Design of quadrature rules for Müntz and Müntz-logarithmic polynomials using monomial transformation

A method for constructing the exact quadratures for Müntz and Müntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz-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 Müntz polynomial

Adaptive quadrature by expansion for layer potential evaluation in two dimensions

When solving partial differential equations using boundary integral equation methods, accurate evaluation of singular and nearly singular integrals in layer potentials is crucial. A recent scheme for this is quadrature by expansion (QBX), which solves the problem by locally approximating the potential using a local expansion centered at some distance from the source boundary. In this paper we introduce an extension of the QBX scheme in 2D denoted AQBX - adaptive quadrature by expansion - which combines QBX with an algorithm for automated selection of parameters, based on a target error tolerance. A key component in this algorithm is the ability to accurately estimate the numerical errors in the coefficients of the expansion. Combining previous results for flat panels with a procedure for taking the panel shape into account, we derive such error estimates for arbitrarily shaped boundaries in 2D that are discretized using panel-based Gauss-Legendre quadrature. Applying our scheme to numerical solutions of Dirichlet problems for the Laplace and Helmholtz equations, and also for solving these equations, we find that the scheme is able to satisfy a given target tolerance to within an order of magnitude, making it useful for practical applications. This represents a significant simplification over the original QBX algorithm, in which choosing a good set of parameters can be hard

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

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators