Astrophysical Journal Letters, ISSN 2041-8205, 08/2015, Volume 809, Issue 1, pp. L9 - 6

Symplectic integrators are widely used for long-term integration of conservative astrophysical problems due to their ability to preserve the constants of motion...

methods: numerical | planets and satellites: dynamical evolution and stability | celestial mechanics | MECHANICS | N-BODY PROBLEM | ASTRONOMY & ASTROPHYSICS | RADIATION | SYMPLECTIC INTEGRATORS | Dynamics | Dissipation | Preserves | Constants | Evolution | Integrators | Formalism | Dynamical systems | PLANETS | GRAVITATIONAL RADIATION | ASTROPHYSICS, COSMOLOGY AND ASTRONOMY | ASTROPHYSICS | SATELLITES | ALGORITHMS | HARMONIC OSCILLATORS | VARIATIONAL METHODS

methods: numerical | planets and satellites: dynamical evolution and stability | celestial mechanics | MECHANICS | N-BODY PROBLEM | ASTRONOMY & ASTROPHYSICS | RADIATION | SYMPLECTIC INTEGRATORS | Dynamics | Dissipation | Preserves | Constants | Evolution | Integrators | Formalism | Dynamical systems | PLANETS | GRAVITATIONAL RADIATION | ASTROPHYSICS, COSMOLOGY AND ASTRONOMY | ASTROPHYSICS | SATELLITES | ALGORITHMS | HARMONIC OSCILLATORS | VARIATIONAL METHODS

Journal Article

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, ISSN 1751-8113, 11/2019, Volume 52, Issue 44, p. 445206

We present geometric numerical integrators for contact flows that stem from a discretization of Herglotz' variational principle...

Herglotz' variational principle | THERMODYNAMICS | PHYSICS, MULTIDISCIPLINARY | geometric integrators | contact geometry | HAMILTONIAN-DYNAMICS | (3+1)-DIMENSIONAL SYSTEMS | VARIABLES | MANIFOLDS | QUANTIZATION | PHYSICS, MATHEMATICAL | GEOMETRY

Herglotz' variational principle | THERMODYNAMICS | PHYSICS, MULTIDISCIPLINARY | geometric integrators | contact geometry | HAMILTONIAN-DYNAMICS | (3+1)-DIMENSIONAL SYSTEMS | VARIABLES | MANIFOLDS | QUANTIZATION | PHYSICS, MATHEMATICAL | GEOMETRY

Journal Article

International Journal for Numerical Methods in Engineering, ISSN 0029-5981, 05/2004, Volume 60, Issue 1, pp. 153 - 212

.... In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs...

discrete mechanics | subcycling | elastodynamics | multi‐time‐step | geometric integration | variational integrators | Subcycling | Variational integrators | Elastodynamics | Multi-time-step | Discrete mechanics | Geometric integration | NEWMARK ALGORITHM | MOLECULAR-DYNAMICS | STEPPING ALGORITHMS | multi-time-step | TRANSIENT ANALYSIS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | NONLINEAR DYNAMICS | ENGINEERING, MULTIDISCIPLINARY | NUMERICAL INTEGRATORS | HAMILTONIAN-SYSTEMS | EXPLICIT FINITE-ELEMENTS | HELICOPTER ROTOR DYNAMICS | STRUCTURAL DYNAMICS | Finite element method | Accuracy | Algorithms | Mathematical analysis | Conservation | Integrators | Mathematical models | Convergence

discrete mechanics | subcycling | elastodynamics | multi‐time‐step | geometric integration | variational integrators | Subcycling | Variational integrators | Elastodynamics | Multi-time-step | Discrete mechanics | Geometric integration | NEWMARK ALGORITHM | MOLECULAR-DYNAMICS | STEPPING ALGORITHMS | multi-time-step | TRANSIENT ANALYSIS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | NONLINEAR DYNAMICS | ENGINEERING, MULTIDISCIPLINARY | NUMERICAL INTEGRATORS | HAMILTONIAN-SYSTEMS | EXPLICIT FINITE-ELEMENTS | HELICOPTER ROTOR DYNAMICS | STRUCTURAL DYNAMICS | Finite element method | Accuracy | Algorithms | Mathematical analysis | Conservation | Integrators | Mathematical models | Convergence

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 09/2016, Volume 321, pp. 435 - 458

.... The resulting integrator preserves important quantities like the total energy, magnetic helicity and cross helicity exactly...

