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  • 1.
    Abdulle, Assyr
    et al.
    ANMC, EPFL.
    Cohen, David
    Institut für Angewandte und Numerische Mathematik, KIT.
    Vilmart, Gilles
    ANMC, EPFL.
    Konstantinos, Zygalakis
    ANMC, EPFL.
    High weak order methods for stochastic differential equations based on modified equations2012Inngår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 34, nr 3, s. A1800-A1823Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (mean-square stable) stochastic problems, and implicit integrators that exactly conserve all quadratic firstintegrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.

  • 2.
    Anton, Rikard
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Department of Mathematics, University of Innsbruck, A–6020 Innsbruck, Austria.
    Exponential integrators for stochastic Schrödinger equations driven by Itô noise2018Inngår i: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 36, nr 2, s. 276-309Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study an explicit exponential scheme for the time discretisation of stochastic Schr¨odinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schr¨odinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.

  • 3.
    Anton, Rikard
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Univ Innsbruck, Dept Math, Innsbruck, Austria.
    Larsson, Stig
    Wang, Xiaojie
    Full discretization of semilinear stochastic wave equations driven by multiplicative noise2016Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 54, nr 2, s. 1093-1119Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space, and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used Stormer-Verlet leapfrog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.

  • 4.
    Anton, Rikard
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Quer-Sardanyons, Lluis
    Department of Mathematics, Universitat Autònoma de Barcelona, Catalonia, Spain.
    A fully discrete approximation of the one-dimensional stochastic heat equation2020Inngår i: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 40, nr 1, s. 247-284Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space–time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz this explicit time integrator allows for error bounds in Lq(Ω), for all q ≥ 2, improving some existing results in the literature. On top of this we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.

  • 5.
    Berg, André
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cohen, David
    Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.
    Dujardin, Guillaume
    Inria Lille Nord-Europe and Laboratoire Paul Painlevé UMR CNRS 8524, Villeneuve d’Asq Cedex, France.
    Numerical study of nonlinear Schrödinger equations with white noise dispersionManuskript (preprint) (Annet vitenskapelig)
  • 6.
    Celledoni, Elena
    et al.
    Department of Mathematical Sciences, NTNU, 7491 Trondheim.
    Cohen, David
    Department of Mathematical Sciences, NTNU, 7491 Trondheim.
    Owren, Brynjulf
    Department of Mathematical Sciences, NTNU, 7491 Trondheim.
    Symmetric exponential integrators with an application to the cubic Schrodinger equation2008Inngår i: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 8, nr 3, s. 303-317Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this article, we derive and study symmetric exponential integrators. Numerical experiments are performed for the cubic Schrodinger equation and comparisons with classical exponential integrators and other geometric methods are also given. Some of the proposed methods preserve the L(2)-norm and/or the energy of the system.

  • 7. Chen, Chuchu
    et al.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    D'Ambrosio, Raffaele
    Lang, Annika
    Drift-preserving numerical integrators for stochastic Hamiltonian systems2020Inngår i: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 46, nr 2, artikkel-id 27Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.

    Fulltekst (pdf)
    fulltext
  • 8. Chen, Chuchu
    et al.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Hong, Jialin
    Conservative methods for stochastic differential equations with a conserved quantity2016Inngår i: International Journal of Numerical Analysis and Modeling, ISSN 1923-2942, Vol. 13, nr 3, s. 435-456Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This paper proposes a novel conservative method for the numerical approximation of general stochastic differential equations in the Stratonovich sense with a conserved quantity. We show that the mean-square order of the method is 1 if noises are commutative and that the weak order is 1 in the general case. Since the proposed method may need the computation of a deterministic integral, we analyse the effect of the use of quadrature formulas on the convergence orders. Furthermore, based on the splitting technique of stochastic vector fields, we construct conservative composition methods with similar orders as the above method. Finally, numerical experiments are presented to support our theoretical results.

  • 9.
    Cohen, David
    Section de Mathématiques, Université de Genève.
    Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems2006Inngår i: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 26, nr 1, s. 34-59Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Modulated Fourier expansion is used to show long-time near-conservation of the total and oscillatory energies of numerical methods for Hamiltonian systems with highly oscillatory solutions. The numerical methods considered are an extension of the trigonometric methods. A brief discussion of conservation properties in the continuous problem and in the multi-frequency case is also given.

