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Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method

E. Celledoni, V. Grimm, R.I. McLachlan, D.I. McLaren, D. O'Neale, B. Owren, G.R.W. Quispel
2012 Journal of Computational Physics  
We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly.  ...  In the Hamiltonian case these include: the sine-Gordon, Korteweg-de Vries, nonlinear Schrodinger, (linear) time-dependent Schrodinger, and Maxwell equations.  ...  This research is supported by the Australian Research Council and by the Marsden Fund of the Royal Society of New Zealand. We are grateful to Will Wright for many useful discussions.  ... 
doi:10.1016/j.jcp.2012.06.022 fatcat:6xswmhrvm5atznekmf76sbvtoy

Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations [article]

Chaolong Jiang and Yushun Wang and Yuezheng Gong
2020 arXiv   pre-print
A novel class of explicit high-order energy-preserving methods are proposed for general Hamiltonian partial differential equations with non-canonical structure matrix.  ...  When the energy is not quadratic, it is firstly done that the original system is reformulated into an equivalent form with a modified quadratic energy conservation law by the energy quadratization approach  ...  Nonlinear Schrödinger equation We consider the nonlinear Schrödinger equation given as follows i∂ t u + ∆u + β|u| 2 u = 0, (4.1) where i = √ −1 is the complex unit, u is the complex-valued wave function  ... 
arXiv:2001.00774v2 fatcat:if3ijuuqrvamvf5jhqvz6vtmsu

Energetic pulses in exciton-phonon molecular chains and conservative numerical methods for quasilinear Hamiltonian systems

Brenton LeMesurier
2013 Physical Review E  
A convenient method is described for construction the highly stable, accurate conservative time discretizations used, with proof of its desirable properties for a large class of Hamiltonian systems, including  ...  In contrast to previous studies based on a proposed long wave approximation by the nonlinear Schrödinger (NLS) equation and focusing on initial data resembling the soliton solutions of that equation, the  ...  A convenient method is described for construction of the highly stable, accurate conservative time discretizations used, with proof of its desirable properties for a large class of Hamiltonian systems,  ... 
doi:10.1103/physreve.88.032707 pmid:24125294 fatcat:j7nah5w2x5cfdfy5xt2a7dt2ja

Book Review: Global solutions of nonlinear Schrödinger equations

Gigliola Staffilani
2002 Bulletin of the American Mathematical Society  
Two well known equations belong to this class: the nonlinear Schrödinger equation (NLS), and the Korteweg-de-Vries equation (KdV).  ...  Unfortunately, this general argument, now known as the energy method, required quite a lot of regularity to start with; see [44] for an overview.  ... 
doi:10.1090/s0273-0979-02-00956-4 fatcat:cvvaypvftrgmnak5pbrm7ddzga

Second Symposium on "Recent Trends in the Numerical Solution of Differential Equations": Preface

Luigi Brugnano, Theodore E. Simos, George Psihoyios, Ch. Tsitouras
2009 AIP Conference Proceedings  
Phys. 53, 102702 (2012) Existence and stability of standing waves for nonlinear fractional Schrödinger equations J. Math.  ...  Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations J. Math.  ...  The authors generalize the line-integral approach which the recently introduced class of energy-conserving methods called Hamiltonian BVMS (HBVMs), for canonical Hamiltonian problems, rely on, thus obtaining  ... 
doi:10.1063/1.3241563 fatcat:a5kushymqfexpni6dcprpmd22u

Non-Abelian BFFT embedding, Schrödinger quantization and the anomaly of the O(N) nonlinear sigma model [article]

E. M. C. Abreu, J. Ananias Neto, W. Oliveira and G. Oliveira-Neto
2000 arXiv   pre-print
Firstly, the quantization is performed with the functional Schrödinger method, for N=2, obtaining the wave functionals for the ground and excited states.  ...  As a first class system, it is quantized using two different approaches: the functional Schrödinger method and the non-local field-antifield procedure.  ...  In [31] it was computed the expected value of the Hamiltonian using a trial wave-functional, in the large N limit. IV.  ... 
arXiv:hep-th/0012115v1 fatcat:woaj2omysfbsxezh437rk3frze

On the Rate of Error Growth in Time for Numerical Solutions of Nonlinear Dispersive Wave Equations [article]

Hendrik Ranocha, Manuel Quezada de Luna, David I. Ketcheson
2021 arXiv   pre-print
A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy  ...  We study the numerical error in solitary wave solutions of nonlinear dispersive wave equations.  ...  Ángel Durán for his insightful and very detailed comments on an early draft that helped us to improve this work, and for helping us learn the theory of relative equilibrium solutions.  ... 
arXiv:2102.07376v3 fatcat:oarh4e46szg6pg2niglmidmdca

A Dirichlet Inhomogenous Boundary Value Problem for 1D Nonlinear Schrödinger Equation

