Simple Quadrature Formula Research Articles

The Numerical Solution of Fractional Black-Scholes-Schrodinger Equation Using the RBFs Method

In this paper, radial basis functions (RBFs) method was used to solve a fractional Black-Scholes-Schrodinger equation in an option pricing of financial problems. The RBFs method is applied in discretizing a spatial derivative process. The approximation of time fractional derivative is interpreted in the Caputo’s sense by a simple quadrature formula. This RBFs approach was theoretically proved with different problems of two numerical examples: time step arbitrage bubble case and time linear arbitrage bubble case. Then, the numerical results were compared with the semiclassical solution in case of fractional order close to 1. As a result, both numerical examples showed that the option prices from RBFs method satisfy the semiclassical solution.

Moment Preserving Local Spline Projection Operators

This article describes an elementary construction of a dual basis for nonuniform B-splines that is local, $$L^\infty $$-stable, and reproduces polynomials of any prescribed degree. This allows one to define local projection operators with near-optimal approximation properties in any $$L^q$$, $$1 \le q \le \infty $$, and high order moment preserving properties. As the dual basis functions share the same piecewise polynomial structure as the underlying splines, simple quadrature formulas can be used to compute the projected spline coefficients.

МЕТОД ІНТЕРПРЕТАЦІЙ ТА КВАДРАТУРИ ГАУССА

Будь-яка математична модель насправді є інтерпретацією природного, технологічного, розумового процесу математичною мовою. В наукових дослідженнях метод інтерпретацій зустрічається на кожному кроці. Достатньо нагадати про теорію графів, аналітичну геометрію, диференціальні рівняння, перетворення Лапласа, швидке перетворення Фур’є, теорію кодування тощо. В методі інтерпретацій, як правило, задача однієї області математики інтерпретується в іншій області, де вона або спрощується, або більше відповідає нашій інтуїції, або дозволяє використання інших підходів і т. ін. Ми звернули увагу на квадратури Гаусса не тільки тому, що саме вони використовуються в сучасних стандартних програмах інтегрування. Ми переконалися, що в квадратурах Гаусса є певний дидактичний потенціал, який може бути корисним для тих, хто вчиться і навчає математичному моделюванню. У роботі розглядається проста квадратурна формула Гаусса (два вузли інтегрування). Наведено приклади задач, в яких існує латентний зв’язок із квадратурою Гаусса. Ці задачі – своєрідна комбінація простоти і нетривіальності, в якій читач може знайти щось цікаве на свій смак. Природно, що кожна задача формулюється на двох "канонічних" інтервалах: [−1, 1] і [0, 1], щоб охопити дві версії квадратури: Гаусса-Лежандра і Гаусса-Бернуллі.

The meshless local Petrov-Galerkin method based on moving Kriging interpolation for solving the time fractional Navier-Stokes equations.

In this paper, we present a numerical scheme used to solve the nonlinear time fractional Navier–Stokes equations in two dimensions. We first employ the meshless local Petrov–Galerkin (MLPG) method based on a local weak formulation to form the system of discretized equations and then we will approximate the time fractional derivative interpreted in the sense of Caputo by a simple quadrature formula. The moving Kriging interpolation which possesses the Kronecker delta property is applied to construct shape functions. This research aims to extend and develop further the applicability of the truly MLPG method to the generalized incompressible Navier–Stokes equations. Two numerical examples are provided to illustrate the accuracy and efficiency of the proposed algorithm. Very good agreement between the numerically and analytically computed solutions can be observed in the verification. The present MLPG method has proved its efficiency and reliability for solving the two-dimensional time fractional Navier–Stokes equations arising in fluid dynamics as well as several other problems in science and engineering.

Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids

In this paper, we derive a general accuracy-preserving boundary flux quadrature formula for the second- and third-order node-centered edge-based finite-volume discretizations on triangular and tetrahedral grids. It is demonstrated that the general formula reduces to some well-known formulas for the second-order scheme in the case of linear fluxes. Some special boundary grids are also examined. In particular, a simple one-point quadrature formula, which is typically used for quadrilateral grids, is shown to be exact for quadratic fluxes on triangular grids with uniformly spaced straight boundaries. Numerical results are presented to demonstrate the accuracy of the general formula, and accuracy deterioration caused by incompatible boundary flux quadrature formulas. In general, the third-order accuracy is lost everywhere in the domain unless the third-order accuracy is maintained at boundary nodes. It is also demonstrated that the third-order scheme does not require high-order curved elements for curved boundaries but requires accurate surface normal vectors defined, which can be estimated by a quadratic interpolation from a given grid, to deliver the designed third-order accuracy.

Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

In the first part of this work, we derive compact numerical quadrature formulas for finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f(x)=g(x)|x?t| β , β being real. Depending on the value of β, these integrals are defined either in the regular sense or in the sense of Hadamard finite part. Assuming that g?C ?[a,b], or g?C ?(a,b) but can have arbitrary algebraic singularities at x=a and/or x=b, and letting h=(b?a)/n, n an integer, we derive asymptotic expansions for ${T}^{*}_{n}[f]=h\sum_{1\leq j\leq n-1,\ x_{j}\neq t}f(x_{j})$ , where x j =a+jh and t?{x 1,?,x n?1}. These asymptotic expansions are based on some recent generalizations of the Euler---Maclaurin expansion due to the author (A. Sidi, Euler---Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities, in Math. Comput., 2012), and are used to construct our quadrature formulas, whose accuracies are then increased at will by applying to them the Richardson extrapolation process. We pay particular attention to the case in which β=?2 and f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ , which arises in the context of periodic hypersingular integral equations. For this case, we propose the remarkably simple and compact quadrature formula $\widehat{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)-\pi^{2} g(t)h^{-1}$ , and show that $\widehat{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h?0 ?μ>0, and that it is exact for a class of singular integrals involving trigonometric polynomials of degree at most n. We show how $\widehat{Q}_{n}[f]$ can be used for solving hypersingular integral equations in an efficient manner. In the second part of this work, we derive the Euler---Maclaurin expansion for integrals $I[f]=\int^{b}_{a} f(x)dx$ , where f(x)=g(x)(x?t) β , with g(x) as before and β=?1,?3,?5,?, from which suitable quadrature formulas can be obtained. We revisit the case of β=?1, for which the known quadrature formula $\widetilde{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)$ satisfies $\widetilde{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h?0 ?μ>0, when f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ . We show that this formula too is exact for a class of singular integrals involving trigonometric polynomials of degree at most n?1. We provide numerical examples involving periodic integrands that confirm the theoretical results.

Endgame Problem Part 1: V-Infinity-Leveraging Technique and the Leveraging Graph

Renewed interest by ESA and NASA in missions to Europa, Ganymede, Enceladus, and Titan poses the question of how to best solve the endgame problem. Endgames typically aim at an inexpensive insertion maneuver into the science orbit and can be designed using either V ∞ -leveraging maneuvers or the multibody dynamics. Although historically linked to insertion maneuvers, the endgame problem is symmetric and equally applies to departure. In this two-part series, we analyze and draw connections between the two apparently separate approaches, providing insight into the dynamics of the multibody gravity-assist problem. In this paper we derive new formulas for the V ∞ -leveraging maneuver and build the leveraging graph to be used as a reference guide for designing endgame tours. We prove that the cost of a V ∞ -leveraging-maneuver sequence decreases when using high-altitude flybys (as done in the multibody technique). Finally, we find a simple quadrature formula to compute the minimum ΔV transfer between moons using V ∞ -leveraging maneuvers, which is the main result of the paper, and a method to estimate transfer times. The leveraging graphs and associated formulas are derived in canonical units and therefore apply to any celestial system with a smaller body in a circular orbit around a primary. Specifically, we demonstrate the new method to provide rapid calculations of the theoretical boundary values for ΔV requirements and estimated transfer times for moon tours in the Saturn and Jupiter systems using the V ∞ -leveraging-maneuver model.