  • 10.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Exponential integrators for stochastic Schrödinger equations driven by Itô noise2017Inngår i: Oberwolfach mini-Workshop: Stochastic Differential Equations:Regularity and Numerical Analysis in Finite and InfiniteDimensions: Workshop ID: 1706b, Oberwolfach, 2017, s. 11-12Konferansepaper (Annet vitenskapelig)
  • 11.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Numerical discretisations of stochastic wave equations2016Inngår i: Geometric Numerical Integration: Workshop Report, Oberwolfach, 2016, Vol. 18, s. 897-899Konferansepaper (Annet vitenskapelig)
  • 12.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Numerical discretisations of stochastic wave equations2018Inngår i: International conference of numerical analysis and applied mathematics (ICNAAM 2017), American Institute of Physics (AIP), 2018, Vol. 1978, s. 1-5, artikkel-id 1Konferansepaper (Fagfellevurdert)
    Abstract [en]

    This extended abstract starts with a brief introduction to stochastic partial differential equations with a particular focus on stochastic wave equations. Various numerical experiments for this stochastic partial differential equation are presented. Finally, we point out results from the literature on the numerical analysis of stochastic wave equations.

    Fulltekst (pdf)
    fulltext
  • 13.
    Cohen, David
    Department of mathematics, University of Basel.
    On the numerical discretisation of stochastic oscillators2012Inngår i: Mathematics and Computers in Simulation, ISSN 0378-4754, E-ISSN 1872-7166, Vol. 82, nr 8, s. 1478-1495Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this article, we propose an approach, based on the variation-of-constants formula, for the numerical discretisation over long-time intervals of several stochastic oscillators. Additive and multiplicative noises are treated separately. The proposed schemes permit the use of large step sizes in the presence of a high frequency in the problem and offer various additional properties. These new numerical integrators can be viewed as a stochastic-generalisation of the trigonometric integrators for highly oscillatory deterministic problems. (C) 2012 IMACS. Published by Elsevier B.V. All rights reserved.

  • 14.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Trigonometric schemes for stiff second-order SDEs2006Konferansepaper (Annet vitenskapelig)
  • 15.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Trigonometric schemes for stiff second-order SDEs2011Konferansepaper (Annet vitenskapelig)
  • 16.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cui, Jianbo
    Hong, Jialin
    Sun, Liying
    Exponential integrators for stochastic Maxwell's equations driven by Itô noise2020Inngår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 410, artikkel-id 109382Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This article presents explicit exponential integrators for stochastic Maxwell's equations driven by both multiplicative and additive noises. By utilizing the regularity estimate of the mild solution, we first prove that the strong order of the numerical approximation is ½ for general multiplicative noise. Combining a proper decomposition with the stochastic Fubini's theorem, the strong order of the proposed scheme is shown to be 1 for additive noise. Moreover, for linear stochastic Maxwell's equation with additive noise, the proposed time integrator is shown to preserve exactly the symplectic structure, the evolution of the energy as well as the evolution of the divergence in the sense of expectation. Several numerical experiments are presented in order to verify our theoretical findings.

  • 17.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.
    Debrabant, Kristian
    Department of Mathematics and Computer Science, University of Southern Denmark, Denmark.
    Rößler, Andreas
    Institut für Mathematik, Universität zu Lübeck, Germany.
    High order numerical integrators for single integrand Stratonovich SDEs2020Inngår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 158, s. 264-270Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We show that applying any deterministic B-series method of order pdwith a random step size to single integrand SDEs gives a numerical method converging in the mean-square and weak sense with order [Pd/2]. As an application, we derive high order energy-preserving methods for stochastic Poisson systems as well as further geometric numerical schemes for this wide class of Stratonovich SDEs.

    Fulltekst (pdf)
    fulltext
  • 18.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Dujardin, Guillaume
    Inria Lille Nord-Europe.
    Energy-preserving integrators for stochastic Poisson systems2014Inngår i: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 12, nr 8, s. 1523-1539Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A new class of energy-preserving numerical schemes for stochastic Hamiltonian systems with non-canonical structure matrix (in the Stratonovich sense) is proposed. These numerical integrators are of mean-square order one and also preserve quadratic Casimir functions. In the deterministic setting, our schemes reduce to methods proposed in [E. Hairer, JNAIAM. J. Numer. Anal. Ind. Appl. Math., 5(1-2), 73–84, 2011] and [D. Cohen, and E. Hairer, BIT, 51(1), 91–101, 2011].

  • 19.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Department of Mathematics, University of Innsbruck, Innsbruck, Austria.
    Dujardin, Guillaume
    Exponential integrators for nonlinear Schrödinger equations with white noise dispersion2017Inngår i: Stochastics and Partial Differential Equations: Analysis and Computations, ISSN 2194-0401, Vol. 5, nr 4, s. 592-613Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L2-norm of the solution.

