Numerische Mathematik, ISSN 0029-599X, 1/2018, Volume 138, Issue 1, pp. 219 - 239

We prove strong convergence of order $$1/4-\epsilon $$ 1 / 4 - ϵ for arbitrarily small $$\epsilon >0$$ ϵ > 0 of the Euler–Maruyama method for multidimensional...

Secondary 65L20 | Stochastic differential equations | Discontinuous drift | Theoretical, Mathematical and Computational Physics | Mathematics | 65C20 | Euler–Maruyama method | Mathematical Methods in Physics | 65C30 | Strong convergence rate | Numerical Analysis | Mathematical and Computational Engineering | Mathematics, general | Degenerate diffusion | Numerical and Computational Physics, Simulation | Primary 60H10 | Euler-Maruyama method | MATHEMATICS, APPLIED | STOCHASTIC DIFFERENTIAL-EQUATIONS | NUMERICAL-METHOD | APPROXIMATIONS | REGULARITY | EXACT SIMULATION

Secondary 65L20 | Stochastic differential equations | Discontinuous drift | Theoretical, Mathematical and Computational Physics | Mathematics | 65C20 | Euler–Maruyama method | Mathematical Methods in Physics | 65C30 | Strong convergence rate | Numerical Analysis | Mathematical and Computational Engineering | Mathematics, general | Degenerate diffusion | Numerical and Computational Physics, Simulation | Primary 60H10 | Euler-Maruyama method | MATHEMATICS, APPLIED | STOCHASTIC DIFFERENTIAL-EQUATIONS | NUMERICAL-METHOD | APPROXIMATIONS | REGULARITY | EXACT SIMULATION

Journal Article

NUMERISCHE MATHEMATIK, ISSN 0029-599X, 02/2020, Volume 144, Issue 2, pp. 357 - 373

We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local...

53C30 | MATHEMATICS, APPLIED | Secondary 22F30 | RUNGE-KUTTA METHODS | Primary 65L20

53C30 | MATHEMATICS, APPLIED | Secondary 22F30 | RUNGE-KUTTA METHODS | Primary 65L20

Journal Article

Numerische Mathematik, ISSN 0029-599X, 3/2017, Volume 135, Issue 3, pp. 833 - 873

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and analyzed for solving the Klein–Gordon–Schrödinger (KGS) equations in the...

65L05 | Mathematical Methods in Physics | 65L70 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Mathematics, general | 65L20 | Mathematics | Numerical and Computational Physics, Simulation

65L05 | Mathematical Methods in Physics | 65L70 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Mathematics, general | 65L20 | Mathematics | Numerical and Computational Physics, Simulation

Journal Article

Numerical Algorithms, ISSN 1017-1398, 2/2015, Volume 68, Issue 2, pp. 337 - 354

In this paper, we present the two-step trigonometrically fitted symmetric Obrechkoff methods with algebraic order of twelve. The method is based on the...

Numeric Computing | Oscillating solution | MSC 65l05 | Theory of Computation | Symmetric multistep methods | Algorithms | Algebra | Numerical Analysis | Computer Science | MSC 65l07 | Trigonometrically-fitting | 65l20 | Obrechkoff methods | Initial value problems | MATHEMATICS, APPLIED | MULTISTEP METHODS | INTEGRATION | P-STABLE METHODS | MULTIDERIVATIVE METHODS | SUPER-IMPLICIT METHODS | 1ST-ORDER | Orbitals | Construction | Numerical analysis | Stability | Mathematical models | Symmetry | Mathematics - Numerical Analysis

Numeric Computing | Oscillating solution | MSC 65l05 | Theory of Computation | Symmetric multistep methods | Algorithms | Algebra | Numerical Analysis | Computer Science | MSC 65l07 | Trigonometrically-fitting | 65l20 | Obrechkoff methods | Initial value problems | MATHEMATICS, APPLIED | MULTISTEP METHODS | INTEGRATION | P-STABLE METHODS | MULTIDERIVATIVE METHODS | SUPER-IMPLICIT METHODS | 1ST-ORDER | Orbitals | Construction | Numerical analysis | Stability | Mathematical models | Symmetry | Mathematics - Numerical Analysis

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 03/2015, Volume 35, Issue 2, pp. 698 - 721

A two-point boundary value problem whose highest order term is a Caputo fractional derivative of order delta is an element of (1, 2) is considered. Al-Refai's...

