123,262 research outputs found
The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction
This paper is a very brief introduction to idempotent mathematics and related
topics.Comment: 24 pages, 2 figures. An introductory paper to the volume "Idempotent
Mathematics and Mathematical Physics" (G.L. Ltvinov, V.P. Maslov, eds.; AMS
Contemporary Mathematics, 2005). More misprints correcte
Informative Words and Discreteness
There are certain families of words and word sequences (words in the
generators of a two-generator group) that arise frequently in the
Teichm{\"u}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian
groups and in the discreteness problem for two generator matrix groups. We
survey some of the families of such words and sequences: the semigroup of so
called {\sl good} words of Gehring-Martin, the so called {\sl killer} words of
Gabai-Meyerhoff-NThurston, the Farey words of Keen-Series and Minsky, the
discreteness-algorithm Fibonacci sequences of Gilman-Jiang and {\sl parabolic
dust} words. We survey connections between the families and establish a new
connection between good words and Farey words.Comment: one pdf fil
Variational integrators for stochastic dissipative Hamiltonian systems
Variational integrators are derived for structure-preserving simulation of
stochastic forced Hamiltonian systems. The derivation is based on a stochastic
discrete Hamiltonian which approximates a type-II stochastic generating
function for the stochastic flow of the Hamiltonian system. The generating
function is obtained by introducing an appropriate stochastic action functional
and considering a stochastic generalization of the deterministic
Lagrange-d'Alembert principle. Our approach presents a general methodology to
derive new structure-preserving numerical schemes. The resulting integrators
satisfy a discrete version of the stochastic Lagrange-d'Alembert principle, and
in the presence of symmetries, they also satisfy a discrete counterpart of
Noether's theorem. Furthermore, mean-square and weak Lagrange-d'Alembert
Runge-Kutta methods are proposed and tested numerically to demonstrate their
superior long-time numerical stability and energy behavior compared to
non-geometric methods. The Vlasov-Fokker-Planck equation is considered as one
of the numerical test cases, and a new geometric approach to collisional
kinetic plasmas is presented.Comment: 54 pages, 11 figures. arXiv admin note: text overlap with
arXiv:1609.0046
Numerical aspects of evolution of plane curves satisfying the fourth order geometric equation
In this review paper we present a stable Lagrangian numerical method for
computing plane curves evolution driven by the fourth order geometric equation.
The numerical scheme and computational examples are presented.Comment: submitted to: Proceedings of Equadiff 2007 Conferenc
Solitons of shallow-water models from energy-dependent spectral problems
The current work investigates the soliton solutions of the Kaup-Boussinesq
equation using the Inverse Scattering Transform method. We outline the
construction of the Riemann-Hilbert problem for a pair energy-dependent
spectral problems for the system, which we then use to construct the solution
of this hydrodynamic system
A Nitsche-eXtended finite element method for distributed optimal control problems of elliptic interface equations
This paper analyzes an interface-unfitted numerical method for distributed
optimal control problems governed by elliptic interface equations. We follow
the variational discretization concept to discretize the optimal control
problems, and apply a Nitsche-eXtended finite element method to discretize the
corresponding state and adjoint equations, where piecewise cut basis functions
around the interface are enriched into the standard linear element space.
Optimal error estimates of the state, co-state and control in a mesh-dependent
norm and the norm are derived. Numerical results are provided to verify
the theoretical results
A Unified Study of Continuous and Discontinuous Galerkin Methods
A unified study is presented in this paper for the design and analysis of
different finite element methods (FEMs), including conforming and nonconforming
FEMs, mixed FEMs, hybrid FEMs,discontinuous Galerkin (DG) methods, hybrid
discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG
and WG are shown to admit inf-sup conditions that hold uniformly with respect
to both mesh and penalization parameters. In addition, by taking the limit of
the stabilization parameters, a WG method is shown to converge to a mixed
method whereas an HDG method is shown to converge to a primal method.