    Fulltekst (pdf)
    fulltext
  • 20.
    Cohen, David
    et al.
    Mathematisches Institut, Universität Basel.
    Gauckler, Ludwig
    Institut für Mathematik, TU Berlin.
    One-stage exponential integrators for nonlinear Schrödinger equations over long times2012Inngår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 52, nr 4, s. 877-903Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Near-conservation over long times of the actions, of the energy, of the mass and of the momentum along the numerical solution of the cubic Schrödinger equation with small initial data is shown. Spectral discretization in space and one-stage exponential integrators in time are used. The proofs use modulated Fourier expansions.

  • 21.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Gauckler, Ludwig
    TU Berlin, Germany.
    Hairer, Ernst
    University of Geneva, Switzerland.
    Lubich, Christian
    University of Tübingen, Germany.
    Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions2015Inngår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 55, nr 3, s. 705-732Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    For trigonometric and modified trigonometric integrators applied to oscillatory Hamiltonian differential equations with one or several constant high frequencies, near-conservation of the total and oscillatory energies are shown over time scales that cover arbitrary negative powers of the step size. This requires non-resonance conditions between the step size and the frequencies, but in contrast to previous results the results do not require any non-resonance conditions among the frequencies. The proof uses modulated Fourier expansions with appropriately modified frequencies.

  • 22.
    Cohen, David
    et al.
    Mathematisches Institut, Universität Basel.
    Hairer, Ernst
    Section de Mathématiques, Université de Genève.
    Linear energy-preserving integrators for Poisson systems2011Inngår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, nr 1, s. 91-101Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    For Hamiltonian systems with non-canonical structure matrix a new class of numerical integrators is proposed. The methods exactly preserve energy, are invariant with respect to linear transformations, and have arbitrarily high order. Those of optimal order also preserve quadratic Casimir functions. The discussion of the order is based on an interpretation as partitioned Runge-Kutta method with infinitely many stages.

  • 23.
    Cohen, David
    et al.
    Department of Mathematical Sciences, NTNU.
    Hairer, Ernst
    Section de Mathématiques, Université de Genève.
    Lubich, Christian
    Mathematisches Institut, Universität Tübingen.
    Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations2008Inngår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 110, nr 2, s. 113-143Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    For classes of symplectic and symmetric time-stepping methods- trigonometric integrators and the Stormer-Verlet or leapfrog method-applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.

  • 24.
    Cohen, David
    et al.
    Department of Mathematical Sciences, NTNU.
    Hairer, Ernst
    Section de Mathématiques, Université de Genève.
    Lubich, Christian
    Mathematisches Institut, Universität Tübingen.
    Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions2008Inngår i: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 187, nr 2, s. 341-368Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A modulated Fourier expansion in time is used to show long-time near-conservation of the harmonic actions associated with spatial Fourier modes along the solutions of nonlinear wave equations with small initial data. The result implies the long-time near-preservation of the Sobolev-type norm that specifies the smallness condition on the initial data.

  • 25.
    Cohen, David
    et al.
    Section de Mathématiques, Université de Genève.
    Hairer, Ernst
    Section de Mathématiques, Université de Genève.
    Lubich, Christian
    Mathematisches Institut, Universität Tübingen.
    Modulated Fourier expansions of highly oscillatory differential equations2003Inngår i: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 3, nr 4, s. 327-345Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Modulated Fourier expansions are developed as a tool for gaining insight into the long-time behavior of Hamiltonian systems with highly oscillatory solutions. Particle systems of Fermi-Pasta-Ulam type with light and heavy masses are considered as an example. It is shown that the harmonic energy of the highly oscillatory part is nearly conserved over times that are exponentially long in the high frequency. Unlike previous approaches to such problems, the technique used here does not employ nonlinear coordinate transforms and can therefore be extended to the analysis of numerical discrelizations.

  • 26.
    Cohen, David
    et al.
    Section de Mathématiques, Université de Genève.
    Hairer, Ernst
    Section de Mathématiques, Université de Genève.
    Lubich, Christian
    Mathematisches Institut, Universität Tübingen.
    Numerical energy conservation for multi-frequency oscillatory differential equations2005Inngår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 45, nr 2, s. 287-305Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the Stormer-Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also included.