fractional differential equation | boundary value problem | convergence proof | Caputo fractional derivative | finite difference method | derivative bounds | PIECEWISE POLYNOMIAL COLLOCATION | MATHEMATICS, APPLIED | EQUATIONS | Intervals | Boundary value problems | Stability | Mathematical analysis | Mathematical models | Derivatives | Convergence | Finite difference method | Mathematics - Numerical Analysis

fractional differential equation | boundary value problem | convergence proof | Caputo fractional derivative | finite difference method | derivative bounds | PIECEWISE POLYNOMIAL COLLOCATION | MATHEMATICS, APPLIED | EQUATIONS | Intervals | Boundary value problems | Stability | Mathematical analysis | Mathematical models | Derivatives | Convergence | Finite difference method | Mathematics - Numerical Analysis

Journal Article

Foundations of Computational Mathematics, ISSN 1615-3375, 2/2016, Volume 16, Issue 1, pp. 151 - 181

We rigorously study a novel type of trigonometric Fourier collocation methods for solving multi-frequency oscillatory second-order ordinary differential...

65L05 | Trigonometric Fourier collocation methods | 65P10 | Symplectic methods | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Quadratic invariant | Second-order ordinary differential equations | Multi-frequency oscillatory Hamiltonian systems | Numerical Analysis | Variation-of-constants formula | 65L20 | Applications of Mathematics | Math Applications in Computer Science | Multi-frequency oscillatory systems | Computer Science, general | Economics, general | 65M20 | MATRIX ARGUMENT | MATHEMATICS, APPLIED | RUNGE-KUTTA METHODS | 2ND-ORDER DIFFERENTIAL-EQUATIONS | CONSTRAINED HAMILTONIAN-SYSTEMS | ENERGY-CONSERVATION | HYPERGEOMETRIC-FUNCTIONS | MATHEMATICS | PRESERVING INTEGRATORS | NYSTROM METHODS | ARKN METHODS | COMPUTER SCIENCE, THEORY & METHODS | PERTURBED OSCILLATORS | Hamiltonian systems | Usage | Trigonometry | Analysis | Differential equations | Mathematical analysis | Ordinary differential equations | Collocation methods | Runge-Kutta method | Hamiltonian functions | Matrix methods | Methods | Texts | Fourier analysis | Mathematical models | Formulas (mathematics) | Oscillators

65L05 | Trigonometric Fourier collocation methods | 65P10 | Symplectic methods | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Quadratic invariant | Second-order ordinary differential equations | Multi-frequency oscillatory Hamiltonian systems | Numerical Analysis | Variation-of-constants formula | 65L20 | Applications of Mathematics | Math Applications in Computer Science | Multi-frequency oscillatory systems | Computer Science, general | Economics, general | 65M20 | MATRIX ARGUMENT | MATHEMATICS, APPLIED | RUNGE-KUTTA METHODS | 2ND-ORDER DIFFERENTIAL-EQUATIONS | CONSTRAINED HAMILTONIAN-SYSTEMS | ENERGY-CONSERVATION | HYPERGEOMETRIC-FUNCTIONS | MATHEMATICS | PRESERVING INTEGRATORS | NYSTROM METHODS | ARKN METHODS | COMPUTER SCIENCE, THEORY & METHODS | PERTURBED OSCILLATORS | Hamiltonian systems | Usage | Trigonometry | Analysis | Differential equations | Mathematical analysis | Ordinary differential equations | Collocation methods | Runge-Kutta method | Hamiltonian functions | Matrix methods | Methods | Texts | Fourier analysis | Mathematical models | Formulas (mathematics) | Oscillators

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 5/2018, Volume 75, Issue 2, pp. 1040 - 1056

Explicit Runge–Kutta methods are standard tools in the numerical solution of ordinary differential equations (ODEs). Applying the method of lines to partial...

Strong-stability preserving | 65L06 | Computational Mathematics and Numerical Analysis | Algorithms | Runge–Kutta methods | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Strong stability | 65L20 | Mathematics | 65M12

Strong-stability preserving | 65L06 | Computational Mathematics and Numerical Analysis | Algorithms | Runge–Kutta methods | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Strong stability | 65L20 | Mathematics | 65M12

Journal Article

JOURNAL OF SCIENTIFIC COMPUTING, ISSN 0885-7474, 08/2019, Volume 80, Issue 2, pp. 784 - 807

In this paper strong stability preserving (SSP) properties of Runge-Kutta methods obtained by composing k different schemes with different step sizes are...

