Journal of Computational Physics, ISSN 0021-9991, 04/2017, Volume 335, pp. 846 - 864

We propose two new classes of time integrators for stiff DEs: the implicit exponential (IMEXP) and the hybrid exponential methods. In contrast to the existing...

Stiff differential equations | Implicit–explicit exponential | Preconditioner | Exponential integrators | RUNGE-KUTTA METHODS | APPROXIMATIONS | ODES | DIFFERENTIAL-EQUATIONS | HIGH-ORDER | PARABOLIC PROBLEMS | PHYSICS, MATHEMATICAL | ROSENBROCK METHODS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | PROPAGATION ITERATIVE METHODS | SYSTEMS | Implicit-explicit exponential | Rain-water (Water-supply)

Stiff differential equations | Implicit–explicit exponential | Preconditioner | Exponential integrators | RUNGE-KUTTA METHODS | APPROXIMATIONS | ODES | DIFFERENTIAL-EQUATIONS | HIGH-ORDER | PARABOLIC PROBLEMS | PHYSICS, MATHEMATICAL | ROSENBROCK METHODS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | PROPAGATION ITERATIVE METHODS | SYSTEMS | Implicit-explicit exponential | Rain-water (Water-supply)

Journal Article

Bulletin des sciences mathématiques, ISSN 0007-4497, 03/2019, Volume 151, pp. 51 - 65

We introduce a notion of p-rough integrator on any Banach manifolds which plays the role of weak geometric Hölder p-rough paths in the usual Banach space...

Banach manifolds | Rough paths theory | Canonical form | MATHEMATICS, APPLIED | SPACES | Differential Geometry | Mathematics | Classical Analysis and ODEs

Banach manifolds | Rough paths theory | Canonical form | MATHEMATICS, APPLIED | SPACES | Differential Geometry | Mathematics | Classical Analysis and ODEs

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. In the...

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

Mathematical and Computer Modelling, ISSN 0895-7177, 2004, Volume 40, Issue 11, pp. 1225 - 1244

An overview of Hamiltonian systems with noncanonical Poisson structures is given. Examples of bi-Hamiltonian ODEs, PDEs, and lattice equations are presented....

Bi-Hamiltonian systems | Symplectic integrators | Hamiltonian ODEs and PDEs | Lie-Poisson systems | Integrable discretizations | Nambu-Hamiltonian systems | MATHEMATICS, APPLIED | NONLINEAR SCHRODINGER-EQUATION | DISCRETIZATIONS | symplectic integrators | GENERALIZED NAMBU MECHANICS | KUTTA COLLOCATION METHODS | GEOMETRIC INTEGRATORS | CONSTRUCTION | MANIFOLDS | integrable discretizations | SCHEMES

Bi-Hamiltonian systems | Symplectic integrators | Hamiltonian ODEs and PDEs | Lie-Poisson systems | Integrable discretizations | Nambu-Hamiltonian systems | MATHEMATICS, APPLIED | NONLINEAR SCHRODINGER-EQUATION | DISCRETIZATIONS | symplectic integrators | GENERALIZED NAMBU MECHANICS | KUTTA COLLOCATION METHODS | GEOMETRIC INTEGRATORS | CONSTRUCTION | MANIFOLDS | integrable discretizations | SCHEMES

Journal Article

Computer Physics Communications, ISSN 0010-4655, 06/2019, Volume 239, pp. 33 - 52

Magnetic quadrupoles are essential components of particle accelerators like the Large Hadron Collider. In order to study numerically the stability of the...

Symplectic methods | High order ODE methods | Particle accelerators | Hamilton equations | Magnetic fields | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MATHEMATICAL | Analysis

Symplectic methods | High order ODE methods | Particle accelerators | Hamilton equations | Magnetic fields | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MATHEMATICAL | Analysis

Journal Article

International Journal of Bifurcation and Chaos, ISSN 0218-1274, 10/2012, Volume 22, Issue 10, pp. 1230033 - 1230033

The reader can find in the literature a lot of different techniques to study the dynamics of a given system and also, many suitable numerical integrators to...