  • 27.
    Cohen, David
    et al.
    Mathematisches Institut, Universität Tübingen.
    Jahnke, Tobias
    Institut für Mathematik II, Freie Universität Berlin.
    Lorenz, Katina
    Mathematisches Institut, Universität Tübingen.
    Lubich, Christian
    Mathematisches Institut, Universität Tübingen .
    Numerical integrators for highly oscillatory Hamiltonian systems: a review2006Inngår i: Analysis, modeling and simulation of multiscale problems, Springer , 2006, s. 553-576Kapittel i bok, del av antologi (Annet vitenskapelig)
  • 28.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Stig
    Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg.
    Sigg, Magdalena
    Mathematisches Institut, Universität Basel.
    A trigonometric method for the linear stochastic wave equation2013Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 51, nr 1, s. 204-222Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented. A standard finite element method is used for the spatial discretization and a stochastic trigonometric scheme for the temporal approximation. This explicit time integrator allows for error bounds independent of the space discretization and thus does not have a step-size restriction as in the often used Störmer--Verlet-leap-frog scheme. Moreover, it enjoys a trace formula as does the exact solution of our problem. These favorable properties are demonstrated with numerical experiments. Read More: http://epubs.siam.org/doi/abs/10.1137/12087030X

  • 29.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Matsuo, Takayasu
    Department of Mathematical Informatics, Graduate School of System of Information Science and Technology, The University of Tokyo, Japan.
    Raynaud, Xavier
    Applied Mathematics, SINTEF ICT, Oslo, Norway ; Department of Mathematical Science, NTNU Trondheim, Norway.
    A multi-symplectic numerical integrator for the two-component Camassa Holm equation2014Inngår i: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 21, nr 3, s. 442-453Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A new multi-symplectic formulation of the two-component Camassa-Holm equation (2CH) is presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. Furthermore, this scheme preserves exactly two discrete versions of the Casimir functions of 2CH. Numerical experiments show that the proposed numerical scheme has good conservation properties.

  • 30.
    Cohen, David
    et al.
    Department of Mathematical Sciences, NTNU.
    Owren, Brynjulf
    Department of Mathematical Sciences, NTNU.
    Raynaud, Xavier
    Department of Mathematical Sciences, NTNU.
    Multi-symplectic integration of the Camassa-Holm equation2008Inngår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 227, nr 11, s. 5492-5512Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The Camassa-Holm equation is rich in geometric structures, it is completely integrable, bi-Hamiltonian, and it represents geodesics for a certain metric in the group of diffeomorphism. Here two new multi-symplectic formulations for the Camassa-Holm equation are presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. The multi-symplectic discretisation of each formulation is exemplified by means of the Euler box scheme. Numerical experiments show that the schemes have good conservative properties, and one of them is designed to handle the conservative continuation of peakon-antipeakon collisions. (c) 2008 Elsevier Inc. All rights reserved.

  • 31.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Quer-Sardanyons, Lluís
    Universitat Autònoma de Barcelona.
    A fully discrete approximation of the one-dimensional stochastic wave equation2016Inngår i: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 36, nr 1, s. 400-420Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A fully discrete approximation of one-dimensional nonlinear stochastic wave equations driven by multiplicative noise is presented. A standard finite difference approximation is used in space and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for error bounds in Lp(Ω)" style="position: relative;" tabindex="0" id="MathJax-Element-1-Frame" class="MathJax">Lp(Ω), uniformly in time and space, in such a way that the time discretization does not suffer from any kind of CFL-type step-size restriction. Moreover, uniform almost sure convergence of the numerical solution is also proved. Numerical experiments are presented and confirm the theoretical results.

  • 32.
    Cohen, David
    et al.
    Mathematisches Institut, Universität Basel.
    Raynaud, Xavier
    CMA, University of Oslo.
    Convergent numerical schemes for the compressible hyperelastic rod wave equation2012Inngår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 122, s. 1-59Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We propose a fully discretised numerical scheme for the hyperelastic rod wave equation on the line. The convergence of the method is established. Moreover, the scheme can handle the blow-up of the derivative which naturally occurs for this equation. By using a time splitting integrator which preserves the invariants of the problem, we can also show that the scheme preserves the positivity of the energy density.

  • 33.
    Cohen, David
    et al.
    Mathematisches Institut, Universität Basel.
    Raynaud, Xavier
    CMA, University of Oslo.
    Geometric finite difference schemes for the generalized hyperelastic-rod wave equation2011Inngår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 235, nr 8, s. 1925-1940Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Geometric integrators are presented for a class of nonlinear dispersive equations which includes the Camassa-Holm equation, the BBM equation and the hyperelastic-rod wave equation. One group of schemes is designed to preserve a global property of the equations: the conservation of energy; while the other one preserves a more local feature of the equations: the multi-symplecticity. (C) 2010 Elsevier B.V. All rights reserved.