Conservation laws | Geometric discretisation | Variational integrators | Noether theorem | Collisionless reconnection | Reduced magnetohydrodynamics | FACTORIZATION | PDES | RECONNECTION | DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | DISCRETIZATION | MECHANICS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MATRIX POLYNOMIALS | FLUID | SYSTEMS | HAMILTONIAN-FORMULATION | Fluid dynamics | Magnetic fields | Analysis | Environmental law | Electric properties | Plasma Physics | Physics | MAGNETIC FIELDS | PLASMA | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | ASTROPHYSICS | CONSERVATION LAWS | EQUATIONS | HAMILTONIANS | MAGNETOHYDRODYNAMICS | LAGRANGIAN FUNCTION | VARIATIONAL METHODS

Conservation laws | Geometric discretisation | Variational integrators | Noether theorem | Collisionless reconnection | Reduced magnetohydrodynamics | FACTORIZATION | PDES | RECONNECTION | DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | DISCRETIZATION | MECHANICS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MATRIX POLYNOMIALS | FLUID | SYSTEMS | HAMILTONIAN-FORMULATION | Fluid dynamics | Magnetic fields | Analysis | Environmental law | Electric properties | Plasma Physics | Physics | MAGNETIC FIELDS | PLASMA | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | ASTROPHYSICS | CONSERVATION LAWS | EQUATIONS | HAMILTONIANS | MAGNETOHYDRODYNAMICS | LAGRANGIAN FUNCTION | VARIATIONAL METHODS

Journal Article

International Journal for Numerical Methods in Engineering, ISSN 0029-5981, 04/2018, Volume 114, Issue 3, pp. 215 - 231

Summary With the postulation of the principle of virtual action, we propose, in this paper, a variational framework for describing the dynamics of finite...

time integration | multibody dynamics | nonsmooth mechanics | variational integrators | time finite element method (TFEM) | FRICTION | STEPPING SCHEMES | MULTIBODY SYSTEMS | GENERALIZED-ALPHA SCHEME | UNILATERAL CONSTRAINTS | NONSMOOTH DYNAMICS | NUMERICAL SCHEME | RIGID-BODY DYNAMICS | MECHANICS | IMPACT | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | Energy conservation | Finite element method | Integrators | Percussion | Inclusions | Mechanical systems | Shape functions

time integration | multibody dynamics | nonsmooth mechanics | variational integrators | time finite element method (TFEM) | FRICTION | STEPPING SCHEMES | MULTIBODY SYSTEMS | GENERALIZED-ALPHA SCHEME | UNILATERAL CONSTRAINTS | NONSMOOTH DYNAMICS | NUMERICAL SCHEME | RIGID-BODY DYNAMICS | MECHANICS | IMPACT | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | Energy conservation | Finite element method | Integrators | Percussion | Inclusions | Mechanical systems | Shape functions

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 01/2014, Volume 257, pp. 1040 - 1061

We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed...

Symplectic methods | Lie group integrators | Integral preserving methods | FREE RIGID-BODY | ENERGY | RUNGE-KUTTA METHODS | VARIATIONAL INTEGRATORS | TIME INTEGRATION | PHYSICS, MATHEMATICAL | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | DYNAMICS | NUMERICAL-INTEGRATION | SYSTEMS | GEOMETRIC INTEGRATION | MOSER-VESELOV ALGORITHM | Integrators | Computation | Bundles | Lie groups | Mathematics - Numerical Analysis

Symplectic methods | Lie group integrators | Integral preserving methods | FREE RIGID-BODY | ENERGY | RUNGE-KUTTA METHODS | VARIATIONAL INTEGRATORS | TIME INTEGRATION | PHYSICS, MATHEMATICAL | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | DYNAMICS | NUMERICAL-INTEGRATION | SYSTEMS | GEOMETRIC INTEGRATION | MOSER-VESELOV ALGORITHM | Integrators | Computation | Bundles | Lie groups | Mathematics - Numerical Analysis

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 10/2011, Volume 31, Issue 4, pp. 1497 - 1532

... leads to a class of generalized Galerkin Hamiltonian variational integrators that includes the symplectic partitioned Runge-Kutta methods...

geometric mechanics | Hamiltonian mechanics | symplectic integrators | geometric numerical integration | variational integrators | MATHEMATICS, APPLIED | MECHANICS | PDES | SYSTEMS | GEOMETRY | SCHEMES | Hamilton-Jacobi equation | Mathematical analysis | Integrators | Mathematical models | Runge-Kutta method | Transformations | Variational principles | Galerkin methods

geometric mechanics | Hamiltonian mechanics | symplectic integrators | geometric numerical integration | variational integrators | MATHEMATICS, APPLIED | MECHANICS | PDES | SYSTEMS | GEOMETRY | SCHEMES | Hamilton-Jacobi equation | Mathematical analysis | Integrators | Mathematical models | Runge-Kutta method | Transformations | Variational principles | Galerkin methods

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 06/2013, Volume 242, pp. 498 - 530

In this contribution, we develop a variational integrator for the simulation of (stochastic and multiscale) electric circuits...

