Quarterly Journal of the Royal Meteorological Society, ISSN 0035-9009, 04/2019, Volume 145, Issue 720, pp. 1070 - 1088

We develop a variational integrator for the shallow‐water equations on a rotating sphere...

variational integrator on sphere | rotating shallow‐water equations | structure‐preserving discretization | DISCRETIZATION | ENERGY | POTENTIAL ENSTROPHY | NUMERICAL-INTEGRATION | rotating shallow-water equations | structure-preserving discretization | METEOROLOGY & ATMOSPHERIC SCIENCES | SCHEMES | Hydrodynamics | Equations | Framework | Mathematics | Nonlinear Sciences

variational integrator on sphere | rotating shallow‐water equations | structure‐preserving discretization | DISCRETIZATION | ENERGY | POTENTIAL ENSTROPHY | NUMERICAL-INTEGRATION | rotating shallow-water equations | structure-preserving discretization | METEOROLOGY & ATMOSPHERIC SCIENCES | SCHEMES | Hydrodynamics | Equations | Framework | Mathematics | Nonlinear Sciences

Journal Article

International Journal for Numerical Methods in Engineering, ISSN 0029-5981, 08/2009, Volume 79, Issue 9, pp. 1147 - 1174

Euler–Lagrange equations and variational integrators are developed for Lagrangian mechanical systems evolving on a product of two‐spheres...

variational integrator | Lagrangian mechanics | geometric integrator | two‐sphere | homogeneous manifold | Geometric integrator | Homogeneous manifold | Variational integrator | Two-sphere | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | two-sphere | GEOMETRIC INTEGRATION | Parametrization | Mathematical analysis | Hamilton's principle | Lie groups | Integrators | Equations of motion | Euler-Lagrange equation | Mechanical systems

variational integrator | Lagrangian mechanics | geometric integrator | two‐sphere | homogeneous manifold | Geometric integrator | Homogeneous manifold | Variational integrator | Two-sphere | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | two-sphere | GEOMETRIC INTEGRATION | Parametrization | Mathematical analysis | Hamilton's principle | Lie groups | Integrators | Equations of motion | Euler-Lagrange equation | Mechanical systems

Journal Article

International Journal for Numerical Methods in Engineering, ISSN 0029-5981, 06/2010, Volume 82, Issue 13, pp. 1609 - 1644

In this paper we present a systematic and general method for developing variational integrators for Lie...

Hamiltonian | Lie‐Poisson systems | variational integrators | Lie-poisson systems | Variational integrators | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | POINCARE | ENGINEERING, MULTIDISCIPLINARY | Lie-Poisson systems | STABILITY | SPHERE | DYNAMICS | RELATIVE EQUILIBRIA | POINT VORTICES | Simulation | Dynamics | Mathematical analysis | Lie groups | Rigid-body dynamics | Integrators | Variational principles | Dynamical systems

Hamiltonian | Lie‐Poisson systems | variational integrators | Lie-poisson systems | Variational integrators | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | POINCARE | ENGINEERING, MULTIDISCIPLINARY | Lie-Poisson systems | STABILITY | SPHERE | DYNAMICS | RELATIVE EQUILIBRIA | POINT VORTICES | Simulation | Dynamics | Mathematical analysis | Lie groups | Rigid-body dynamics | Integrators | Variational principles | Dynamical systems

Journal Article

Journal of Nonlinear Science, ISSN 0938-8974, 2/2014, Volume 24, Issue 1, pp. 1 - 37

.... We then use the isomorphism of the 3-sphere with the Lie group SU(2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint...

Symplectic integration | Point vortices | Variational methods | 37M15 | Theoretical, Mathematical and Computational Physics | 76B47 | Mathematics | 70H03 | Hopf fibration | Analysis | Appl.Mathematics/Computational Methods of Engineering | Mechanics | Economic Theory | MATHEMATICS, APPLIED | MECHANICS | VORTICES | MOTION | VARIATIONAL INTEGRATORS | BACKWARD ERROR ANALYSIS | PHYSICS, MATHEMATICAL | RELATIVE EQUILIBRIA

Symplectic integration | Point vortices | Variational methods | 37M15 | Theoretical, Mathematical and Computational Physics | 76B47 | Mathematics | 70H03 | Hopf fibration | Analysis | Appl.Mathematics/Computational Methods of Engineering | Mechanics | Economic Theory | MATHEMATICS, APPLIED | MECHANICS | VORTICES | MOTION | VARIATIONAL INTEGRATORS | BACKWARD ERROR ANALYSIS | PHYSICS, MATHEMATICAL | RELATIVE EQUILIBRIA

Journal Article

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2007, Volume 30, Issue 1, pp. 134 - 147

We consider numerical methods for resolving the dynamics of a Hamiltonian N-body problem subject to hard-sphere inequality constraints...

