23 research outputs found
Canonical and Noncanonical Hamiltonian Operator Inference
A method for the nonintrusive and structure-preserving model reduction of
canonical and noncanonical Hamiltonian systems is presented. Based on the idea
of operator inference, this technique is provably convergent and reduces to a
straightforward linear solve given snapshot data and gray-box knowledge of the
system Hamiltonian. Examples involving several hyperbolic partial differential
equations show that the proposed method yields reduced models which, in
addition to being accurate and stable with respect to the addition of basis
modes, preserve conserved quantities well outside the range of their training
data
A conservative fully-discrete numerical method for the regularised shallow water wave equations
The paper proposes a new, conservative fully-discrete scheme for the
numerical solution of the regularised shallow water Boussinesq system of
equations in the cases of periodic and reflective boundary conditions. The
particular system is one of a class of equations derived recently and can be
used in practical simulations to describe the propagation of weakly nonlinear
and weakly dispersive long water waves, such as tsunamis. Studies of
small-amplitude long waves usually require long-time simulations in order to
investigate scenarios such as the overtaking collision of two solitary waves or
the propagation of transoceanic tsunamis. For long-time simulations of
non-dissipative waves such as solitary waves, the preservation of the total
energy by the numerical method can be crucial in the quality of the
approximation. The new conservative fully-discrete method consists of a
Galerkin finite element method for spatial semidiscretisation and an explicit
relaxation Runge--Kutta scheme for integration in time. The Galerkin method is
expressed and implemented in the framework of mixed finite element methods. The
paper provides an extended experimental study of the accuracy and convergence
properties of the new numerical method. The experiments reveal a new
convergence pattern compared to standard Galerkin methods
Numerical simulation of a solitonic gas in KdV and KdV-BBM equations
19 pages, 11 figures, 47 references. Other author's papers can be found at http://www.denys-dutykh.com/The collective behaviour of soliton ensembles (i.e. the solitonic gas) is studied using the methods of the direct numerical simulation. Traditionally this problem was addressed in the context of integrable models such as the celebrated KdV equation. We extend this analysis to non-integrable KdV-BBM type models. Some high resolution numerical results are presented in both integrable and nonintegrable cases. Moreover, the free surface elevation probability distribution is shown to be quasi-stationary. Finally, we employ the asymptotic methods along with the Monte-Carlo simulations in order to study quantitatively the dependence of some important statistical characteristics (such as the kurtosis and skewness) on the Stokes-Ursell number (which measures the relative importance of nonlinear effects compared to the dispersion) and also on the magnitude of the BBM term
On Algebraic Singularities, Finite Graphs and D-Brane Gauge Theories: A String Theoretic Perspective
In this writing we shall address certain beautiful inter-relations between
the construction of 4-dimensional supersymmetric gauge theories and resolution
of algebraic singularities, from the perspective of String Theory. We review in
some detail the requisite background in both the mathematics, such as
orbifolds, symplectic quotients and quiver representations, as well as the
physics, such as gauged linear sigma models, geometrical engineering,
Hanany-Witten setups and D-brane probes.
We investigate aspects of world-volume gauge dynamics using D-brane
resolutions of various Calabi-Yau singularities, notably Gorenstein quotients
and toric singularities. Attention will be paid to the general methodology of
constructing gauge theories for these singular backgrounds, with and without
the presence of the NS-NS B-field, as well as the T-duals to brane setups and
branes wrapping cycles in the mirror geometry. Applications of such diverse and
elegant mathematics as crepant resolution of algebraic singularities,
representation of finite groups and finite graphs, modular invariants of affine
Lie algebras, etc. will naturally arise. Various viewpoints and generalisations
of McKay's Correspondence will also be considered.
The present work is a transcription of excerpts from the first three volumes
of the author's PhD thesis which was written under the direction of Prof. A.
Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of
MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac
vice; it is his sincerest wish that the ensuing pages might be of some small
use to the beginning student.Comment: 513 pages, 71 figs, Edited Excerpts from the first 3 volumes of the
author's PhD Thesi
An explicit finite difference scheme for the Camassa-Holm equation
We put forward and analyze an explicit finite difference scheme for the
Camassa-Holm shallow water equation that can handle general initial data
and thus peakon-antipeakon interactions. Assuming a specified condition
restricting the time step in terms of the spatial discretization parameter, we
prove that the difference scheme converges strongly in towards a
dissipative weak solution of Camassa-Holm equation.Comment: 45 pages, 6 figure
Recommended from our members
On algebraic singularities, finite graphs and D-brane gauge theories: A String theoretic perspective
In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes.
We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered.
The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student