103,075 research outputs found
A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations
In this paper, we develop regularized discrete least squares collocation and
finite volume methods for solving two-dimensional nonlinear time-dependent
partial differential equations on irregular domains. The solution is
approximated using tensor product cubic spline basis functions defined on a
background rectangular (interpolation) mesh, which leads to high spatial
accuracy and straightforward implementation, and establishes a solid base for
extending the computational framework to three-dimensional problems. A
semi-implicit time-stepping method is employed to transform the nonlinear
partial differential equation into a linear boundary value problem. A key
finding of our study is that the newly proposed mesh-free finite volume method
based on circular control volumes reduces to the collocation method as the
radius limits to zero. Both methods produce a large constrained least-squares
problem that must be solved at each time step in the advancement of the
solution. We have found that regularization yields a relatively
well-conditioned system that can be solved accurately using QR factorization.
An extensive numerical investigation is performed to illustrate the
effectiveness of the present methods, including the application of the new
method to a coupled system of time-fractional partial differential equations
having different fractional indices in different (irregularly shaped) regions
of the solution domain
Multimonopoles and closed vortices in SU(2) Yang-Mills-Higgs theory
We review classical monopole solutions of the SU(2) Yang-Mills-Higgs theory.
The first part is a pedagogical introduction into to the basic features of the
celebrated 't Hooft - Polyakov monopole. In the second part we describe new
classes of static axially symmetric solutions which generalise 't Hooft -
Polyakov monopole. These configurations are either deformations of the
topologically trivial sector or the sectors with different topological charges.
In both situations we construct the solutions representing the chains of
monopoles and antimonopoles in static equilibrium. The solutions of another
type are closed vortices which are centred around the symmetry axis and form
different bound systems. Configurations of the third type are monopoles bounded
with vortices. We suggest classification of these solutions which is related
with 2d Poincare index.Comment: 34 pages, 8 figures. Invited contribution prepared for Review Volume
"Etudes on Theoretical Physics" dedicated to 65th anniversary of the
Department of Theoretical Physics, Belarus State University, Minsk. Based on
a work in collaboration with Jutta Kunz and Burkhard Kleihaus. Any comments
and suggestions, especially with respect to references, are welcom
Arc-Length Continuation and Multigrid Techniques for Nonlinear Elliptic Eigenvalue Problems
We investigate multi-grid methods for solving linear systems arising from arc-length continuation techniques applied to nonlinear elliptic eigenvalue problems. We find that the usual multi-grid methods diverge in the neighborhood of singular points of the solution branches. As a result, the continuation method is unable to continue past a limit point in the Bratu problem. This divergence is analyzed and a modified multi-grid algorithm has been devised based on this analysis. In principle, this new multi-grid algorithm converges for elliptic systems, arbitrarily close to singularity and has been used successfully in conjunction with arc-length continuation procedures on the model problem. In the worst situation, both the storage and the computational work are only about a factor of two more than the unmodified multi-grid methods
A soliton menagerie in AdS
We explore the behaviour of charged scalar solitons in asymptotically global
AdS4 spacetimes. This is motivated in part by attempting to identify under what
circumstances such objects can become large relative to the AdS length scale.
We demonstrate that such solitons generically do get large and in fact in the
planar limit smoothly connect up with the zero temperature limit of planar
scalar hair black holes. In particular, for given Lagrangian parameters we
encounter multiple branches of solitons: some which are perturbatively
connected to the AdS vacuum and surprisingly, some which are not. We explore
the phase space of solutions by tuning the charge of the scalar field and
changing scalar boundary conditions at AdS asymptopia, finding intriguing
critical behaviour as a function of these parameters. We demonstrate these
features not only for phenomenologically motivated gravitational Abelian-Higgs
models, but also for models that can be consistently embedded into eleven
dimensional supergravity.Comment: 62 pages, 21 figures. v2: added refs and comments and updated
appendice
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