Hamiltonian systems | Impact dynamics | Backward error analysis | Hard spheres | Collision Verlet | Inequality constraints | Geometric integration | MATHEMATICS, APPLIED | MOLECULAR-DYNAMICS | inequality constraints | VARIATIONAL INTEGRATORS | ALGORITHM | POTENTIALS | hard spheres | geometric integration | collision Verlet | impact dynamics | MECHANICS | backward error analysis | PRIMITIVE MODEL | SIMULATIONS

Hamiltonian systems | Impact dynamics | Backward error analysis | Hard spheres | Collision Verlet | Inequality constraints | Geometric integration | MATHEMATICS, APPLIED | MOLECULAR-DYNAMICS | inequality constraints | VARIATIONAL INTEGRATORS | ALGORITHM | POTENTIALS | hard spheres | geometric integration | collision Verlet | impact dynamics | MECHANICS | backward error analysis | PRIMITIVE MODEL | SIMULATIONS

Journal Article

Nonlinearity, ISSN 0951-7715, 04/2015, Volume 28, Issue 4, pp. 871 - 900

In this paper, we will discuss new developments regarding the geometric nonholonomic integrator (GNI...

70-08 | 65P10 | affine constraints Mathematics Subject Classification: 70F25 | discrete variational calculus | 37M15 | geometric nonholonomic integrator | 37N05 | 37J60 | reduction by symmetries | nonholonomic mechanics | BALL | MATHEMATICS, APPLIED | MOLECULAR-DYNAMICS | RIGID-BODY | LAGRANGIAN SYSTEMS | PHYSICS, MATHEMATICAL | HAMILTONIZATION | MECHANICS | MOTION | MODELS | affine constraints | PLANE | HIERARCHY | Preservation | Dynamics | Mathematical analysis | Tables (data) | Nonlinearity | Integrators | Dynamical systems | Convergence

70-08 | 65P10 | affine constraints Mathematics Subject Classification: 70F25 | discrete variational calculus | 37M15 | geometric nonholonomic integrator | 37N05 | 37J60 | reduction by symmetries | nonholonomic mechanics | BALL | MATHEMATICS, APPLIED | MOLECULAR-DYNAMICS | RIGID-BODY | LAGRANGIAN SYSTEMS | PHYSICS, MATHEMATICAL | HAMILTONIZATION | MECHANICS | MOTION | MODELS | affine constraints | PLANE | HIERARCHY | Preservation | Dynamics | Mathematical analysis | Tables (data) | Nonlinearity | Integrators | Dynamical systems | Convergence

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

Journal of Computational Physics, ISSN 0021-9991, 01/2014, Volume 257, pp. 1423 - 1443

This paper formulates variational integrators for finite element discretizations of deformable bodies with heat conduction in the form of finite speed thermal waves...

Symplectic methods | Variational integrators | Thermo-elasticity | Geometric integration | PRINCIPLE | FORMULATION | PHYSICS, MATHEMATICAL | LAGRANGIAN MECHANICS | MOMENTUM CONSERVING ALGORITHMS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | NONLINEAR DYNAMICS | THERMODYNAMICS | SYSTEMS | MEDIA | ENERGY-DISSIPATION | HEAT-CONDUCTION | Thermodynamics | Algorithms | Finite element method | Construction | Discretization | Mathematical analysis | Integrators | Entropy | Three dimensional | Laws