variational equations | Chaos indicators | Hamiltonian systems | numerical integrators | ODEs | MULTIDISCIPLINARY SCIENCES | FAST LYAPUNOV INDICATOR | DIFFERENTIAL-EQUATIONS | PHASE-SPACE STRUCTURE | PERIODIC-ORBITS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | CHARACTERISTIC EXPONENTS | MOTION | GLOBAL DYNAMICS | HAMILTONIAN-SYSTEMS | TWIST ANGLES | RESONANCES | Analysis | Algorithms | Chaos theory | Dynamics | Mathematical analysis | Integrators | Mathematical models | Spectra | Dynamical systems | Indicators

variational equations | Chaos indicators | Hamiltonian systems | numerical integrators | ODEs | MULTIDISCIPLINARY SCIENCES | FAST LYAPUNOV INDICATOR | DIFFERENTIAL-EQUATIONS | PHASE-SPACE STRUCTURE | PERIODIC-ORBITS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | CHARACTERISTIC EXPONENTS | MOTION | GLOBAL DYNAMICS | HAMILTONIAN-SYSTEMS | TWIST ANGLES | RESONANCES | Analysis | Algorithms | Chaos theory | Dynamics | Mathematical analysis | Integrators | Mathematical models | Spectra | Dynamical systems | Indicators

Journal Article

ACM Transactions on Mathematical Software (TOMS), ISSN 0098-3500, 11/2012, Volume 39, Issue 1, pp. 1 - 28

This article introduces the software package TIDES and revisits the use of the Taylor series method for the numerical integration of ODEs. The package TIDES...

variational equations | automatic differentiation | high precision | numerical integration of ODEs | Taylor series method | Taylor series method, automatic differentiation, high precision, variational equations, numerical integration of ODEs | MATHEMATICS, APPLIED | IVP | PERFORMANCE | ODES | ALGEBRAIC EQUATIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | Algorithms | DAES | NUMERICAL-INTEGRATION | Usage | Research | Mathematical software | Series, Taylor's | Differential equations

variational equations | automatic differentiation | high precision | numerical integration of ODEs | Taylor series method | Taylor series method, automatic differentiation, high precision, variational equations, numerical integration of ODEs | MATHEMATICS, APPLIED | IVP | PERFORMANCE | ODES | ALGEBRAIC EQUATIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | Algorithms | DAES | NUMERICAL-INTEGRATION | Usage | Research | Mathematical software | Series, Taylor's | Differential equations

Journal Article

Symmetry, ISSN 2073-8994, 02/2019, Volume 11, Issue 2, p. 246

The primary contribution of this work is to develop direct processes of explicit Runge-Kutta type (RKT) as solutions for any fourth-order ordinary differential...

Runge-Kutta type methods | Order conditions | Quad-colored trees | B-series | Fourth-order ODEs | NUMERICAL-METHODS | LINEAR MULTISTEP METHOD | order conditions | MULTIDISCIPLINARY SCIENCES | quad-colored trees | fourth-order ODEs | ORDINARY DIFFERENTIAL-EQUATIONS | INITIAL-VALUE PROBLEMS

Runge-Kutta type methods | Order conditions | Quad-colored trees | B-series | Fourth-order ODEs | NUMERICAL-METHODS | LINEAR MULTISTEP METHOD | order conditions | MULTIDISCIPLINARY SCIENCES | quad-colored trees | fourth-order ODEs | ORDINARY DIFFERENTIAL-EQUATIONS | INITIAL-VALUE PROBLEMS

Journal Article

Foundations of Computational Mathematics, ISSN 1615-3375, 12/2010, Volume 10, Issue 6, pp. 673 - 693

B-series are a powerful tool in the analysis of Runge–Kutta numerical integrators and some of their generalizations (“B-series methods”). A general goal is to...