  • 34.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Schweitzer, Julia
    Karlsruhe Institute of Technology.
    High order numerical methods for highly oscillatory problem2015Inngår i: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 49, nr 3, s. 695-711Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This paper is concerned with the numerical solution of nonlinear Hamiltonian highly oscillatory systems of second-order differential equations of a special form. We present numerical methods of high asymptotic as well as time stepping order based on the modulated Fourier expansion of the exact solution. In particular we obtain time stepping orders higher than 2 with only a finite energy assumption on the initial values of the problem. In addition, the stepsize of these new numerical integrators is not restricted by the high frequency of the problem. Furthermore, numerical experiments on the modified Fermi–Pasta–Ulam problem as well as on a one dimensional model of a diatomic gas with short-range interaction forces support our investigations.

  • 35.
    Cohen, David
    et al.
    Department of mathematics, University of Basel.
    Sigg, Magdalena
    Department of mathematics, University of Basel.
    Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations2012Inngår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 121, nr 1, s. 1-29Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study a class of numerical methods for a system of second-order SDE driven by a linear fast force generating high frequency oscillatory solutions. The proposed schemes permit the use of large step sizes, have uniform global error bounds in the position (i.e. independent of the large frequencies present in the SDE) and offer various additional properties. This new family of numerical integrators for SDE can be viewed as a stochastic generalisation of the trigonometric integrators for highly oscillatory deterministic problems.

  • 36.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Department of Mathematics, University of Innsbruck, Austria.
    Verdier, Olivier
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Department of Computing, Mathematics and Physics, Bergen University College, Norway.
    MultiSymplectic discretisation of wave map equations2016Inngår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, nr 2, s. A953-A972Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We present a new multisymplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multisymplectic discretization based on this new formulation is exemplified by means of the Euler box scheme. When applied to the wave map equation, this numerical scheme is explicit, preserves the constraint, and can be seen as a generalization of the \smaller SHAKE algorithm for constrained mechanical systems. Furthermore, numerical experiments show excellent conservation properties of the numerical solutions.

  • 37.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Department of Mathematical Sciences, Chalmers University of Technology, University of Gothenburg, Gothenburg, Sweden.
    Vilmart, Gilles
    Section de mathématiques, Université de Genève, Geneva, Switzerland.
    Drift-preserving numerical integrators for stochastic Poisson systems2022Inngår i: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 99, nr 1, s. 4-20Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments.

    Fulltekst (pdf)
    fulltext
  • 38. Komori, Yoshio
    et al.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Innsbruck university.
    Burrage, Kevin
    Weak second order explicit exponential Runge–Kutta methods for stochastic differential equations2017Inngår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, nr 6, s. A2857-A2878Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We propose new explicit exponential Runge--Kutta methods for the weak approximation of solutions of stiff Itô stochastic differential equations (SDEs). We also consider the use of exponential Runge--Kutta methods in combination with splitting methods. These methods have weak order 2 for multidimensional, noncommutative SDEs with a semilinear drift term, whereas they are of order 2 or 3 for semilinear ordinary differential equations. These methods are A-stable in the mean square sense for a scalar linear test equation whose drift and diffusion terms have complex coefficients. We carry out numerical experiments to compare the performance of these methods with an existing explicit stabilized method of weak order 2.

  • 39. Miyatake, Yuto
    et al.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Furihata, Daisuke
    Matsuo, Takayasu
    Geometric numerical integrators for Hunter–Saxton-like equations2017Inngår i: Japan journal of industrial and applied mathematics, ISSN 0916-7005, E-ISSN 1868-937X, Vol. 34, nr 2, s. 441-472Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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. We consider the Hunter-Saxton equation, the modified Hunter-Saxton equation, and the two-component Hunter-Saxton equation. Multi-symplectic discretisations based on these new formulations of the problems are exemplified by means of the explicit Euler box scheme, and Hamiltonian-preserving discretisations are exemplified by means of the discrete variational derivative method. We explain and justify the correct treatment of boundary conditions in a unified manner. This is necessary for a proper numerical implementation of these equations and was never explicitly clarified in the literature before, to the best of our knowledge. Finally, numerical experiments demonstrate the favourable behaviour of the proposed numerical integrators.

1 - 39 of 39
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