Symplectic methods | Variational integrators | Thermo-elasticity | Geometric integration | PRINCIPLE | FORMULATION | PHYSICS, MATHEMATICAL | LAGRANGIAN MECHANICS | MOMENTUM CONSERVING ALGORITHMS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | NONLINEAR DYNAMICS | THERMODYNAMICS | SYSTEMS | MEDIA | ENERGY-DISSIPATION | HEAT-CONDUCTION | Thermodynamics | Algorithms | Finite element method | Construction | Discretization | Mathematical analysis | Integrators | Entropy | Three dimensional | Laws

Journal Article

International Journal for Numerical Methods in Engineering, ISSN 0029-5981, 08/2013, Volume 95, Issue 7, pp. 562 - 586

SUMMARYThis article presents asynchronous collision integrators and a simple asynchronous method treating nodal restraints...

variational methods | Hamiltonian | nonlinear dynamics | time integration, explicit | contact | Nonlinear dynamics | Variational methods | Time integration, explicit | Contact | MOLECULAR-DYNAMICS | VARIATIONAL INTEGRATORS | CONSERVING FRAMEWORK | STABILITY | ALGORITHM | explicit | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | TRANSIENT IMPACT PROBLEMS | DISCRETE MECHANICS | time integration | SYSTEMS | IMPROVED IMPLICIT INTEGRATORS | Analysis | Algorithms | Numerical analysis | Constraints | Friction | Mathematical analysis | Projection | Mathematical models | Integrators

variational methods | Hamiltonian | nonlinear dynamics | time integration, explicit | contact | Nonlinear dynamics | Variational methods | Time integration, explicit | Contact | MOLECULAR-DYNAMICS | VARIATIONAL INTEGRATORS | CONSERVING FRAMEWORK | STABILITY | ALGORITHM | explicit | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | TRANSIENT IMPACT PROBLEMS | DISCRETE MECHANICS | time integration | SYSTEMS | IMPROVED IMPLICIT INTEGRATORS | Analysis | Algorithms | Numerical analysis | Constraints | Friction | Mathematical analysis | Projection | Mathematical models | Integrators

Journal Article

Computational Mechanics, ISSN 0178-7675, 11/2008, Volume 42, Issue 6, pp. 825 - 836

Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere...

Engineering | Mechanics, Fluids, Thermodynamics | Rigid body | Variational method | Computational Science and Engineering | Theoretical and Applied Mechanics | Time integration | Rotations | Geometric integration | ENERGY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | rigid body | variational method | MOMENTUM | DYNAMICS | time integration | SYSTEMS | ALGORITHMS | rotations | geometric integration | Rigid structures | Algorithms | Lagrange multipliers | Rotating bodies | Angular momentum | Nonlinearity | Integrators | Mechanical systems | Configurations

Engineering | Mechanics, Fluids, Thermodynamics | Rigid body | Variational method | Computational Science and Engineering | Theoretical and Applied Mechanics | Time integration | Rotations | Geometric integration | ENERGY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | rigid body | variational method | MOMENTUM | DYNAMICS | time integration | SYSTEMS | ALGORITHMS | rotations | geometric integration | Rigid structures | Algorithms | Lagrange multipliers | Rotating bodies | Angular momentum | Nonlinearity | Integrators | Mechanical systems | Configurations

Journal Article

Japan Journal of Industrial and Applied Mathematics, ISSN 0916-7005, 8/2017, Volume 34, Issue 2, pp. 441 - 472

We present novel geometric numerical integrators for Hunter–Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems...

Computational Mathematics and Numerical Analysis | 37M15 | 65M06 | Modified Hunter–Saxton equation | Mathematics | 35Q53 | 37K10 | Geometric numerical integration | Multi-symplectic formulation | Numerical discretisation | Discrete variational derivative method | Two-component Hunter–Saxton equation | Multi-symplectic schemes | Applications of Mathematics | Hunter–Saxton equation | Euler box scheme | SYSTEM | DISSIPATION | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | CAMASSA-HOLM EQUATION | DISPERSION LIMITS | PDES | DIFFERENCE-SCHEMES | Modified Hunter-Saxton equation | HYPERBOLIC VARIATIONAL EQUATION | Hunter-Saxton equation | Two-component Hunter-Saxton equation | ZERO-VISCOSITY | MULTI-SYMPLECTIC INTEGRATION | Information science | Matematik