Trees | 65P10 | Economics general | B-series methods | 37M15 | Linear and Multilinear Algebras, Matrix Theory | Mathematics | 65D30 | Numerical Analysis | Energy preservation | Symplectic integration | Math Applications in Computer Science | Applications of Mathematics | Computer Science, general | 05C05 | Conjugate methods | Symplectic integration | Energy preservation | B-series methods | MATHEMATICS, APPLIED | RUNGE-KUTTA METHODS | ODES | MATHEMATICS | CONSERVATION | NUMERICAL INTEGRATORS | SYSTEMS | COMPUTER SCIENCE, THEORY & METHODS | SYMPLECTIC INTEGRATORS | Computational mathematics | Numerical analysis | Vector space | Linear algebra | Conjugates | Mathematical analysis | Preserves | Tools | Mathematical models | Runge-Kutta method | Integrators | Vectors (mathematics)

Trees | 65P10 | Economics general | B-series methods | 37M15 | Linear and Multilinear Algebras, Matrix Theory | Mathematics | 65D30 | Numerical Analysis | Energy preservation | Symplectic integration | Math Applications in Computer Science | Applications of Mathematics | Computer Science, general | 05C05 | Conjugate methods | Symplectic integration | Energy preservation | B-series methods | MATHEMATICS, APPLIED | RUNGE-KUTTA METHODS | ODES | MATHEMATICS | CONSERVATION | NUMERICAL INTEGRATORS | SYSTEMS | COMPUTER SCIENCE, THEORY & METHODS | SYMPLECTIC INTEGRATORS | Computational mathematics | Numerical analysis | Vector space | Linear algebra | Conjugates | Mathematical analysis | Preserves | Tools | Mathematical models | Runge-Kutta method | Integrators | Vectors (mathematics)

Journal Article

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Collisional N-body numerical integrator with applications to charged particle dynamics

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2018, Volume 40, Issue 6, pp. B1517 - B1540

We developed a novel collisional N-body numerical integrator, which we termed the Simo integrator, designed to model the electrostatic N-body problem of...

Coulomb collisions | N-body | Adaptive time stepping | Dense output | Picard iteration | Numerical methods | Odes | Differential algebra | Optimal time stepsize | Optimal order | Variable order | Taylor method | optimal time stepsize | MATHEMATICS, APPLIED | numerical methods | ODEs | ALGORITHM | TIME | adaptive time stepping | ORDER | TAYLOR-SERIES | dense output | variable order | SOLAR-SYSTEM | optimal order | differential algebra | ZEROS | Mathematics

Coulomb collisions | N-body | Adaptive time stepping | Dense output | Picard iteration | Numerical methods | Odes | Differential algebra | Optimal time stepsize | Optimal order | Variable order | Taylor method | optimal time stepsize | MATHEMATICS, APPLIED | numerical methods | ODEs | ALGORITHM | TIME | adaptive time stepping | ORDER | TAYLOR-SERIES | dense output | variable order | SOLAR-SYSTEM | optimal order | differential algebra | ZEROS | Mathematics

Journal Article

11.
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Geometric integrators for multiplicative viscoplasticity: Analysis of error accumulation

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 2010, Volume 199, Issue 9, pp. 700 - 711

The inelastic incompressibility is a typical feature of metal plasticity/viscoplasticity. Over the last decade, there has been a great amount of research...

Integration algorithm | Inelastic incompressibility | Viscoplasticity | Finite strains | Exponential stability | Contractivity | Error accumulation | FINITE THERMOVISCOPLASTICITY | ALGORITHM | TIME INTEGRATION | IMPLEMENTATION | INCOMPRESSIBILITY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | CONSTITUTIVE-EQUATIONS | PLASTIC-DEFORMATION | STRAIN

Integration algorithm | Inelastic incompressibility | Viscoplasticity | Finite strains | Exponential stability | Contractivity | Error accumulation | FINITE THERMOVISCOPLASTICITY | ALGORITHM | TIME INTEGRATION | IMPLEMENTATION | INCOMPRESSIBILITY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | CONSTITUTIVE-EQUATIONS | PLASTIC-DEFORMATION | STRAIN

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 2011, Volume 230, Issue 9, pp. 3413 - 3429

Implicit-explicit (IMEX) multistep methods are very useful for the time discretization of convection diffusion PDE problems such as the Burgers equations and...

