73,043 research outputs found
On solving norm equations in global function fields
Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.The potential of solving norm equations is crucial for a variety of applications of algebraic number theory, especially in cryptography. In this article we develop general effective methods for that task in global function fields for the first time
RANS Equations with Explicit Data-Driven Reynolds Stress Closure Can Be Ill-Conditioned
Reynolds-averaged Navier--Stokes (RANS) simulations with turbulence closure
models continue to play important roles in industrial flow simulations.
However, the commonly used linear eddy viscosity models are intrinsically
unable to handle flows with non-equilibrium turbulence. Reynolds stress models,
on the other hand, are plagued by their lack of robustness. Recent studies in
plane channel flows found that even substituting Reynolds stresses with errors
below 0.5% from direct numerical simulation (DNS) databases into RANS equations
leads to velocities with large errors (up to 35%). While such an observation
may have only marginal relevance to traditional Reynolds stress models, it is
disturbing for the recently emerging data-driven models that treat the Reynolds
stress as an explicit source term in the RANS equations, as it suggests that
the RANS equations with such models can be ill-conditioned. So far, a rigorous
analysis of the condition of such models is still lacking. As such, in this
work we propose a metric based on local condition number function for a priori
evaluation of the conditioning of the RANS equations. We further show that the
ill-conditioning cannot be explained by the global matrix condition number of
the discretized RANS equations. Comprehensive numerical tests are performed on
turbulent channel flows at various Reynolds numbers and additionally on two
complex flows, i.e., flow over periodic hills and flow in a square duct.
Results suggest that the proposed metric can adequately explain observations in
previous studies, i.e., deteriorated model conditioning with increasing
Reynolds number and better conditioning of the implicit treatment of Reynolds
stress compared to the explicit treatment. This metric can play critical roles
in the future development of data-driven turbulence models by enforcing the
conditioning as a requirement on these models.Comment: 35 pages, 18 figure
Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics
In this article we develop some new existence results for the Einstein
constraint equations using the Lichnerowicz-York conformal rescaling method.
The mean extrinsic curvature is taken to be an arbitrary smooth function
without restrictions on the size of its spatial derivatives, so that it can be
arbitrarily far from constant. The rescaled background metric belongs to the
positive Yamabe class, and the freely specifiable part of the data given by the
traceless-transverse part of the rescaled extrinsic curvature and the matter
fields are taken to be sufficiently small, with the matter energy density not
identically zero. Using topological fixed-point arguments and global barrier
constructions, we then establish existence of solutions to the constraints. Two
recent advances in the analysis of the Einstein constraint equations make this
result possible: A new type of topological fixed-point argument without
smallness conditions on spatial derivatives of the mean extrinsic curvature,
and a new construction of global super-solutions for the Hamiltonian constraint
that is similarly free of such conditions on the mean extrinsic curvature. For
clarity, we present our results only for strong solutions on closed manifolds.
However, our results also hold for weak solutions and for other cases such as
compact manifolds with boundary; these generalizations will appear elsewhere.
The existence results presented here for the Einstein constraints are
apparently the first such results that do not require smallness conditions on
spatial derivatives of the mean extrinsic curvature.Comment: 4 pages, no figures, accepted for publication in Physical Review
Letters. (Abstract shortenned and other minor changes reflecting v4 version
of arXiv:0712.0798
A spectral solver for evolution problems with spatial S3-topology
We introduce a single patch collocation method in order to compute solutions
of initial value problems of partial differential equations whose spatial
domains are 3-spheres. Besides the main ideas, we discuss issues related to our
implementation and analyze numerical test applications. Our main interest lies
in cosmological solutions of Einstein's field equations. Motivated by this, we
also elaborate on problems of our approach for general tensorial evolution
equations when certain symmetries are assumed. We restrict to U(1)- and Gowdy
symmetry here.Comment: 29 pages, 11 figures, uses psfrag and hyperref, large parts rewritten
in order to match to the requirements of the journal, conclusions unchanged;
J. Comput. Phys. (2009
Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains
We describe a multidomain spectral-tau method for solving the
three-dimensional helically reduced wave equation on the type of two-center
domain that arises when modeling compact binary objects in astrophysical
applications. A global two-center domain may arise as the union of Cartesian
blocks, cylindrical shells, and inner and outer spherical shells. For each such
subdomain, our key objective is to realize certain (differential and
multiplication) physical-space operators as matrices acting on the
corresponding set of modal coefficients. We achieve sparse banded realizations
through the integration "preconditioning" of Coutsias, Hagstrom, Hesthaven, and
Torres. Since ours is the first three-dimensional multidomain implementation of
the technique, we focus on the issue of convergence for the global solver, here
the alternating Schwarz method accelerated by GMRES. Our methods may prove
relevant for numerical solution of other mixed-type or elliptic problems, and
in particular for the generation of initial data in general relativity.Comment: 37 pages, 3 figures, 12 table
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