65L05 | 65L06 | MATHEMATICS, APPLIED | Radius of absolute monotonicity | STEPSIZE RESTRICTIONS | Initial value problem | Strong stability preserving | ABSOLUTE MONOTONICITY | CONTRACTIVITY | SSP | NUMERICAL-SOLUTION | 65L20 | Runge-Kutta | Composition method | Monotonicity | 65M20

65L05 | 65L06 | MATHEMATICS, APPLIED | Radius of absolute monotonicity | STEPSIZE RESTRICTIONS | Initial value problem | Strong stability preserving | ABSOLUTE MONOTONICITY | CONTRACTIVITY | SSP | NUMERICAL-SOLUTION | 65L20 | Runge-Kutta | Composition method | Monotonicity | 65M20

Journal Article

Calcolo, ISSN 0008-0624, 3/2017, Volume 54, Issue 1, pp. 117 - 140

This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge–Kutta–Nyström (ERKN) schemes for solving...

65L05 | Sixth-order extended Runge–Kutta–Nyström schemes | 34C15 | 65P10 | Multi-frequency oscillatory nonlinear Hamiltonian equations | Structure-preserving algorithms | Numerical Analysis | 65L20 | 65N40 | Mathematics | Theory of Computation | Symplectic and symmetric explicit schemes | MATHEMATICS | ORDER | 2ND-ORDER DIFFERENTIAL-EQUATIONS | STABILITY | KUTTA-NYSTROM METHODS | NUMERICAL-INTEGRATION | SYSTEMS | Sixth-order extended Runge-Kutta-Nystrom schemes | Analysis | Algorithms | Robustness (mathematics) | Efficiency | Mathematical analysis | Wave equations | Nonlinearity | Mathematical models | Runge-Kutta method | Symmetry

65L05 | Sixth-order extended Runge–Kutta–Nyström schemes | 34C15 | 65P10 | Multi-frequency oscillatory nonlinear Hamiltonian equations | Structure-preserving algorithms | Numerical Analysis | 65L20 | 65N40 | Mathematics | Theory of Computation | Symplectic and symmetric explicit schemes | MATHEMATICS | ORDER | 2ND-ORDER DIFFERENTIAL-EQUATIONS | STABILITY | KUTTA-NYSTROM METHODS | NUMERICAL-INTEGRATION | SYSTEMS | Sixth-order extended Runge-Kutta-Nystrom schemes | Analysis | Algorithms | Robustness (mathematics) | Efficiency | Mathematical analysis | Wave equations | Nonlinearity | Mathematical models | Runge-Kutta method | Symmetry

Journal Article

Journal of Difference Equations and Applications, ISSN 1023-6198, 03/2013, Volume 19, Issue 3, pp. 466 - 490

For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the classical explicit...

tamed Milstein method | one-sided Lipschitz condition | commutative noise | strong convergence | superlinearly growing coefficient | 65C20 | SCHEME | MATHEMATICS, APPLIED | 65L20 | EULER | 60H35 | STRONG-CONVERGENCE | Difference equations | Mathematical analysis | Differential equations | Exact solutions | Mathematical models | Computational efficiency | Stochasticity | Convergence

tamed Milstein method | one-sided Lipschitz condition | commutative noise | strong convergence | superlinearly growing coefficient | 65C20 | SCHEME | MATHEMATICS, APPLIED | 65L20 | EULER | 60H35 | STRONG-CONVERGENCE | Difference equations | Mathematical analysis | Differential equations | Exact solutions | Mathematical models | Computational efficiency | Stochasticity | Convergence

Journal Article

BIT Numerical Mathematics, ISSN 0006-3835, 03/2016, Volume 56, Issue 1, pp. 151 - 162

To access, purchase, authenticate, or subscribe to the full-text of this article, please visit this link: http://dx.doi.org/10.1007/s10543-015-0549-x In this...

Numerical methods for stochastic differential equations | Discontinuous drift | Stochastic differential equations | Monte Carlo method | Methods | Mathematics - Probability

Numerical methods for stochastic differential equations | Discontinuous drift | Stochastic differential equations | Monte Carlo method | Methods | Mathematics - Probability

Journal Article

Numerical Algorithms, ISSN 1017-1398, 8/2019, Volume 81, Issue 4, pp. 1253 - 1274

We show how to compute the optimal relative backward error for the numerical solution of the Dahlquist test problem by one-step methods. This is an example of...