Multistep methods | Semi-Lagrangian | Exponential integrators | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | NAVIER-STOKES EQUATIONS | RUNGE-KUTTA METHODS | APPROXIMATIONS | ODES | PHYSICS, MATHEMATICAL | SCHEMES | Plasma physics | Analysis | Discretization | Partial differential equations | Mathematical analysis | Integrators | Mathematical models | Burgers equation | Diffusion | Convection | Navier-Stokes equations

Multistep methods | Semi-Lagrangian | Exponential integrators | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | NAVIER-STOKES EQUATIONS | RUNGE-KUTTA METHODS | APPROXIMATIONS | ODES | PHYSICS, MATHEMATICAL | SCHEMES | Plasma physics | Analysis | Discretization | Partial differential equations | Mathematical analysis | Integrators | Mathematical models | Burgers equation | Diffusion | Convection | Navier-Stokes equations

Journal Article

Mathematics and Computers in Simulation, ISSN 0378-4754, 02/2016, Volume 120, pp. 104 - 119

Zhang-gradient (ZG) method is a combination of Zhang dynamics (ZD) and gradient dynamics (GD) methods which are two powerful methods for online time-varying...

Options | Zhang-gradient (ZG) controllers | Tracking control | Double-integrator system | MATLAB ODE solver | COMPUTER SCIENCE, SOFTWARE ENGINEERING | SYNCHRONIZATION | MATHEMATICS, APPLIED | DESIGN | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | NEURAL-NETWORKS | DYNAMICS

Options | Zhang-gradient (ZG) controllers | Tracking control | Double-integrator system | MATLAB ODE solver | COMPUTER SCIENCE, SOFTWARE ENGINEERING | SYNCHRONIZATION | MATHEMATICS, APPLIED | DESIGN | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | NEURAL-NETWORKS | DYNAMICS

Journal Article

Annals of Functional Analysis, ISSN 2008-8752, 2014, Volume 5, Issue 1, pp. 77 - 93

In this paper several new inequalities of Gruss' type for Riemann-Stieltjes integral with monotonic nondecreasing integrators under various assumptions for...

Functions of selfadjoint operators | Selfadjoint operators | Hölder continuous functions | Riemann-stieltjes integral | Functions of bounded variation | Grüss inequality | Gruss inequality | MATHEMATICS | MATHEMATICS, APPLIED | BOUNDS | Riemann-Stieltjes integral | Holder continuous functions

Functions of selfadjoint operators | Selfadjoint operators | Hölder continuous functions | Riemann-stieltjes integral | Functions of bounded variation | Grüss inequality | Gruss inequality | MATHEMATICS | MATHEMATICS, APPLIED | BOUNDS | Riemann-Stieltjes integral | Holder continuous functions

Journal Article

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2011, Volume 33, Issue 2, pp. 488 - 511

A new algorithm is developed for computing e(tA)B, where A is an n x n matrix and B is n x n(0) with n(0) << n. The algorithm works for any A, its...

φ functions | Chebyshev polynomial | Backward error analysis | Exponential integrator | Taylor series | Laguerre polynomial | Overscaling | Ordinary differential equation | MATLAB | Matrix exponential | Expm | ODE | Krylov method | Condition number | exponential integrator | MATHEMATICS, APPLIED | matrix exponential | APPROXIMATION | ordinary differential equation | ALGORITHM | SQUARING METHOD | phi functions | backward error analysis | overscaling | condition number | expm | Studies | Algorithms | Matrix

φ functions | Chebyshev polynomial | Backward error analysis | Exponential integrator | Taylor series | Laguerre polynomial | Overscaling | Ordinary differential equation | MATLAB | Matrix exponential | Expm | ODE | Krylov method | Condition number | exponential integrator | MATHEMATICS, APPLIED | matrix exponential | APPROXIMATION | ordinary differential equation | ALGORITHM | SQUARING METHOD | phi functions | backward error analysis | overscaling | condition number | expm | Studies | Algorithms | Matrix

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 11/2012, Volume 36, Issue 11, pp. 5466 - 5481

In this work, we investigate the application of polynomial integrators in a high-order discontinuous Galerkin method for solving the time-domain Maxwell...

