(α,y)this implies that

α(/0) =0, α( ) =1, α(τ) =0 for 2≤ |τ_{| ≤}p. (1.14) With a view to studying the conjugate symplecticity of numerical integrators in terms of their modified differential equation we introduce the notation (for u,v∈T )

α(u,v) =α(u◦v) +α(v◦u).

The same notation is used for the coefficients β of the modified differential equation in the transformed coordinates and for the coefficients c of the transformation.

Lemma 1.1. In addition to (1.14) assume that the B-series B(c,y)satisfies

c(/0) =1, c(τ) =0 for 1≤ |τ_{| ≤}^p₋1, (1.15) so that B(c,y) =y+O(h^p), and letβ be given by (1.13). For u,v∈T with|u|+|v| ≤2p we then have

α(u,v) =β(u,v)−

∑

ˆ v∈SP_∗(v)

c(u,v)ˆ −

∑

ˆ u∈SP_∗(u)

c(u,ˆ v), (1.16) where SP_∗(τ) ={θ _∈SP(τ);|θ_|=|τ_{| −}¹_} is the set of splittings that separate only one tree with one vertex. By convention SP_∗( )is the empty set, so that the corresponding sums are zero.

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Proof. The assumption on c implies that the conjugated method is also of order p, so that the coefficientsβ satisfy the same relations (1.14) asα. We now write the equation (1.13) as

∂αc=cβ. The assumptions onαand c imply that only the terms withθ ⁼^{/0 and}|θ|=|τ|−1 give rise to non-vanishing terms in (1.8). Those onβ and c imply that in the composition law for cβ only the terms withθ =τ and|θ_|=1 have to be considered. The relation∂_αc=cβ thus yields

α(τ) +

∑

θ∈SP(τ),|θ|=|τ|−1

c(θ) = β(τ) +

∑

θ∈SP(τ),|θ|=1

c(τ_\θ) for |τ_{| ≤}2p. (1.17) Forτ =u◦v, the sum in the right-hand side of (1.17) is empty if |u| ≥2, and the set {θ _∈ SP(τ);|θ_|=|τ_{| −}¹_}is in one-to-one correspondence with SP_∗(u)∪SP_∗(v)if|v| ≥2. This proves

α(u◦v) =β(u◦v)−

∑

ˆ v∈SP_∗(v)

c(u◦v)ˆ −

∑

ˆ u∈SP_∗(u)

c(uˆ◦v), and the statement of the lemma follows for|u| ≥2 and|v| ≥2.

Forτ= ◦v the sum on the right-hand side of (1.17) reduces to c(v). Forτ =u◦ the set{θ _∈SP(τ);|θ_|=|τ_|−¹_}is in one-to-one correspondence with SP_∗(u)∪{u}, so that the sum on the left-hand side has an additional term c(u). In the sumα⁽ ^,^{v) =}α⁽ ◦v)+α^(v◦ ) these terms cancel and we get (1.16) also in this case.

Example 1.1. For p≥2 and|u|+|v|=3 we have

α( , ) =β( , )−c( , ).

For p≥2 and|u|+|v|=4 we have

α( , ) = β( , )−2 c( , ), α( , ) = β( , )−c( , ), α( , ) = β( , )−2 c( , ).

Equations (1.16) can be considered as a linear system for the coefficients c(u,v). For its formulation we let(T×T)r ={(u,v);|u|+|v|=r} for r≥2, and we consider the vector space of mappings on(T×T)r,

V_r={c :(T×T)r →R; c(u,v) =c(v,u)}.

To compute the dimension of this vector space we consider the formal series

N(ζ) =n₁ζ+n₂ζ²+n₃ζ³+···=ζ(1−ζ)⁻ⁿ¹(1−ζ²)⁻ⁿ²(1−ζ³)⁻ⁿ³···, M(ζ) =m₂ζ²+m₃ζ³+m₄ζ⁴+···=1

2

N(ζ)²+N(ζ²) .