Charles Bu
2022 Journal of Applied Mathematics and Physics  
Pure initial value problems for important nonlinear evolution equations such as nonlinear Schrödinger equation (NLS) and the Ginzburg-Landau equation (GL) have been extensively studied.  ...  In this paper, we investigate the mixed initial-boundary condition problem for the nonlinear Schrödinger equation 1 , , 3 p t xx iu u g u u g R p − = − ∈ > on a semi-infinite strip.  ...  Professorship, the Brachman Hoffman Small Grant and a Wellesley College Faculty Award.  ... 
doi:10.4236/jamp.2022.103047 fatcat:34c7cxiiqze4hoxrok5uh4d47m

Page 5547 of Mathematical Reviews Vol. , Issue 90J [page]

1990 Mathematical Reviews  
Fluid mechanics and nonlinear waves: Ben Yu Guo and J. A. C. Weideman, Solitary solution of an initial-boundary value problem of the Korteweg-de Vries equation (pp. 696-702); P. N. Kaloni and A. M.  ...  Miscellaneous: Gian Luigi Agnoli, A theorem of existence and uniqueness for a class of nonlinear equations in applied mathemat- ics (pp. 1160-1164); P. C. Das and A. K.  ... 

High-order linearly implicit structure-preserving exponential integrators for the nonlinear Schrödinger equation [article]

Chaolong Jiang and Jin Cui and Xu Qian and Songhe Song
2021 arXiv   pre-print
A novel class of high-order linearly implicit energy-preserving integrating factor Runge-Kutta methods are proposed for the nonlinear Schr\"odinger equation.  ...  We show that, under certain circumstances for the coefficients of a Runge-Kutta method, the proposed scheme can produce numerical solutions along which the modified energy is precisely conserved, as is  ...  Acknowledgments The authors would like to express sincere gratitude to the referees for their insightful comments and suggestions.  ... 
arXiv:2103.00390v2 fatcat:7gbqrze5qffhpl43mwieu2l25y

Localization of Bose-Einstein condensates in optical lattices

Roberto Franzosi, Salvatore Giampaolo, Fabrizio Illuminati, Roberto Livi, Gian-Luca Oppo, Antonio Politi
2011 Open Physics  
Such dynamics is governed by a discrete nonlinear Schrödinger equation.  ...  Thus, this kind of states, reproduce a novel quantum localization phenomenon due to the interplay between bounded energy spectrum and non-linearity.  ...  The superfluid regime for a system of BECs in an optical lattice is described by a discrete nonlinear Schrödinger equation (DNSE).  ... 
doi:10.2478/s11534-011-0035-2 fatcat:hbh344i5abgtfpwfc6cyzyfcy4

Dynamics, numerical analysis, and some geometry [article]

Ludwig Gauckler, Ernst Hairer, Christian Lubich
2017 arXiv   pre-print
Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations.  ...  Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical low-rank approximation of time-dependent large matrices and  ...  We thank Balázs Kovács, Frank Loose, Hanna Walach, and Gerhard Wanner for helpful comments.  ... 
arXiv:1710.03946v1 fatcat:q35wnhffuzehhjtxb6htzs2uxi

Structure-preserving algorithms for multi-dimensional fractional Klein-Gordon-Schrödinger equation [article]

Yayun Fu Wenjun Cai, Yushun Wang
2019 arXiv   pre-print
First, we derive an equivalent equation, and reformulate the equation as a canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian.  ...  Further applying the partitioned averaged vector field methods on the temporal direction gives a class of fully-discrete schemes that can preserve the mass and energy exactly.  ...  a class of conservative schemes for the system based on its Hamiltonian formulation.  ... 
arXiv:1911.10845v2 fatcat:a5ffnywksza6jbzzgah37m57b4

On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations

Hendrik Ranocha, Manuel Quezada de Luna, David I. Ketcheson
2021 SN Partial Differential Equations and Applications  
A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy  ...  We provide numerical experiments suggesting that this result extends to a very broad class of equations and numerical methods.  ...  Ángel Durán for his insightful and very detailed comments on an early draft that helped us to improve this work, and for helping us learn the theory of relative equilibrium solutions.  ... 
doi:10.1007/s42985-021-00126-3 fatcat:ywnh5nwjgfh4jbpemewbrhyzci

Nonlinear Schrödinger equations and generalized Heisenberg uncertainty principle from estimation schemes violating the principle of estimation independence

Agung Budiyono, Hermawan K. Dipojono
2020 Physical Review A  
The nonlinearity of the Schr\"odinger equation and the deviation from the Heisenberg uncertainty principle thus have a common transparent operational origin in terms of generalizations of estimation errors  ...  In the present work, keeping the Born's quadratic law intact, we construct a class of nonlinear variants of Schr\"odinger equation and generalized Heisenberg uncertainty principle within the estimation  ...  Acknowledgments This work is partially supported by the Ministry of Education and Culture, and the  ... 
doi:10.1103/physreva.102.012205 fatcat:q437pws6fzclrfuzb7zefwouwe
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