Discontinuous Galerkin method | Time-stepping schemes | Faber polynomials | Maxwell’s equations | Maxwell's equations | APPROXIMATION | EQUATIONS | SOLVE | FABER | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | COMPUTATION | PROPAGATION | Electromagnetism | Algorithms | Analysis | Investigations | Methods | Differential equations | Mathematical analysis | Integrators | Mathematical models | Matrices | Electromagnetic fields | Galerkin methods | Modeling and Simulation | Mathematical Physics | Numerical Analysis | Computer Science | Complex Variables | Mathematics | Engineering Sciences | Classical Analysis and ODEs

Discontinuous Galerkin method | Time-stepping schemes | Faber polynomials | Maxwell’s equations | Maxwell's equations | APPROXIMATION | EQUATIONS | SOLVE | FABER | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | COMPUTATION | PROPAGATION | Electromagnetism | Algorithms | Analysis | Investigations | Methods | Differential equations | Mathematical analysis | Integrators | Mathematical models | Matrices | Electromagnetic fields | Galerkin methods | Modeling and Simulation | Mathematical Physics | Numerical Analysis | Computer Science | Complex Variables | Mathematics | Engineering Sciences | Classical Analysis and ODEs

Journal Article

Comptes rendus - Mécanique, ISSN 1631-0721, 10/2014, Volume 342, Issue 10-11, pp. 595 - 609

We change a previous time-stepping algorithm for solving a multi-scale Vlasov–Poisson system within a Particle-In-Cell method, in order to perform accurate...

Paraxial approximation | Particle-In-Cell method | Exponential time differencing | Highly oscillatory ODEs | Long-time simulation | Vlasov–Poisson system | Vlasov-Poisson system | MECHANICS | Algorithms | Modeling and Simulation | Mathematics | Plasma Physics | Numerical Analysis | Physics | Computer Science

Paraxial approximation | Particle-In-Cell method | Exponential time differencing | Highly oscillatory ODEs | Long-time simulation | Vlasov–Poisson system | Vlasov-Poisson system | MECHANICS | Algorithms | Modeling and Simulation | Mathematics | Plasma Physics | Numerical Analysis | Physics | Computer Science

Journal Article

Communications in Computational Physics, ISSN 1815-2406, 07/2015, Volume 18, Issue 2, pp. 263 - 296

With the aim of solving in a four dimensional phase space a multi-scale Vlasov-Poisson system, we propose in a Particle-In-Cell framework a robust...

highly oscillatory ODEs | Vlasov-Poisson system | Particle-In-Cell method | long-time simulation | Guiding-Center | SIMULATION | PHYSICS, MATHEMATICAL | EQUATION | SCHEMES | Modeling and Simulation | Analysis of PDEs | Numerical Analysis | Computer Science | Mathematics | Plasma Physics | Physics

highly oscillatory ODEs | Vlasov-Poisson system | Particle-In-Cell method | long-time simulation | Guiding-Center | SIMULATION | PHYSICS, MATHEMATICAL | EQUATION | SCHEMES | Modeling and Simulation | Analysis of PDEs | Numerical Analysis | Computer Science | Mathematics | Plasma Physics | Physics

Journal Article

SIAM JOURNAL ON SCIENTIFIC COMPUTING, ISSN 1064-8275, 2019, Volume 41, Issue 5, pp. A2911 - A2937

We present a methodology for constructing time integrators for solving systems of first-order ordinary differential equations by using interpolating...

approximation theory | MATHEMATICS, APPLIED | parallelism | high-order | ODES | polynomial interpolation | time integration

approximation theory | MATHEMATICS, APPLIED | parallelism | high-order | ODES | polynomial interpolation | time integration

Journal Article

Atmospheric Environment, ISSN 1352-2310, 2009, Volume 43, Issue 40, pp. 6287 - 6295

Air quality models compute the transformation of species in the atmosphere undergoing chemical and physical changes. The numerical algorithms used to predict...

Operator splitting | Chemical integrator | Mass conservation | Chemical kinetics | Positive definiteness | CHEMISTRY PROBLEMS | SCHEME | ENVIRONMENTAL SCIENCES | SYSTEMS | METEOROLOGY & ATMOSPHERIC SCIENCES | EQUATION | STIFF ODE SOLVERS

Operator splitting | Chemical integrator | Mass conservation | Chemical kinetics | Positive definiteness | CHEMISTRY PROBLEMS | SCHEME | ENVIRONMENTAL SCIENCES | SYSTEMS | METEOROLOGY & ATMOSPHERIC SCIENCES | EQUATION | STIFF ODE SOLVERS

Journal Article

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