ON FULLY DISCRETE COLLOCATION METHODS FOR SOLVING WEAKLY SINGULAR INTEGRO‐DIFFERENTIAL EQUATIONS

In order to find approximate solutions of Volterra and Fredholm integro‐differential equations by collocation methods it is necessary to compute certain integrals that determine the required algebraic systems. Those integrals usually can not be computed exactly and if the kernels of the integral operators are not smooth, simple quadrature formula approximations of the integrals do not preserve the convergence rate of the collocation method. In the present paper fully discrete analogs of collocation methods where non‐smooth integrals are replaced by appropriate quadrature formulas approximations, are considered and corresponding error estimates are derived. Presented numerical examples display that theoretical results are in a good accordance with the actual convergence rates of the proposed algorithms.

Fast computation of Toeplitz forms and some multidimensional integrals

A fast algorithm is proposed for computation of certain multilevel Toeplitz forms. It helps to implement some simple quadrature formulas in an efficient way, allowing for the use of more nodes. An application example is given with the volume integral equations in electromagnetics.

A mixed–FEM and BEM coupling for a three-dimensional eddy current problem

We study in this paper the electromagnetic field generated in an infinite cylindrical conductor by an alternating current density. The resulting interface problem (see [1]) between the metal and the dielectric medium is treated by a mixed-FEM and BEM coupling method. We prove that our BEM-FEM formulation is well posed and leads to a convergent Galerkin method with optimal error estimates. Furthermore, we introduce a fully discrete version of our Galerkin scheme based on simple quadrature formulas. We show that, if the parameter of discretization is sufficiently small, the fully discrete method is well posed and the error estimates remain unaltered.

A precise computation of stress intensity factor on the front of a convex planar crack

AbstractIn this paper, we consider a general integral expression for mode I stress intensity factor along the fronts of convex planar cracks. For this integral approximation, we develop a simple numerical quadrature formula on every convex set Ω which allows a precise estimation of the error. This permits the use of extrapolation techniques for the accurate computation of the integral. Copyright © 2002 John Wiley & Sons, Ltd.

An Efficient Algorithm for Calculating the Heat Capacity of a Large-Scale Molecular System

We present an efficient algorithm for computing the heat capacity of a large-scale molecular system. The new algorithm is based on a special Gaussian quadrature whose knots and weights are obtained by a simple Lanczos iteration. Our numerical results have indicated that this new computational scheme is quite accurate. We have also shown that this method is considerably faster than the earlier approach that is based on estimating the density of states and integrating with a simple quadrature formula.

A fully discrete BEM–FEM method for an exterior elasticity system in the plane

We present a modified version of the usual BEM–FEM coupling for the exterior elasticity problem in the plane (cf. Costabel and Stephan, SIAM J. Numer. Anal. 27 (1990) 1212–1226). This new formulation allows us to take advantage of techniques from Hsiao et al., Computing 25 (1980) 557–566 to compute the boundary integral terms using simple quadrature formulas. We provide error estimates for the Galerkin method and prove that the corresponding fully discrete scheme preserves the optimal rates of convergence.

A MIXED-FEM AND BEM COUPLING FOR A TWO-DIMENSIONAL EDDY CURRENT PROBLEM

We study in this paper the electromagnetic field generated in an infinite cylindrical conductor by an alternating current density. The resulting interface problem (see [1]) between the metal and the dielectric medium is treated by a mixed-FEM and BEM coupling method. We prove that our BEM-FEM formulation is well posed and leads to a convergent Galerkin method with optimal error estimates. Furthermore, we introduce a fully discrete version of our Galerkin scheme based on simple quadrature formulas. We show that, if the parameter of discretization is sufficiently small, the fully discrete method is well posed and the error estimates remain unaltered.