65L05 | 65L04 | Numeric Computing | Backward error | Theory of Computation | Stiff IVP | Residual | Algorithms | Algebra | Numerical Analysis | Optimal control | Computer Science | 65L20 | 49M05 | NUMERICAL-METHODS | MATHEMATICS, APPLIED | DEFECT CONTROL | TOLERANCE PROPORTIONALITY | ERROR-CONTROL | Analysis

65L05 | 65L04 | Numeric Computing | Backward error | Theory of Computation | Stiff IVP | Residual | Algorithms | Algebra | Numerical Analysis | Optimal control | Computer Science | 65L20 | 49M05 | NUMERICAL-METHODS | MATHEMATICS, APPLIED | DEFECT CONTROL | TOLERANCE PROPORTIONALITY | ERROR-CONTROL | Analysis

Journal Article

Numerical Algorithms, ISSN 1017-1398, 8/2019, Volume 81, Issue 4, pp. 1547 - 1571

Many physical phenomena contain different scales. These phenomena can be modeled using partial differential equations (PDEs). Often, these PDEs can be split...

65D30 | 65L05 | Multirate schemes | Algorithms | Algebra | Numerical Analysis | Computer Science | Numeric Computing | 65L20 | Theory of Computation | Numerical method | Time integration | MATHEMATICS, APPLIED | Analysis | Methods | Differential equations

65D30 | 65L05 | Multirate schemes | Algorithms | Algebra | Numerical Analysis | Computer Science | Numeric Computing | 65L20 | Theory of Computation | Numerical method | Time integration | MATHEMATICS, APPLIED | Analysis | Methods | Differential equations

Journal Article

Journal of Numerical Mathematics, ISSN 1570-2820, 03/2019, Volume 27, Issue 1, pp. 23 - 36

In engineering, it is a common desire to couple existing simulation tools together into one big system by passing information from subsystems as parameters...

coupled problems | 65L06 | extrapolation of signals | convergence | 65L20 | 65G99 | balance correction | 65Y05 | co-simulation | stability | MATHEMATICS | MATHEMATICS, APPLIED | DYNAMIC ITERATION | Data exchange | Simulation | Convergence | Subsystems

coupled problems | 65L06 | extrapolation of signals | convergence | 65L20 | 65G99 | balance correction | 65Y05 | co-simulation | stability | MATHEMATICS | MATHEMATICS, APPLIED | DYNAMIC ITERATION | Data exchange | Simulation | Convergence | Subsystems

Journal Article

BIT Numerical Mathematics, ISSN 0006-3835, 3/2014, Volume 54, Issue 1, pp. 171 - 188

The dynamical low-rank approximation of time-dependent matrices is a low-rank factorization updating technique. It leads to differential equations for factors...

65L05 | Computational Mathematics and Numerical Analysis | Low-rank approximation | 15A23 | 65F30 | Numeric Computing | Matrix differential equations | Mathematics, general | 65L20 | Mathematics | Time-dependent matrices | Numerical integrator | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | Projectors | Rankings | Algorithms | Mathematical optimization | Analysis

65L05 | Computational Mathematics and Numerical Analysis | Low-rank approximation | 15A23 | 65F30 | Numeric Computing | Matrix differential equations | Mathematics, general | 65L20 | Mathematics | Time-dependent matrices | Numerical integrator | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | Projectors | Rankings | Algorithms | Mathematical optimization | Analysis

Journal Article

16.
Full Text
Strong convergence rates of probabilistic integrators for ordinary differential equations

Statistics and Computing, ISSN 0960-3174, 11/2019, Volume 29, Issue 6, pp. 1265 - 1283

Probabilistic integration of a continuous dynamical system is a way of systematically introducing discretisation error, at scales no larger than errors...

Statistics and Computing/Statistics Programs | 68W20 | Artificial Intelligence | Convergence rates | Uncertainty quantification | Statistical Theory and Methods | 37H10 | Statistics | 65C99 | Ordinary differential equations | 65L20 | Probabilistic numerical methods | Probability and Statistics in Computer Science | INVERSE PROBLEMS | STATISTICS & PROBABILITY | COMPUTER SCIENCE, THEORY & METHODS

Statistics and Computing/Statistics Programs | 68W20 | Artificial Intelligence | Convergence rates | Uncertainty quantification | Statistical Theory and Methods | 37H10 | Statistics | 65C99 | Ordinary differential equations | 65L20 | Probabilistic numerical methods | Probability and Statistics in Computer Science | INVERSE PROBLEMS | STATISTICS & PROBABILITY | COMPUTER SCIENCE, THEORY & METHODS

Journal Article

Numerical Algorithms, ISSN 1017-1398, 02/2019, Volume 80, Issue 2, pp. 469 - 483

This paper concerns the discrete time waveform relaxation (DWR) methods for ordinary differential equations (ODEs). We present a general algorithm of...

