Computing the demagnetizing tensor for finite difference micromagnetic simulations via numerical integration

In the finite difference method which is commonly used in computational micromagnetics, the demagnetizing field is usually computed as a convolution of the magnetization vector field with the demagnetizing tensor that describes the magnetostatic field of a cuboidal cell with constant magnetization. An analytical expression for the demagnetizing tensor is available, however at distances far from the cuboidal cell, the numerical evaluation of the analytical expression can be very inaccurate. Due to this large-distance inaccuracy numerical packages such as OOMMF compute the demagnetizing tensor using the explicit formula at distances close to the originating cell, but at distances far from the originating cell a formula based on an asymptotic expansion has to be used. In this work, we describe a method to calculate the demagnetizing field by numerical evaluation of the multidimensional integral in the demagnetization tensor terms using a sparse grid integration scheme. This method improves the accuracy of computation at intermediate distances from the origin. We compute and report the accuracy of (i) the numerical evaluation of the exact tensor expression which is best for short distances, (ii) the asymptotic expansion best suited for large distances, and (iii) the new method based on numerical integration, which is superior to methods (i) and (ii) for intermediate distances. For all three methods, we show the measurements of accuracy and execution time as a function of distance, for calculations using single precision (4-byte) and double precision (8-byte) floating point arithmetic. We make recommendations for the choice of scheme order and integrating coefficients for the numerical integration method (iii)

Harmonic Shears and Numerical Conformal Mappings

In this article we introduce a numerical algorithm for finding harmonic mappings by using the shear construction introduced by Clunie and Sheil-Small in 1984. The MATLAB implementation of the algorithm is based on the numerical conformal mapping package, the Schwarz-Christoffel toolbox, by T. Driscoll. Several numerical examples are given. In addition, we discuss briefly the minimal surfaces associated with harmonic mappings and give a numerical example of minimal surfaces.Comment: 15 pages, 6 figure

Rapid evaluation of radial basis functions

Over the past decade, the radial basis function method has been shown to produce high quality solutions to the multivariate scattered data interpolation problem. However, this method has been associated with very high computational cost, as compared to alternative methods such as finite element or multivariate spline interpolation. For example. the direct evaluation at M locations of a radial basis function interpolant with N centres requires O(M N) floating-point operations. In this paper we introduce a fast evaluation method based on the Fast Gauss Transform and suitable quadrature rules. This method has been applied to the Hardy multiquadric, the inverse multiquadric and the thin-plate spline to reduce the computational complexity of the interpolant evaluation to O(M + N) floating point operations. By using certain localisation properties of conditionally negative definite functions this method has several performance advantages against traditional hierarchical rapid summation methods which we discuss in detail

Numerical Integration in S-PLUS or R: A Survey

This paper reviews current quadrature methods for approximate calculation of integrals within S-Plus or R. Starting with the general framework, Gaussian quadrature will be discussed first, followed by adaptive rules and Monte Carlo methods. Finally, a comparison of the methods presented is given. The aim of this survey paper is to help readers, not expert in computing, to apply numerical integration methods and to realize that numerical analysis is an art, not a science.

A new representation of generalized averaged Gauss quadrature rules

Gauss quadrature rules associated with a nonnegative measure with support on (part of) the real axis find many applications in Scientific Computing. It is important to be able to estimate the quadrature error when replacing an integral by an l-node Gauss quadrature rule in order to choose a suitable number of nodes. A classical approach to estimate this error is to evaluate the associated (2l + 1)-node Gauss-Kronrod rule. However, Gauss-Kronrod rules with 2l + 1 real nodes might not exist. The (2l + 1)-node generalized averaged Gauss formula associated with the l-node Gauss rule described in Spalevic (2007) [16] is guaranteed to exist and provides an attractive alternative to the (2l + 1)-node Gauss-Kronrod rule. This paper describes a new representation of generalized averaged Gauss formulas that is cheaper to evaluate than the available representation

Rational Averaged Gauss Quadrature Rules

It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules

Highly accurate quadrature-based Scharfetter--Gummel schemes for charge transport in degenerate semiconductors

We introduce a family of two point flux expressions for charge carrier transport described by drift-diffusion problems in degenerate semiconductors with non-Boltzmann statistics which can be used in Vorono"i finite volume discretizations. In the case of Boltzmann statistics, Scharfetter and Gummel derived such fluxes by solving a linear two point boundary value problem yielding a closed form expression for the flux. Instead, a generalization of this approach to the nonlinear case yields a flux value given implicitly as the solution of a nonlinear integral equation. We examine the solution of this integral equation numerically via quadrature rules to approximate the integral as well as Newton's method to solve the resulting approximate integral equation. This approach results into a family of quadrature-based Scharfetter-Gummel flux approximations. We focus on four quadrature rules and compare the resulting schemes with respect to execution time and accuracy. A convergence study reveals that the solution of the approximate integral equation converges exponentially in terms of the number of quadrature points. With very few integration nodes they are already more accurate than a state-of-the-art reference flux, especially in the challenging physical scenario of high nonlinear diffusion. Finally, we show that thermodynamic consistency is practically guaranteed

A new representation of generalized averaged Gauss quadrature rules

Gauss quadrature rules associated with a nonnegative measure with support on (part of) the real axis find many applications in Scientific Computing. It is important to be able to estimate the quadrature error when replacing an integral by an l-node Gauss quadrature rule in order to choose a suitable number of nodes. A classical approach to estimate this error is to evaluate the associated (2l + 1)-node Gauss-Kronrod rule. However, Gauss-Kronrod rules with 2l + 1 real nodes might not exist. The (2l + 1)-node generalized averaged Gauss formula associated with the l-node Gauss rule described in Spalevic (2007) [16] is guaranteed to exist and provides an attractive alternative to the (2l + 1)-node Gauss-Kronrod rule. This paper describes a new representation of generalized averaged Gauss formulas that is cheaper to evaluate than the available representation