Simulating Hamiltonian dynamics 1st Edition by Benedict Leimkuhler, Sebastian Reich ISBN 9780521772907 by Leimkuhler B., Reich S. 9780521772907, 0521772907 instant download after payment.

Simulating Hamiltonian dynamics 1st Edition by Benedict Leimkuhler, Sebastian Reich - Ebook PDF Instant Download/Delivery: 9780521772907
Full download Simulating Hamiltonian dynamics 1st Edition after payment

Product details:

ISBN 13: 9780521772907
Author: Benedict Leimkuhler, Sebastian Reich

Geometric integrators are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations. In this book the authors outline the principles of geometric integration and demonstrate how they can be applied to provide efficient numerical methods for simulating conservative models. Beginning from basic principles and continuing with discussions regarding the advantageous properties of such schemes, the book introduces methods for the N-body problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include high-order and variable stepsize methods, schemes for treating problems involving multiple time-scales, and applications to molecular dynamics and partial differential equations. The emphasis is on providing a unified theoretical framework as well as a practical guide for users. The inclusion of examples, background material and exercises enhance the usefulness of the book for self-instruction or as a text for a graduate course on the subject.

Simulating Hamiltonian dynamics 1st Table of contents:

1 Introduction
1.1 N-body problems
1.2 Problems and applications
1.3 Constrained dynamics
1.4 Exercises
2 Numerical methods
2.1 One-step methods
2.1.1 Derivation of one-step methods
2.1.2 Error analysis
2.2 Numerical example: the Lennard?Jones oscillator
2.3 Higher-order methods
2.4 Runge?Kutta methods
2.5 Partitioned Runge?Kutta methods
2.6 Stability and eigenvalues
2.7 Exercises
3 Hamiltonian mechanics
3.1 Canonical and noncanonical Hamiltonian systems
3.2 Examples of Hamiltonian systems
3.2.1 Linear systems
3.2.2 Single-degree-of-freedom problems
3.2.3 Central forces
3.2.4 Charged particle in a magnetic field
3.2.5 Lagrange?s equation
3.2.6 N-body problem
3.3 First integrals
3.4 The flow map and variational equations
3.5 Symplectic maps and Hamiltonian flow maps
3.5.1 One-degree-of-freedom systems
3.5.2 The symplectic structure of phase space
3.6 Differential forms and the wedge product
3.7 Exercises
4 Geometric integrators
4.1 Symplectic maps and methods
4.2 Construction of symplectic methods by Hamiltonian splitting
4.2.1 Separable Hamiltonian systems
4.2.2 A second-order splitting method
4.3 Time-reversal symmetry and reversible discretizations
4.3.1 Time-reversible maps
4.3.2 Linear-reversible maps
4.3.3 Time-reversible methods by symmetric composition
4.4 First integrals
4.4.1 Preservation of first integrals by splitting methods
4.4.2 Implicit midpoint preserves quadratic first integrals
4.5 Case studies
4.5.1 Application to N-body systems: a molecular dynamics model problem
4.5.2 Particle in a magnetic field
Numerical experiment
4.5.3 Weakly coupled systems
4.5.4 Linear/nonlinear splitting
4.6 Exercises
5 The modified equations
5.1 Forward v. backward error analysis
5.1.1 Linear systems
5.1.2 The nearby Hamiltonian
5.2 The modified equations
5.2.1 Asymptotic expansion of the modified equations
5.2.2 Conservation of energy for symplectic methods
5.2.3 Applications
Integrable systems
Hyperbolic systems
Adiabatic invariants
5.3 Geometric integration and modified equations
5.4 Modified equations for composition methods
5.5 Exercises
6 Higher-order methods
6.1 Construction of higher-order methods
6.2 Composition methods
6.2.1 Composition methods for separable Hamiltonian systems
6.2.2 Composition methods based on second-order symmetric methods
6.2.3 Post-processing of composition methods