Discrete time waveform relaxation methods | Interpolation polynomial | Convergence | 65L05 | 65L06 | MATHEMATICS, APPLIED | 65L20 | SYSTEMS | Methods | Algorithms | Differential equations

Discrete time waveform relaxation methods | Interpolation polynomial | Convergence | 65L05 | 65L06 | MATHEMATICS, APPLIED | 65L20 | SYSTEMS | Methods | Algorithms | Differential equations

Journal Article

Numerische Mathematik, ISSN 0029-599X, 4/2018, Volume 138, Issue 4, pp. 839 - 867

We consider a damped linear hyperbolic system modeling the propagation of pressure waves in a network of pipes. Well-posedness is established via semi-group...

Mathematical Methods in Physics | 35L50 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | 35L05 | Mathematics, general | 65L20 | Mathematics | 65M60 | Numerical and Computational Physics, Simulation | SPACE | SEMIGROUPS | MATHEMATICS, APPLIED | DISCRETIZATION | DECAY | HYPERBOLIC-EQUATIONS | GALERKIN APPROXIMATIONS

Mathematical Methods in Physics | 35L50 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | 35L05 | Mathematics, general | 65L20 | Mathematics | 65M60 | Numerical and Computational Physics, Simulation | SPACE | SEMIGROUPS | MATHEMATICS, APPLIED | DISCRETIZATION | DECAY | HYPERBOLIC-EQUATIONS | GALERKIN APPROXIMATIONS

Journal Article

Statistics and Computing, ISSN 0960-3174, 7/2017, Volume 27, Issue 4, pp. 1065 - 1082

In this paper, we present a formal quantification of uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical...

Statistics and Computing/Statistics Programs | Numerical analysis | Inverse problems | Artificial Intelligence (incl. Robotics) | Probabilistic numerics | Uncertainty quantification | Statistical Theory and Methods | 65L20 | Statistics | 62F15 | Probability and Statistics in Computer Science | 65N75 | PARAMETER-ESTIMATION | MODELS | STATISTICS & PROBABILITY | ERROR | COMPUTER SCIENCE, THEORY & METHODS | Differential equations

Statistics and Computing/Statistics Programs | Numerical analysis | Inverse problems | Artificial Intelligence (incl. Robotics) | Probabilistic numerics | Uncertainty quantification | Statistical Theory and Methods | 65L20 | Statistics | 62F15 | Probability and Statistics in Computer Science | 65N75 | PARAMETER-ESTIMATION | MODELS | STATISTICS & PROBABILITY | ERROR | COMPUTER SCIENCE, THEORY & METHODS | Differential equations

Journal Article

20.
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Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes

Journal of Scientific Computing, ISSN 0885-7474, 3/2017, Volume 70, Issue 3, pp. 1042 - 1077

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme,...

Computational Mathematics and Numerical Analysis | Stochastic differential equations | Strong convergence | Split-step backward Milstein method | C-stability | Theoretical, Mathematical and Computational Physics | Mathematics | Algorithms | 65C30 | Appl.Mathematics/Computational Methods of Engineering | Projected Milstein method | Global monotonicity condition | Mean-square convergence | 65L20 | B-consistency | MATHEMATICS, APPLIED | DIFFERENTIAL-EQUATIONS | ITERATED ITO INTEGRALS | SIMULATION | LIPSCHITZ CONTINUOUS COEFFICIENTS | STRONG-CONVERGENCE | Analysis | Differential equations

Computational Mathematics and Numerical Analysis | Stochastic differential equations | Strong convergence | Split-step backward Milstein method | C-stability | Theoretical, Mathematical and Computational Physics | Mathematics | Algorithms | 65C30 | Appl.Mathematics/Computational Methods of Engineering | Projected Milstein method | Global monotonicity condition | Mean-square convergence | 65L20 | B-consistency | MATHEMATICS, APPLIED | DIFFERENTIAL-EQUATIONS | ITERATED ITO INTEGRALS | SIMULATION | LIPSCHITZ CONTINUOUS COEFFICIENTS | STRONG-CONVERGENCE | Analysis | Differential equations

Journal Article

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