A FULLY DISCRETE GALERKIN SCHEME FOR A TWO-FOLD SADDLE POINT FORMULATION OF AN EXTERIOR NONLINEAR PROBLEM *

We analyze a fully discrete Galerkin method for the coupling of mixed finite elements and boundary elements as applied to an exterior nonlinear transmission problem arising in potential theory. We first show that the corresponding continuous formulation becomes a well posed two-fold saddle point problem. Our discrete approach uses Raviart-Thomas elements of lowest order and is based on simple quadrature formulas for the interior and boundary terms. We prove that, if the parameter of discretization is sufficiently small, the fully discrete Galerkin scheme is uniquely solvable and leads to optimal error estimates. *This research was partially supported by Fondecyt-Chile through the FONDAP Program in Applied Mathematics, by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program, and by D.G.E.S. through the project PB98-1564.

On a FEM--BEM Formulation for an Exterior Quasilinear Problem in the Plane

We use a version of the FEM--BEM method introduced by Costabel [ Boundary Elements IX, Vol. 1, C. A. Brebbia et al., eds., Springer-Verlag, 1987] and Han [ J. Comput. Math., 8 (1990), pp. 223--232] to discretize an exterior quasilinear problem. We provide error estimates for the Galerkin method and propose a fully discrete scheme based on simple quadrature formulas. Furthermore, we show that these numerical integration schemes preserve the optimal rates of convergence. Finally, we present results of numerical experiments involving our discretization method.

An Optimal Iterative Process for the Johnson--Nedelec Method of Coupling Boundary and Finite Elements

We reformulate the discretization of the Johnson--Nedelec method [11] of coupling boundary elements and finite elements for an exterior bidimensional Laplacian. This new formulation leads to optimal error estimates and allows the use of simple quadrature formulas for calculation of the boundary element matrix. We show that if the parameter of discretization is sufficiently small, the fully discrete scheme is well posed and the error estimates remain unaltered. The rest of the paper is devoted to the study of an efficient algorithm for solving the resulting discrete linear systems.

Unbounded divergence of simple quadrature formulas

Given a sequence of real or complex coefficients c i and a sequence of distinct nodes t i in a compact interval T, we prove the divergence and the unbounded divergence on superdense sets in the space C( T) of the simple quadrature formulas ∝ T x( t) du( t) = Q n ( x) + R n ( x) and ∝ T w( t) x( t) dt = Q n ( x) + R n ( x), where Q n ( x)=∑ i=1 m n c i x( t i ), ϵ C( T). The divergence (not certainly unbounded) for at most one continuous function of the first simple quadrature formula, with m n = n and u( t) = t, was established by P. J. Davis in 1953.

Relativistic continuum normalizations and phase shifts in a screened Coulomb potential from a relativistic generalized WKB approximation.

Simple quadrature formulas are obtained for relativistic continuum normalizations and phase shifts in a screened spherically symmetric central potential, utilizing a relativistic generalized WKB approximation based on the use of Coulomb wave functions as comparison functions. The derivation is based on our similar results for the nonrelativistic case, combined with our recent transformation of the Dirac equation into a modified Schr\"odinger equation. Excellent agreement is obtained in comparison with exact numerical calculations.

Algorithm 655

We present FORTRAN subroutines that implement the method described in [3] for the stable evaluation of the weights of interpolatory quadratures with prescribed simple or multiple knots. Given a set of knots and their multiplicities, the package generates the weights by using the zeroth moment μ 0 of w , the weight function in the integrand, and the (symmetric tridiagonal) Jacobi matrix J associated with the polynomials orthogonal on ( a, b ) with respect to w . There are utility routines that generate μ 0 and J for classical weight functions, but quadratures can be generated for any μ 0 and J supplied by the user. Utility routines are also provided that (1) evaluate a computed quadrature, applied to a user-supplied integrand, (2) check the polynomial order of precision of a quadrature formula, and (3) compute the knots and weights of simple Gaussian quadrature formula.