Variational integrators | Electric circuits | Multiscale integration | Structure-preserving integration | Degenerate systems | Noisy systems | HIGH OSCILLATION | ALGORITHM | EQUATIONS | PHYSICS, MATHEMATICAL | LAGRANGIAN MECHANICS | INTERCONNECTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | EXPLICIT TOPOLOGICAL FORMULATION | DISCRETE MECHANICS | HAMILTONIAN-SYSTEMS | DIRAC STRUCTURES | DYNAMICS | Electric potential | Computer simulation | Dynamics | Voltage | Mathematical models | Integrators | Dynamical systems | ELECTRIC POTENTIAL | STOCHASTIC PROCESSES | EQUATIONS OF MOTION | RESISTORS | COMPARATIVE EVALUATIONS | SIMULATION | MATHEMATICAL METHODS AND COMPUTING | GAIN | LAGRANGIAN FUNCTION | VARIATIONAL METHODS | CONTROL

Variational integrators | Electric circuits | Multiscale integration | Structure-preserving integration | Degenerate systems | Noisy systems | HIGH OSCILLATION | ALGORITHM | EQUATIONS | PHYSICS, MATHEMATICAL | LAGRANGIAN MECHANICS | INTERCONNECTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | EXPLICIT TOPOLOGICAL FORMULATION | DISCRETE MECHANICS | HAMILTONIAN-SYSTEMS | DIRAC STRUCTURES | DYNAMICS | Electric potential | Computer simulation | Dynamics | Voltage | Mathematical models | Integrators | Dynamical systems | ELECTRIC POTENTIAL | STOCHASTIC PROCESSES | EQUATIONS OF MOTION | RESISTORS | COMPARATIVE EVALUATIONS | SIMULATION | MATHEMATICAL METHODS AND COMPUTING | GAIN | LAGRANGIAN FUNCTION | VARIATIONAL METHODS | CONTROL

Journal Article

BIT Numerical Mathematics, ISSN 0006-3835, 12/2018, Volume 58, Issue 4, pp. 1009 - 1048

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics...

Computational Mathematics and Numerical Analysis | Stochastic differential equations | 65C30 | Geometric numerical integration methods | Numeric Computing | Mathematics, general | Mathematics | Variational integrators | Geometric mechanics | Stochastic Hamiltonian systems | MATHEMATICS, APPLIED | QUADRATIC-INVARIANTS | RUNGE-KUTTA METHODS | DIFFERENTIAL-EQUATIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | SYMPLECTIC SCHEMES | SYSTEMS | ORDER CONDITIONS | Differential equations

Computational Mathematics and Numerical Analysis | Stochastic differential equations | 65C30 | Geometric numerical integration methods | Numeric Computing | Mathematics, general | Mathematics | Variational integrators | Geometric mechanics | Stochastic Hamiltonian systems | MATHEMATICS, APPLIED | QUADRATIC-INVARIANTS | RUNGE-KUTTA METHODS | DIFFERENTIAL-EQUATIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | SYMPLECTIC SCHEMES | SYSTEMS | ORDER CONDITIONS | Differential equations

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 4/2009, Volume 29, Issue 2, pp. 421 - 443

.... These are easy consequences of the fact that the stochastic action is intrinsically defined. Stochastic variational integrators (SVIs...

Ornstein-Uhlenbeck process | Variational integrators | Stochastic Hamiltonian systems | MATHEMATICS, APPLIED | MECHANICS | RIGID-BODY | MOTION | DIRAC STRUCTURES | EQUATIONS | stochastic Hamiltonian systems | SYSTEMS | variational integrators | Mathematics - Probability

Ornstein-Uhlenbeck process | Variational integrators | Stochastic Hamiltonian systems | MATHEMATICS, APPLIED | MECHANICS | RIGID-BODY | MOTION | DIRAC STRUCTURES | EQUATIONS | stochastic Hamiltonian systems | SYSTEMS | variational integrators | Mathematics - Probability

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 01/2018, Volume 38, Issue 1, pp. 377 - 398

Abstract Discrete Hamiltonian variational integrators are derived from type II and type III generating functions for symplectic maps, and in this article, we establish a variational error analysis...