The coefficient n_r denotes the number of trees with r vertices (this formula is due to Cayley and can be found in [21, p. 95]), and a straight-forward computation shows that the coefficient m_r is the dimension ofV_r. These numbers are given in Table 1.1 for r≤12. They have also been computed in [12].

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Table 1.1: Number of rooted trees of order r, and dimension mrof the vector spaceV_r.

r 1 2 3 4 5 6 7 8 9 10 11 12

nr 1 1 2 4 9 20 48 115 286 719 1842 4766

mr 0 1 1 3 6 16 37 96 239 622 1607 4235

We consider the linear mapping A on∪^r≥2V_r, whose restriction A :V_r→V_r+1is defined by

(Ac)(u,v) =

∑

ˆ v∈SP_∗(v)

c(u,v) +ˆ

∑

ˆ u∈SP_∗(u)

c(u,ˆ v), (1.18)

so that the condition (1.16) becomesα(u,v) =β(u,v)−(Ac)(u,v). Since symplecticity of the transformed method is equivalent toβ(u,v) =0 for all u,v∈T , the integratorΦh(y) =B(a,y) is conjugate symplectic up to order p+r if and only if there exists a B-series B(c,y)satisfying (1.15) such that α(u,v) =−(Ac)(u,v)for all pairs of trees with|u|+|v| ≤ p+r. Before we state this as a theorem we show that the conditions are independent, and we prove that the assumption (1.15) can be removed.

Lemma 1.2. The mapping A :V_r→V_r+1of (1.18) is injective.

Proof. For a mapping c :(T×T)r→Rwe have to prove that the condition Ac=0 implies c=0. It is sufficient to consider pairs of trees satisfying|u| ≤ |v|.

We start with u= , we assume that(Ac)( ,v) =0 for trees v with|v|=r, and we prove by induction on the height of the tree ˆv that c( ,v) =ˆ 0 for trees with|vˆ|=r−1. We denote byµk

the unique tree with k vertices and maximal height k. The equation(Ac)( ,µr) =c( ,µr−1) then proves c( ,v) =ˆ 0 for ˆv=µr−1. Assume that this relation holds for ˆv with height at least h. For an arbitrary tree ˆv with r−1 vertices and height h−1 we choose a tree v of height h such that ˆv∈SP_∗(v). We have(Ac)( ,v) =c( ,v), because further terms in the sum (1.18)ˆ vanish by the induction hypothesis. Consequently, c( ,v) =ˆ 0 for all trees with|vˆ|=r−1.

We next put u= , and apply the same induction argument as above on the height of the tree ˆv. Then we consider trees of order 3 for u, etc.

Lemma 1.3. Consider a B-series integrator B(a,y)of order p which is conjugate symplectic up to order p+r with r≥0. Then, there exists a change of coordinates z=B(c,y)satisfying B(c,y) =y+O(h^p), such that in the z coordinates the method is symplectic up to order p+r.

Proof. Since the method is conjugate symplectic up to order p+r, there exists a change of coordinates z=B(c,y) that makes the method symplectic up to order p+r. Let ρ = min{|τ_|^;τ _∈T,c(τ)6=0}. If ρ _≥ p, nothing has to be proved. Therefore, let us assume ρ<p. For trees(u,v)satisfying|u|+|v|=ρ+1, we haveα(u,v) =0 (as a consequence of order p) andβ(u,v) =0 (as a consequence of symplecticity). Lemma 1.1 (withρ in place of p) shows that(Ac)(u,v) =0 for all such pairs of trees, and Lemma 1.2 implies c(u^∗,v^∗) =0 for all trees with|u^∗|+|v^∗|=ρ. Consequently, there exists a symplectic mapping B(c_ρ,y) such that B(c_ρ,y) =B(c,y) +O(h^ρ+1). The transformation B(c c⁻_ρ¹,y)is O(h^ρ⁺¹)close to the identity, and leaves the transformed method symplectic up to order p+r. The proof can be repeated untilρ _≥p is reached.