6.3 Runge?Kutta methods
6.3.1 Implicit Runge?Kutta methods
6.3.2 Partitioned Runge?Kutta methods
6.4 Generating functions
6.5 Numerical experiments
6.5.1 Arenstorf orbits
6.5.2 Solar system
6.6 Exercises
7 Constrained mechanical systems
7.1 N-body systems with holonomic constraints
7.2 Numerical methods for constraints
7.2.1 Direct discretization: SHAKE and RATTLE
7.2.2 Implementation
7.2.3 Numerical experiment
7.3 Transition to Hamiltonian mechanics
7.4 The symplectic structure with constraints
7.5 Direct symplectic discretization
7.5.1 Second-order methods
7.5.2 Higher-order methods
7.6 Alternative approaches to constrained integration
7.6.1 Parametrization of manifolds ? local charts
7.6.2 The Hamiltonian case
7.6.3 Numerical methods based on local charts
7.6.4 Methods based on projection
7.7 Exercises
8 Rigid body dynamics
8.1 Rigid bodies as constrained systems
8.1.1 Hamiltonian formulation
8.1.2 Linear and planar bodies
8.1.3 Symplectic discretization using SHAKE
8.1.4 Numerical experiment: a symmetric top
8.2 Angular momentum and the inertia tensor
8.3 The Euler equations of rigid body motion
8.3.1 Symplectic discretization of the Euler equations
8.3.2 Numerical experiment: the Lagrangian top
8.3.3 Integrable discretization: RATTLE and the scheme of MOSER AND VESELOV
8.4 Order 4 and order 6 variants of RATTLE for the free rigid body
8.5 Freely moving rigid bodies
8.6 Other formulations for rigid body motion
8.6.1 Euler angles
8.6.2 Quaternions
8.7 Exercises
9 Adaptive geometric integrators
9.1 Sundman and Poincar? transformations
9.2 Reversible variable stepsize integration
9.2.1 Local error control as a time transformation
9.2.2 Semi-explicit methods based on generalized leapfrog
9.2.3 Differentiating the control
9.2.4 The Adaptive Verlet method
9.3 Sundman transformations
9.3.1 Arclength parameterization
9.3.2 Rescaling for the N-body problem
9.3.3 Stepsize bounds
9.4 Backward error analysis
9.5 Generalized reversible adaptive methods
9.5.1 Switching
9.6 Poincar? transformations
9.7 Exercises
10 Highly oscillatory problems
10.1 Large timestep methods
10.1.1 A single oscillatory degree of freedom
10.1.2 Slow-fast systems
10.1.3 Adiabatic invariants
10.1.4 The effect of numerical resonances
A linear model problem
A nonlinear model problem
10.2 Averaging and reduced equations
10.3 The reversible averaging (RA) method
10.3.1 Numerical experiments
A linear test problem
A nonliner model problem
10.4 The mollified impulse (MOLLY) method
10.5 Multiple frequency systems
10.6 Exercises
11 Molecular dynamics
11.1 From liquids to biopolymers
11.2 Statistical mechanics from MD trajectories
11.2.1 Ensemble computations
11.3 Dynamical formulations for the NVT, NPT and other ensembles
11.3.1 Coordinate transformations: the separated form
11.3.2 Time-reparameterization and the Nos??Hoover method
11.4 A symplectic approach: the Nos??Poincar? method
11.4.1 Generalized baths
11.4.2 Simulation in other ensembles
11.5 Exercises
12 Hamiltonian PDEs
12.1 Examples of Hamiltonian PDEs
12.1.1 The nonlinear wave equation
12.1.2 Soliton solutions
12.1.3 The two-dimensional rotating shallow-water equations
12.1.4 Noncanonical Hamiltonian wave equations
12.2 Symplectic discretizations
12.2.1 Grid-based methods
12.2.2 Particle-based methods
12.3 Multi-symplectic PDEs
12.3.1 Conservation laws
12.3.2 Traveling waves and dispersion
12.3.3 Multi-symplectic imtegrators

People also search for Simulating Hamiltonian dynamics 1st:

simulating hamiltonian dynamics
    
simulating hamiltonian dynamics with a truncated taylor series
    
hamiltonian simulation in the interaction picture
    
xy hamiltonian
    
what is a hamiltonian system
    
time dependent hamiltonian symplectic
    
quantum computing hamiltonian

Tags: Benedict Leimkuhler, Sebastian Reich, Simulating, Hamiltonian