Hamiltonian mechanics | Symplectic integrators | Variational integrators | Geometric mechanics | Geometric numerical integration | MATHEMATICS, APPLIED | symplectic integrators | MECHANICS | geometric numerical integration | geometric mechanics | variational integrators

Hamiltonian mechanics | Symplectic integrators | Variational integrators | Geometric mechanics | Geometric numerical integration | MATHEMATICS, APPLIED | symplectic integrators | MECHANICS | geometric numerical integration | geometric mechanics | variational integrators

Journal Article

Physica D: Nonlinear Phenomena, ISSN 0167-2789, 08/2015, Volume 310, pp. 37 - 71

Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods...

Conservation laws | Geometric discretisation | Variational integrators | Noether theorem | Lagrangian field theory | Linear and nonlinear PDEs | Linear and nonlinear pdes | MATHEMATICS, APPLIED | FACTORIZATION | PHYSICS, MULTIDISCIPLINARY | MULTISYMPLECTIC GEOMETRY | PHYSICS, MATHEMATICAL | PRINCIPLES | DISCRETIZATION | CONTINUUM-MECHANICS | MATRIX POLYNOMIALS | SYSTEMS | Differential equations | Fluid dynamics | Plasma physics

Conservation laws | Geometric discretisation | Variational integrators | Noether theorem | Lagrangian field theory | Linear and nonlinear PDEs | Linear and nonlinear pdes | MATHEMATICS, APPLIED | FACTORIZATION | PHYSICS, MULTIDISCIPLINARY | MULTISYMPLECTIC GEOMETRY | PHYSICS, MATHEMATICAL | PRINCIPLES | DISCRETIZATION | CONTINUUM-MECHANICS | MATRIX POLYNOMIALS | SYSTEMS | Differential equations | Fluid dynamics | Plasma physics

Journal Article

International Journal for Numerical Methods in Engineering, ISSN 0029-5981, 03/2017, Volume 109, Issue 11, pp. 1549 - 1581

Summary This article presents a family of variational integrators from a continuous time point of view...

numerical integration methods | symplectic integrators | multibody dynamics | discontinuous Galerkin methods | variational integrators | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | FINITE-ELEMENTS | ENGINEERING, MULTIDISCIPLINARY | DYNAMICS | Concretes | Numerical analysis | Pendulums | Runge-Kutta method | Integrators | Regularization | Cartesian coordinates

numerical integration methods | symplectic integrators | multibody dynamics | discontinuous Galerkin methods | variational integrators | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | FINITE-ELEMENTS | ENGINEERING, MULTIDISCIPLINARY | DYNAMICS | Concretes | Numerical analysis | Pendulums | Runge-Kutta method | Integrators | Regularization | Cartesian coordinates

Journal Article

Journal of Nonlinear Science, ISSN 0938-8974, 10/2017, Volume 27, Issue 5, pp. 1399 - 1434

... (Jacobs and Yoshimura 2014) and discrete Dirac variational integrators (Leok and Ohsawa 2011). We test our results by simulating some of the continuous examples given in Jacobs and Yoshimura 2014.

65P10 | 37J05 | 70Q05 | Theoretical, Mathematical and Computational Physics | Classical Mechanics | Economic Theory/Quantitative Economics/Mathematical Methods | 37N05 | 37J60 | Dirac structures | Hamiltonian DAEs | 70H05 | Mathematics | 70F25 | 70H45 | Geometric integration | 70G45 | Lagrange–Dirac systems | Interconnection | Analysis | 93B27 | Mathematical and Computational Engineering | Variational integrators | 93A30 | MATHEMATICS, APPLIED | PART I | PHYSICS, MATHEMATICAL | LIE-GROUPS | MECHANICS | MECHANICAL SYSTEMS | Lagrange-Dirac systems | GEOMETRY

65P10 | 37J05 | 70Q05 | Theoretical, Mathematical and Computational Physics | Classical Mechanics | Economic Theory/Quantitative Economics/Mathematical Methods | 37N05 | 37J60 | Dirac structures | Hamiltonian DAEs | 70H05 | Mathematics | 70F25 | 70H45 | Geometric integration | 70G45 | Lagrange–Dirac systems | Interconnection | Analysis | 93B27 | Mathematical and Computational Engineering | Variational integrators | 93A30 | MATHEMATICS, APPLIED | PART I | PHYSICS, MATHEMATICAL | LIE-GROUPS | MECHANICS | MECHANICAL SYSTEMS | Lagrange-Dirac systems | GEOMETRY

Journal Article

Foundations of Computational Mathematics, ISSN 1615-3375, 2/2017, Volume 17, Issue 1, pp. 199 - 257

We present a new class of high-order variational integrators on Lie groups. We show that these integrators are symplectic and momentum-preserving, can be constructed to be of arbitrarily high order, or can be made to converge geometrically...

