2,716 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
Machine Precision Evaluation of Singular and Nearly Singular Potential Integrals by Use of Gauss Quadrature Formulas for Rational Functions
A new technique for machine precision evaluation of singular and nearly singular potential integrals with 1/R singularities is presented. The numerical quadrature scheme is based on a new rational expression for the integrands, obtained by a cancellation procedure. In particular, by using library routines for Gauss quadrature of rational functions readily available in the literature, this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions. The rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed, and several numerical examples are provide
Resolving velocity space dynamics in continuum gyrokinetics
Many plasmas of interest to the astrophysical and fusion communities are
weakly collisional. In such plasmas, small scales can develop in the
distribution of particle velocities, potentially affecting observable
quantities such as turbulent fluxes. Consequently, it is necessary to monitor
velocity space resolution in gyrokinetic simulations. In this paper, we present
a set of computationally efficient diagnostics for measuring velocity space
resolution in gyrokinetic simulations and apply them to a range of plasma
physics phenomena using the continuum gyrokinetic code GS2. For the cases
considered here, it is found that the use of a collisionality at or below
experimental values allows for the resolution of plasma dynamics with
relatively few velocity space grid points. Additionally, we describe
implementation of an adaptive collision frequency which can be used to improve
velocity space resolution in the collisionless regime, where results are
expected to be independent of collision frequency.Comment: 20 pages, 11 figures, submitted to Phys. Plasma
- ā¦