Computational Mathematics and Numerical Analysis | 37M15 | 65M06 | Modified Hunter–Saxton equation | Mathematics | 35Q53 | 37K10 | Geometric numerical integration | Multi-symplectic formulation | Numerical discretisation | Discrete variational derivative method | Two-component Hunter–Saxton equation | Multi-symplectic schemes | Applications of Mathematics | Hunter–Saxton equation | Euler box scheme | SYSTEM | DISSIPATION | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | CAMASSA-HOLM EQUATION | DISPERSION LIMITS | PDES | DIFFERENCE-SCHEMES | Modified Hunter-Saxton equation | HYPERBOLIC VARIATIONAL EQUATION | Hunter-Saxton equation | Two-component Hunter-Saxton equation | ZERO-VISCOSITY | MULTI-SYMPLECTIC INTEGRATION | Information science | Matematik

Journal Article

Frontiers of Mathematics in China, ISSN 1673-3452, 4/2012, Volume 7, Issue 2, pp. 273 - 303

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact...

variational integrator | 65P10 | geometric mechanics | Mathematics, general | Mathematics | Lagrangian mechanics | symplectic integrator | 37J10 | Geometric numerical integration | 70H25 | MATHEMATICS | FULL BODY PROBLEM | NUMERICAL INTEGRATORS | EQUATIONS | BACKWARD ERROR ANALYSIS | Studies | Mathematical analysis | Lagrange multiplier | Accuracy | Construction | Approximation | Hamilton-Jacobi equation | Function space | Integrators | Mathematical models | Quadratures

variational integrator | 65P10 | geometric mechanics | Mathematics, general | Mathematics | Lagrangian mechanics | symplectic integrator | 37J10 | Geometric numerical integration | 70H25 | MATHEMATICS | FULL BODY PROBLEM | NUMERICAL INTEGRATORS | EQUATIONS | BACKWARD ERROR ANALYSIS | Studies | Mathematical analysis | Lagrange multiplier | Accuracy | Construction | Approximation | Hamilton-Jacobi equation | Function space | Integrators | Mathematical models | Quadratures

Journal Article

International journal for numerical methods in engineering, ISSN 0029-5981, 2012, Volume 90, Issue 3, pp. 390 - 402

In the context of Hamiltonian ODEs, a necessary condition for an integrator to be symplectic or conjugate-symplectic is that it nearly preserves the exact Hamiltonian...

variational methods | geometric methods | electromechanical | long‐time stability | rigid body dynamics | Geometric methods | Electromechanical | Long-time stability | Rigid body dynamics | Variational methods | RUNGE-KUTTA METHODS | long-time stability | MOMENTUM | VARIATIONAL INTEGRATORS | ROTATIONAL-DYNAMICS | ALGORITHMS | SPLITTING METHODS | LIE-GROUPS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | HAMILTONIAN-SYSTEMS | SYMPLECTIC INTEGRATION | MANIFOLDS

variational methods | geometric methods | electromechanical | long‐time stability | rigid body dynamics | Geometric methods | Electromechanical | Long-time stability | Rigid body dynamics | Variational methods | RUNGE-KUTTA METHODS | long-time stability | MOMENTUM | VARIATIONAL INTEGRATORS | ROTATIONAL-DYNAMICS | ALGORITHMS | SPLITTING METHODS | LIE-GROUPS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | HAMILTONIAN-SYSTEMS | SYMPLECTIC INTEGRATION | MANIFOLDS

Journal Article

SIAM Journal on Applied Dynamical Systems, ISSN 1536-0040, 08/2003, Volume 2, Issue 3, pp. 381 - 416

.... This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory...

Collisions | Discrete mechanics | Variational integrators | MATHEMATICS, APPLIED | ENERGY | discrete mechanics | collisions | APPROXIMATION | ALGORITHM | TIME | MODEL | FORMULATION | PHYSICS, MATHEMATICAL | variational integrators | DRY FRICTION | RIGID-BODY DYNAMICS | CONTACT PROBLEMS | CONSTRAINTS

Collisions | Discrete mechanics | Variational integrators | MATHEMATICS, APPLIED | ENERGY | discrete mechanics | collisions | APPROXIMATION | ALGORITHM | TIME | MODEL | FORMULATION | PHYSICS, MATHEMATICAL | variational integrators | DRY FRICTION | RIGID-BODY DYNAMICS | CONTACT PROBLEMS | CONSTRAINTS

Journal Article

Celestial Mechanics and Dynamical Astronomy, ISSN 0923-2958, 6/2007, Volume 98, Issue 2, pp. 121 - 144

.... The discrete equations of motion, referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method for simulating full body dynamics in orbital mechanics...

Lie group method | Full rigid body problem | Mathematics, general | Variational integrator | Symplectic integrator | Physics | Astronomy | variational integrator | full rigid body problem | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ASTRONOMY & ASTROPHYSICS | DYNAMICS | symplectic integrator | Studies | Algorithms

Lie group method | Full rigid body problem | Mathematics, general | Variational integrator | Symplectic integrator | Physics | Astronomy | variational integrator | full rigid body problem | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ASTRONOMY & ASTROPHYSICS | DYNAMICS | symplectic integrator | Studies | Algorithms

Journal Article

Journal of Nonlinear Science, ISSN 0938-8974, 4/2009, Volume 19, Issue 2, pp. 153 - 177

...-convergence of variational integrators to the corresponding continuum action functional and the convergence properties of solutions of the discrete Euler...

65P99 | 37M15 | Mathematical and Computational Physics | Mathematics | 37J45 | 49J45 | 70E55 | Analysis | 70F20 | Appl.Mathematics/Computational Methods of Engineering | Mechanics | Economic Theory | Constrained systems | Variational integrators | Gamma-convergence | ENERGY-CONSISTENT INTEGRATION | MATHEMATICS, APPLIED | CONSERVING INTEGRATION | MECHANICS | MECHANICAL SYSTEMS | DYNAMICS | NULL SPACE METHOD | PHYSICS, MATHEMATICAL

65P99 | 37M15 | Mathematical and Computational Physics | Mathematics | 37J45 | 49J45 | 70E55 | Analysis | 70F20 | Appl.Mathematics/Computational Methods of Engineering | Mechanics | Economic Theory | Constrained systems | Variational integrators | Gamma-convergence | ENERGY-CONSISTENT INTEGRATION | MATHEMATICS, APPLIED | CONSERVING INTEGRATION | MECHANICS | MECHANICAL SYSTEMS | DYNAMICS | NULL SPACE METHOD | PHYSICS, MATHEMATICAL

Journal Article

Multiscale Modeling and Simulation, ISSN 1540-3459, 2004, Volume 2, Issue 1, pp. 1 - 21

.... However, current time stepping integrators are not able to address the time scale problems...

Long molecular dynamics simulations | Targeted Langevin stabilization | Verlet-I/reversible REference System Propagator Algorithm | Mollified impulse method | Multiple time stepping | Self-consistent dissipative leapfrog | ELECTROSTATIC INTERACTIONS | targeted Langevin stabilization | multiple time stepping | DISSIPATIVE PARTICLE DYNAMICS | mollified impulse method | BOUNDARY-CONDITIONS | VARIATIONAL INTEGRATORS | long molecular dynamics simulations | LARGE SYSTEMS | PHYSICS, MATHEMATICAL | MESH EWALD | OSCILLATORY DIFFERENTIAL-EQUATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | LANGEVIN DYNAMICS | self-consistent dissipative leapfrog | COMPUTER-SIMULATION | SYMPLECTIC INTEGRATORS

Long molecular dynamics simulations | Targeted Langevin stabilization | Verlet-I/reversible REference System Propagator Algorithm | Mollified impulse method | Multiple time stepping | Self-consistent dissipative leapfrog | ELECTROSTATIC INTERACTIONS | targeted Langevin stabilization | multiple time stepping | DISSIPATIVE PARTICLE DYNAMICS | mollified impulse method | BOUNDARY-CONDITIONS | VARIATIONAL INTEGRATORS | long molecular dynamics simulations | LARGE SYSTEMS | PHYSICS, MATHEMATICAL | MESH EWALD | OSCILLATORY DIFFERENTIAL-EQUATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | LANGEVIN DYNAMICS | self-consistent dissipative leapfrog | COMPUTER-SIMULATION | SYMPLECTIC INTEGRATORS

Journal Article