58,884 research outputs found
Space-Time Isogeometric Analysis of Parabolic Evolution Equations
We present and analyze a new stable space-time Isogeometric Analysis (IgA)
method for the numerical solution of parabolic evolution equations in fixed and
moving spatial computational domains. The discrete bilinear form is elliptic on
the IgA space with respect to a discrete energy norm. This property together
with a corresponding boundedness property, consistency and approximation
results for the IgA spaces yields an a priori discretization error estimate
with respect to the discrete norm. The theoretical results are confirmed by
several numerical experiments with low- and high-order IgA spaces
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
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
The Novikov Conjecture
We give a survey on recent development of the Novikov conjecture and its
applications to topological rigidity and non-rigidity. .Comment: 16 pages. Dedicated to Sergei Novikov on the occasion of his 80th
birthday. To appear in Russian Math Survey, 2019. arXiv admin note: text
overlap with arXiv:1811.0208
Construction and analysis of higher order Galerkin variational integrators
In this work we derive and analyze variational integrators of higher order
for the structure-preserving simulation of mechanical systems. The construction
is based on a space of polynomials together with Gauss and Lobatto quadrature
rules to approximate the relevant integrals in the variational principle. The
use of higher order schemes increases the accuracy of the discrete solution and
thereby decrease the computational cost while the preservation properties of
the scheme are still guaranteed. The order of convergence of the resulting
variational integrators are investigated numerically and it is discussed which
combination of space of polynomials and quadrature rules provide optimal
convergence rates. For particular integrators the order can be increased
compared to the Galerkin variational integrators previously introduced in
Marsden & West 2001. Furthermore, linear stability properties, time
reversibility, structure-preserving properties as well as efficiency for the
constructed variational integrators are investigated and demonstrated by
numerical examples.Comment: 27 page
High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids
This paper is concerned with the numerical solution of the unified first
order hyperbolic formulation of continuum mechanics recently proposed by
Peshkov & Romenski, denoted as HPR model. In that framework, the viscous
stresses are computed from the so-called distortion tensor A, which is one of
the primary state variables. A very important key feature of the model is its
ability to describe at the same time the behavior of inviscid and viscous
compressible Newtonian and non-Newtonian fluids with heat conduction, as well
as the behavior of elastic and visco-plastic solids. This is achieved via a
stiff source term that accounts for strain relaxation in the evolution
equations of A. Also heat conduction is included via a first order hyperbolic
evolution equation of the thermal impulse, from which the heat flux is
computed. The governing PDE system is hyperbolic and fully consistent with the
principles of thermodynamics. It is also fundamentally different from first
order Maxwell-Cattaneo-type relaxation models based on extended irreversible
thermodynamics. The connection between the HPR model and the classical
hyperbolic-parabolic Navier-Stokes-Fourier theory is established via a formal
asymptotic analysis in the stiff relaxation limit. From a numerical point of
view, the governing partial differential equations are very challenging, since
they form a large nonlinear hyperbolic PDE system that includes stiff source
terms and non-conservative products. We apply the successful family of one-step
ADER-WENO finite volume and ADER discontinuous Galerkin finite element schemes
in the stiff relaxation limit, and compare the numerical results with exact or
numerical reference solutions obtained for the Euler and Navier-Stokes
equations. To show the universality of the model, the paper is rounded-off with
an application to wave propagation in elastic solids
Application of multivariate splines to discrete mathematics
Using methods developed in multivariate splines, we present an explicit
formula for discrete truncated powers, which are defined as the number of
non-negative integer solutions of linear Diophantine equations. We further use
the formula to study some classical problems in discrete mathematics as
follows. First, we extend the partition function of integers in number theory.
Second, we exploit the relation between the relative volume of convex polytopes
and multivariate truncated powers and give a simple proof for the volume
formula for the Pitman-Stanley polytope. Third, an explicit formula for the
Ehrhart quasi-polynomial is presented.Comment: Box splines; Number of integer points in polytopes; Pitman-Stanley
polytop
Data-driven approximations of dynamical systems operators for control
The Koopman and Perron Frobenius transport operators are fundamentally
changing how we approach dynamical systems, providing linear representations
for even strongly nonlinear dynamics. Although there is tremendous potential
benefit of such a linear representation for estimation and control, transport
operators are infinite-dimensional, making them difficult to work with
numerically. Obtaining low-dimensional matrix approximations of these operators
is paramount for applications, and the dynamic mode decomposition has quickly
become a standard numerical algorithm to approximate the Koopman operator.
Related methods have seen rapid development, due to a combination of an
increasing abundance of data and the extensibility of DMD based on its simple
framing in terms of linear algebra. In this chapter, we review key innovations
in the data-driven characterization of transport operators for control,
providing a high-level and unified perspective. We emphasize important recent
developments around sparsity and control, and discuss emerging methods in big
data and machine learning.Comment: 37 pages, 4 figure
A three-level multi-continua upscaling method for flow problems in fractured porous media
Traditional two level upscaling techniques suffer from a high offline cost
when the coarse grid size is much larger than the fine grid size. Thus,
multilevel methods are desirable for problems with complex heterogeneities and
high contrast. In this paper, we propose a novel three-level upscaling method
for flow problems in fractured porous media. Our method starts with a fine grid
discretization for the system involving fractured porous media. In the next
step, based on the fine grid model, we construct a nonlocal multi-continua
upscaling (NLMC) method using an intermediate grid. The system resulting from
NLMC gives solutions that have physical meaning. In order to enhance locality,
the grid size of the intermediate grid needs to be relatively small, and this
motivates using such an intermediate grid. However, the resulting NLMC upscaled
system has a relatively large dimension. This motivates a further step of
dimension reduction. In particular, we will apply the idea of the Generalized
Multiscale Finite Element Method (GMsFEM) to the NLMC system to obtain a final
reduced model. We present simulation results for a two-dimensional model
problem with a large number of fractures using the proposed three-level method
Distance sets of two subsets of vector spaces over finite fields
We investigate the size of the distance set determined by two subsets of
finite dimensional vector spaces over finite fields. A lower bound of the size
is given explicitly in terms of cardinalities of the two subsets. As a result,
we improve upon the results by Rainer Dietmann. In the case that one of the
subsets is a product set, we obtain further improvement on the estimate.Comment: Changed title and structure, main results unchange
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