3,052 research outputs found
Design of quadrature rules for MĆ¼ntz and MĆ¼ntz-logarithmic polynomials using monomial transformation
A method for constructing the exact quadratures for MĆ¼ntz and MĆ¼ntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of MĆ¼ntz-logarithmic polynomials in terms of the number of Gauss-Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end-point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run-time evaluation than the existing methods for MĆ¼ntz polynomial
GPU-accelerated discontinuous Galerkin methods on hybrid meshes
We present a time-explicit discontinuous Galerkin (DG) solver for the
time-domain acoustic wave equation on hybrid meshes containing vertex-mapped
hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable
formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto
(Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions
for hybrid meshes are derived by bounding the spectral radius of the DG
operator using order-dependent constants in trace and Markov inequalities.
Computational efficiency is achieved under a combination of element-specific
kernels (including new quadrature-free operators for the pyramid), multi-rate
timestepping, and acceleration using Graphics Processing Units.Comment: Submitted to CMAM
A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
We propose a positivity preserving entropy decreasing finite volume scheme
for nonlinear nonlocal equations with a gradient flow structure. These
properties allow for accurate computations of stationary states and long-time
asymptotics demonstrated by suitably chosen test cases in which these features
of the scheme are essential. The proposed scheme is able to cope with
non-smooth stationary states, different time scales including metastability, as
well as concentrations and self-similar behavior induced by singular nonlocal
kernels. We use the scheme to explore properties of these equations beyond
their present theoretical knowledge
Support-type properties of convex functions of higher order and Hadamard-type inequalities
It is well-known that every convex function admits an affine support at every
interior point of a domain. Convex functions of higher order (precisely of an
odd order) have a similar property: they are supported by the polynomials of
degree no greater than the order of convexity. In this paper the attaching
method is developed. It is applied to obtain the general result Theorem 2, from
which the mentioned above support theorem and some related properties of convex
functions of higher (both odd and even) order are derived. They are applied to
obtain some known and new Hadamard-type inequalities between the quadrature
operators and the integral approximated by them. It is also shown that the
error bounds of quadrature rules follow by inequalities of this kind.Comment: In the journal version of the paper an example given in Remark 4 was
not correct. Here we give a proper on
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