导语

  

内容提要

  

    E.海尔、C.卢比希、G.万纳著的《几何数值积分（(第2版）（英文版）》是一部教科书，旨在向大学3-4年级学生介绍哈密顿系统，可逆系统流型上的微分方程和高频振荡解问题。书中全面论述了辛理论和对称法。第2版主要增加了非典型哈密顿系统，高频振荡机械系统和多步法动力学。本书各章有习题。
    读者对象：数学等相关专业的大学高年级本科书和低年研究生。

I.Examples and Numerical Experiments
  I.1 First Problems and Methods
    I.1.1  The Lotka-Volterra Model
    I.1.2  First Numerical Methods
    I.1.3  The Pendulum as a Hamiltonian System
    I.1.4  The St6rmer-Verlet Scheme
  I.2  The Kepler Problem and the Outer Solar System
    I.2.1  Angular Momentum and Kepler's Second Law
    I.2.2  Exact Integration of the Kepler Problem
    I.2.3  Numerical Integration of the Kepler Problem
    I.2.4  The Outer Solar System
  I.3  The Henon-Heiles Model
  I.4  Molecular Dynamics
  I.5  Highly Oscillatory Problems
    I.5.1  A Fermi-Pasta-Ulam Problem
    I.5.2  Application of Classical Integrators
  I.6  Exercises
II.Numerical Integrators
  II.1  Runge-Kutta and Collocation Methods
    II.1.1  Runge-Kutta Methods
    II.1.2  Collocation Methods
    II.1.3  Gauss and Lobatto Collocation
    II.1.4  Discontinuous Collocation Methods
  II.2  Partitioned Runge-Kutta Methods
    II.2.1  Definition and First Examples
    II.2.2  Lobatto IIIA-IIIB Pairs
    II.2.3  Nystr6m Methods
  II.3  The Adjoint of a Method
  II.4  Composition Methods
  II.5  Splitting Methods
  II.6  Exercises
III.Order Conditions, Trees and B-Series
  III.1  Runge-Kutta Order Conditions and B-Series
    III.1.1  Derivation of the Order Conditions
    III.1.2  B-Series
    III.1.3  Composition of Methods
    III.1.4  Composition of B-Series
    III.l.5  The Butcher Group
  III.2  Order Conditions for Partitioned Runge-Kutta Methods
    III.2.1  Bi-Coloured Trees and P-Series
    III.2.2  Order Conditions for Partitioned Runge-Kutta Methods
    III.2.3  Order Conditions for Nystrom Methods
    III.3  Order Conditions for Composition Methods
    III.3.1  Introduction
    III.3.2  The General Case
    III.3.3  Reduction of the Order Conditions
    III.3.4  Order Conditions for Splitting Methods
  III.4  The Baker-Campbell-Hausdorff Formula
    III.4.1  Derivative of the Exponential and Its Inverse
    III.4.2  The BCH Formula
  II1.5  Order Conditions via the BCH Formula
    1II.5.1  Calculus of Lie Derivatives
    III.5.2  Lie Brackets and Commutativity
    III.5.3  Splitting Methods
    II1.5.4  Composition Methods
  III.6  Exercises
IV.Conservation of First Integrals and Methods on Manifolds
  IV.1  Examples of First Integrals
  IV.2  Quadratic Invariants
    IV.2.1  Runge-Kutta Methods
    IV.2.2  Partitioned Runge-Kutta Methods
    IV.2.3  Nystrom Methods
  IV.3  Polynomial Invariants
    IV.3.1  The Determinant as a First Integral
    IV.3.2  Isospectral Flows
  IV.4  Projection Methods
  IV.5  Numerical Methods Based on Local Coordinates
    IV.5.1  Manifolds and the Tangent Space
    IV.5.2  Differential Equations on Manifolds
    IV.5.3  Numerical Integrators on Manifolds
  IV.6  Differential Equations on Lie Groups
  1V.7  Methods Based on the Magnus Series Expansion
  IV.8  Lie Group Methods
    IV.8.1  Crouch-Grossman Methods
    IV.8.2  Munthe-Kaas Methods
    IV.8.3  Further Coordinate Mappings
  IV.9  Geometric Numerical Integration Meets Geometric Numerical Linear Algebra
    IV.9.1  Numerical Integration on the Stiefel Manifold
    IV.9.2  Differential Equations on the Grassmann Manifold
    IV.9.3  Dynamical Low-Rank Approximation
  IV.10  Exercises
V.Symmetric Integration and Reversibility
VI.Symplectic Integration of Hamiltonian Systems
VII.Non-Canonical Hamiitonian Systems
VIII.Structure-Preserving Implementation
IX.Backward Error Analysis and Structure Preservation
X.Hamiltonian Perturbation Theory and Symplectic Integrators
XI.Reversible Perturbation Theory and Symmetric Integrators
XII.Dissivativeiv Perturbed Hamiltonian and Reversible Systems
XIII.Oscillatory Differential Equations with Constant High Frequencies
XIV.Oscillatory Differential Equations with Varying High Frequencies
XV.Dynamics of Multistep Methods
Bibliography
Index