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Lemma 1.4. Consider a symmetric B-series integrator B(a,y)of order p which is conjugate symplectic up to order p+r with 0 ≤r ≤ p. Then, there exists a change of coordinates z=χh(y) =B(c,y) satisfyingχ₋h(y) =χh(y), such that in the z coordinates the method is symplectic up to order p+r.

Proof. By Lemma 1.3 it is sufficient to consider transformations B(c,y)that areO(h^p)close to the identity. For pairs of trees(u,v)with even|u|+|v|we haveα(u,v) =0 by the symmetry of the method. Lemma 1.1 thus implies(Ac)(u,v) =0 for all such pairs of trees if|u|+|v| ≤ p+r, and Lemma 1.2 implies c(u^∗,v^∗) =0 for all trees with odd |u^∗|+|v^∗| ≤ p+r−1.

The same argument as in the proof of Lemma 1.3 shows that non-zero terms c(τ)with odd

|τ_{| ≤}^p+r−1 can be removed from the transformation. Non-zero terms c(τ)with|τ_{| ≥}^p+r can also be removed, because they do not affect conjugate symplecticity up to order p+r.

Theorem 1.5. A B-series integrator B(a,y) of order p is conjugate symplectic up to order p+r (with 0≤r≤ p), if and only if there exist coefficients c(u^∗,v^∗) such that the B-series coefficients of its modified differential equation h ˙y=B(α,y)satisfy

α(u,v) = −

∑

ˆ v∈SP_∗(v)

c(u,v)ˆ −

∑

ˆ u∈SP_∗(u)

c(u,ˆ v) for p<|u|+|v| ≤p+r. (1.19) Elimination of the coefficients c(u^∗,v^∗)gives exactly m_p+r−m_plinear relations between the expressions α(u,v). If the integrator is symmetric, the conditions (1.19) are automatically satisfied for trees with even|u|+|v|.

Proof. It follows from Lemma 1.3 that (1.15) can be assumed without loss of generality.

The equivalence of conjugate symplecticity with (1.19) is then a consequence of Lemma 1.1.

Finally, Lemma 1.2 yields the number of additional order conditions, and Lemma 1.4 the statement for symmetric methods.

1.4 Conjugate symplecticity in terms of the B-series integrator

The aim of this section is to translate the criterion of Theorem 1.5 into conditions on the coefficients a(τ)of the integrator y_n+1=B(a,y_n). To this end, we introduce the expression (for u,v∈T )

a(u,v) =a(u◦v) +a(v◦u)−a(u)a(v).

The coefficientsα(τ)of the modified differential equation are related to the coefficients a(τ) of the method by (1.9). We have to find a similar relation between the expressions α(u,v) for the modified equation and a(u,v) of the B-series, so that the conditions for conjugate symplecticity can be expressed in terms of a(u,v).

The following notation will be convenient: a j-fold splitting of a tree τ is a chain of ordered subtrees

θ0<θ1<···<θj−1<θj=τ ^{such that} θl−1∈SP(θl), l=1, . . . ,j.

We denote such a j-fold splitting byΘ=θ0<θ1<···<θj−1<θj, and we let SP^j(τ)be the set of all j-fold splittings ofτ. For a mapping on the set of trees satisfyingα(/0) =0, we define

α(Θ) =α(θ0)α(θ1\θ0)···α(θj\θj−1).

Design and analysis of integrators for stiff and Hamiltonian problems

Thesis

Reference

Design and analysis of integrators for stiff and Hamiltonian problems

Design and analysis of integrators for stiff and Hamiltonian problems

Remerciements

Contents

Résumé

Summary

Chapter 0

Overview

0.1 Chapter 1

0.2 Chapter 2

0.3 Chapter 3

∑

∑

∑

Chapter 1

On conjugate symplecticity of B-series integrators

1.1 Introduction

1.2 B-series theory

∑

∑

∑

∑

1.3 Conjugate symplecticity in terms of the modified equation

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

1.4 Conjugate symplecticity in terms of the B-series integrator