Furthermore, a special class of DG methods, known as the mixed DG methods, is
presented to fill a gap revealed in the unified framework.Comment: 39 page
Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting
In this paper we present a novel arbitrary high order accurate discontinuous
Galerkin (DG) finite element method on space-time adaptive Cartesian meshes
(AMR) for hyperbolic conservation laws in multiple space dimensions, using a
high order \aposteriori sub-cell ADER-WENO finite volume \emph{limiter}.
Notoriously, the original DG method produces strong oscillations in the
presence of discontinuous solutions and several types of limiters have been
introduced over the years to cope with this problem. Following the innovative
idea recently proposed in \cite{Dumbser2014}, the discrete solution within the
troubled cells is \textit{recomputed} by scattering the DG polynomial at the
previous time step onto a suitable number of sub-cells along each direction.
Relying on the robustness of classical finite volume WENO schemes, the sub-cell
averages are recomputed and then gathered back into the DG polynomials over the
main grid. In this paper this approach is implemented for the first time within
a space-time adaptive AMR framework in two and three space dimensions, after
assuring the proper averaging and projection between sub-cells that belong to
different levels of refinement. The combination of the sub-cell resolution with
the advantages of AMR allows for an unprecedented ability in resolving even the
finest details in the dynamics of the fluid. The spectacular resolution
properties of the new scheme have been shown through a wide number of test
cases performed in two and in three space dimensions, both for the Euler
equations of compressible gas dynamics and for the magnetohydrodynamics (MHD)
equations.Comment: Computers and Fluids 118 (2015) 204-22
A divergence-free semi-implicit finite volume scheme for ideal, viscous and resistive magnetohydrodynamics
In this paper we present a novel pressure-based semi-implicit finite volume
solver for the equations of compressible ideal, viscous and resistive
magnetohydrodynamics (MHD). The new method is conservative for mass, momentum
and total energy and in multiple space dimensions it is constructed in such a
way as to respect the divergence-free condition of the magnetic field exactly,
also in the presence of resistive effects. This is possible via the use of
multi-dimensional Riemann solvers on an appropriately staggered grid for the
time evolution of the magnetic field and a double curl formulation of the
resistive terms. The new semi-implicit method for the MHD equations proposed
here discretizes all terms related to the pressure in the momentum equation and
the total energy equation implicitly, making again use of a properly staggered
grid for pressure and velocity. The time step of the scheme is restricted by a
CFL condition based only on the fluid velocity and the Alfv\'en wave speed and
is not based on the speed of the magnetosonic waves. Our new method is
particularly well-suited for low Mach number flows and for the incompressible
limit of the MHD equations, for which it is well-known that explicit
density-based Godunov-type finite volume solvers become increasingly
inefficient and inaccurate due to the increasingly stringent CFL condition and
the wrong scaling of the numerical viscosity in the incompressible limit. We
show a relevant MHD test problem in the low Mach number regime where the new
semi-implicit algorithm is a factor of 50 faster than a traditional explicit
finite volume method, which is a very significant gain in terms of
computational efficiency. However, our numerical results confirm that our new
method performs well also for classical MHD test cases with strong shocks. In
this sense our new scheme is a true all Mach number flow solver.Comment: 26 pages, 12 figures,1 tabl
Particle Swarm Optimization: A survey of historical and recent developments with hybridization perspectives
Particle Swarm Optimization (PSO) is a metaheuristic global optimization
paradigm that has gained prominence in the last two decades due to its ease of
application in unsupervised, complex multidimensional problems which cannot be
solved using traditional deterministic algorithms. The canonical particle swarm
optimizer is based on the flocking behavior and social co-operation of birds
and fish schools and draws heavily from the evolutionary behavior of these
organisms. This paper serves to provide a thorough survey of the PSO algorithm
with special emphasis on the development, deployment and improvements of its
most basic as well as some of the state-of-the-art implementations. Concepts
and directions on choosing the inertia weight, constriction factor, cognition
and social weights and perspectives on convergence, parallelization, elitism,
niching and discrete optimization as well as neighborhood topologies are
outlined. Hybridization attempts with other evolutionary and swarm paradigms in
selected applications are covered and an up-to-date review is put forward for
the interested reader.Comment: 34 pages, 7 table
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