65P10 | Lie group integrators | 37M15 | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Symplectic integrators | Geometric numerical integration | 70H25 | 70G75 | 65M70 | Numerical Analysis | Applications of Mathematics | Math Applications in Computer Science | Variational integrators | Computer Science, general | Economics, general | MATHEMATICS, APPLIED | RIGID-BODY | RUNGE-KUTTA METHODS | HAMILTON-JACOBI THEORY | LAGRANGIAN MECHANICS | MATHEMATICS | REDUCTION | DYNAMICS | SYSTEMS | COMPUTER SCIENCE, THEORY & METHODS | Convergence (Mathematics) | Numerical analysis | Research | Mathematical research | Foundations | Computation | Lie groups | Rigid-body dynamics | Mathematical models | Integrators | Spectra

65P10 | Lie group integrators | 37M15 | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Symplectic integrators | Geometric numerical integration | 70H25 | 70G75 | 65M70 | Numerical Analysis | Applications of Mathematics | Math Applications in Computer Science | Variational integrators | Computer Science, general | Economics, general | MATHEMATICS, APPLIED | RIGID-BODY | RUNGE-KUTTA METHODS | HAMILTON-JACOBI THEORY | LAGRANGIAN MECHANICS | MATHEMATICS | REDUCTION | DYNAMICS | SYSTEMS | COMPUTER SCIENCE, THEORY & METHODS | Convergence (Mathematics) | Numerical analysis | Research | Mathematical research | Foundations | Computation | Lie groups | Rigid-body dynamics | Mathematical models | Integrators | Spectra

Journal Article

Physics of Plasmas, ISSN 1070-664X, 10/2017, Volume 24, Issue 10, p. 102311

...Metriplectic integrators for the Landau collision operator Michael Kraus 1,2,a) and Eero Hirvijoki 3,b) 1 Max-Planck-Institut f€ ur Plasmaphysik...

IN-CELL ALGORITHM | ENERGY | MAXWELL-VLASOV EQUATIONS | PHYSICS, FLUIDS & PLASMAS | DISCONTINUOUS GALERKIN METHODS | PLASMA SIMULATIONS | POISSON SYSTEM | VARIATIONAL FORMULATION | BRACKET FORMULATION | CONTINUOUS HAMILTONIAN SYSTEM | IMPLICIT | 70 PLASMA PHYSICS AND FUSION TECHNOLOGY

IN-CELL ALGORITHM | ENERGY | MAXWELL-VLASOV EQUATIONS | PHYSICS, FLUIDS & PLASMAS | DISCONTINUOUS GALERKIN METHODS | PLASMA SIMULATIONS | POISSON SYSTEM | VARIATIONAL FORMULATION | BRACKET FORMULATION | CONTINUOUS HAMILTONIAN SYSTEM | IMPLICIT | 70 PLASMA PHYSICS AND FUSION TECHNOLOGY

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 07/2012, Volume 32, Issue 3, pp. 1194 - 1216

We introduce a novel technique for constructing higher-order variational integrators for Hamiltonian systems of ordinary differential equations...

variational order | Hamiltonian ODEs | Lagrangian variational integrators | Euler-Lagrange equations | MATHEMATICS, APPLIED | MECHANICS | DISCRETE

variational order | Hamiltonian ODEs | Lagrangian variational integrators | Euler-Lagrange equations | MATHEMATICS, APPLIED | MECHANICS | DISCRETE

Journal Article

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 08/2018, Volume 338, pp. 333 - 361

Lie-group variational integrators of arbitrary order are developed using the Galerkin method, based on unit quaternion interpolation...

Lie group method | Higher order | Variational integrators | RIGID-BODY DYNAMICS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | CONSTRAINED MECHANICAL SYSTEMS | ALGORITHM | TIME INTEGRATION | NULL SPACE METHOD | ENERGY CONSISTENT INTEGRATION | Control systems

Lie group method | Higher order | Variational integrators | RIGID-BODY DYNAMICS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | CONSTRAINED MECHANICAL SYSTEMS | ALGORITHM | TIME INTEGRATION | NULL SPACE METHOD | ENERGY CONSISTENT INTEGRATION | Control systems

Journal Article