Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Advanced Mathematical Methods for Scientists and Engineers

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

Want to get experience? Want to get any ideas to create new things in your life? Read methods of numerical integration now! By reading this book as soon as possible, you can renew the situation to get the inspirations. Yeah, this way will lead you to always think more and more. In this case, this book will be always right for you. When you can observe more about the book, you will know why you need this.

Methods of Numerical Integration

THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

We propose a variant of the numerical method of steepest descent for oscillatory integrals by using a low-cost explicit polynomial approximation of the paths of steepest descent. A loss of asymptotic order is observed, but in the most relevant cases the overall asymptotic order remains higher than a truncated asymptotic expansion at similar computational effort. Theoretical results based on number theory underpinning the mechanisms behind this effect are presented.

/pdf/asymptotic-analysis-of-numerical-steepest-descent-with-path-49m482kfpy.pdf

Asymptotic Analysis of Numerical Steepest Descent with Path Approximations

The classical theory of Gaussian quadrature assumes a positive weight function. We will show that in some cases Gaussian rules can be constructed with respect to an oscillatory weight, yielding methods with complex quadrature nodes and positive weights. These rules are well suited for highly oscillatory integrals because they attain optimal asymptotic order. We show that for the Fourier oscillator this approach yields the numerical method of steepest descent, a method with optimal asymptotic order that has previously been proposed for this class of integrals. However, the approach readily extends to more general kernels, such as Bessel functions that appear as the kernel of the Hankel transform.

Complex Gaussian quadrature for oscillatory integral transforms

Several recent numerical schemes for high frequency scattering simulations are based on the extraction of known phase functions from an oscillatory solution. The remaining function is typically no longer oscillatory, and as such it can be approximated numerically with a number of degrees of freedom that does not depend on the frequency of the original problem. Knowledge of the phase of a solution typically comes from asymptotic analysis, for example, from geometrical optics. We consider integral equation formulations of time-harmonic scattering by a smooth and convex obstacle and focus on the so-called shadow boundaries. They are the points where the incoming waves are tangential to the boundary of the scatterer. We devise a numerical method that incorporates advanced results from asymptotic analysis which describe the frequency-dependent transitional behavior of the solution uniformly across these points. We describe and resolve an apparent conflict between two theories that describe the asymptotic behav...

/pdf/extraction-of-uniformly-accurate-phase-functions-across-4arqyg3uxc.pdf

Extraction of Uniformly Accurate Phase Functions Across Smooth Shadow Boundaries in High Frequency Scattering Problems

https://www.math.ntnu.no/preprint/numerics/2008/N5-2008.pdf

Local solutions to high-frequency 2D scattering problems

A formulation of the problem of scattering from obstacles with edges is presented. The formulation is based on decomposing the field into geometrical acoustics, first-order, and multiple-order edge diffraction components. An existing secondary-source model for edge diffraction from finite edges is extended to handle multiple diffraction of all orders. It is shown that the multiple-order diffraction component can be found via the solution to an integral equation formulated on pairs of edge points. This gives what can be called an edge source signal. In a subsequent step, this edge source signal is propagated to yield a multiple-order diffracted field, taking all diffraction orders into account. Numerical experiments demonstrate accurate response for frequencies down to 0 for thin plates and a cube. No problems with irregular frequencies, as happen with the Kirchhoff–Helmholtz integral equation, are observed for this formulation. For the axisymmetric scattering from a circular disc, a highly effective symmetric formulation results, and results agree with reference solutions across the entire frequency range.

Andreas Asheim

Papers

Asymptotic Analysis of Numerical Steepest Descent with Path Approximations

Complex Gaussian quadrature for oscillatory integral transforms

Extraction of Uniformly Accurate Phase Functions Across Smooth Shadow Boundaries in High Frequency Scattering Problems

Local solutions to high-frequency 2D scattering problems

An integral equation formulation for the diffraction from convex plates and polyhedra.