Gauss-Legendre Quadrature Rule Research Articles

Gravity field forward modelling using tesseroids accelerated by Taylor series expansion and symmetry relations

SUMMARY In this study, we developed a new method that can significantly accelerate the forward modelling of gravity fields generated by large-scale tesseroids while keeping the computational accuracy as high as possible. The cost of the high efficiency is that the method only works under the assumptions that (1) all tesseroids in the same latitude band have the same horizontal dimension, (2) the computation points are located at the same surface level and aligned with the horizontal centres of tesseroids and (3) each tesseroid has a constant or linearly varying density. The new method first integrates the kernel function of the Newton’s volume integral analytically in the radial direction to eliminate its dependence on the vertical dimension of the tesseroid, and then expands the integrated kernel function into a Taylor series up to a certain order. Because the Taylor series expansion term of the integrated kernel function is an odd or even function of the difference between the longitudes of the tesseroid and computation point, there exist shifting or swapping symmetry relations among the gravity field of tesseroids. Consequently, the shifting or swapping symmetry is extended to the tesseroids with unequal vertical dimensions. Numerical experiments using the spherical shell model are conducted to verify the effectiveness of the new method. The results show that the computational speed of the new method is about 30 times faster than that of the traditional method, which employs the Gauss–Legendre quadrature rule and a 2-D adaptive subdivision approach, while keeping almost the same computational accuracy. When applying the new method to an ice shell with unequal thicknesses, the results reveal that the relative errors of calculating V, Vz and Vzz are smaller than 10−8, 10−6 and 10−4, respectively if the Taylor series expansion is truncated at order 4, while the computational time consumed by the new method is about 7 times less than that of the traditional method. Finally, the influence of the truncation order on the computational accuracy and the strategies for dividing the latitude band into several parts to further improve the accuracy are discussed.

A regularization of the parametric integral equation system applied to 2D boundary problems for Laplace’s equation with stability evaluation

The paper proposes the regularization of the parametric integral equation system (PIES) for 2D potential problems governed by Laplace’s equation. The regularization eliminates all singularities (weak and strong) in PIES associated with its kernels as a result of their modification and is based on introducing a regularizing function with unknown regularization coefficients. Hence, all integrals appearing in the regularized PIES formulation are non-singular and can be evaluated numerically with a standard low order Gauss–Legendre quadrature rule. Using this formulation, the stability of solutions for different distances between collocation points and quadrature nodes is investigated. The influence of the number of integration and collocation points on the accuracy and stability of solutions for the proposed regularization in comparison with dedicated procedures for evaluation of strongly and weakly singular integrals is also studied. Furthermore, we analyze the impact of the regularization in PIES on the results compared with other methods for evaluating singular integrals in non-regularized PIES.

A new approach for direct discretization of fractional order operator in delta domain

The fractional order system (FOS) comprises fractional order operator. In order to obtain the discretized version of the fractional order system, the first step is to discretize the fractional order operator, commonly expressed as s?m, 0 < m < 1. The fractional order operator can be used as fractional order differentiator or integrator, depending upon the values of . In general, there are two approaches for discretization of fractional order operator, one is indirect method of discretization and another is direct method of discretization. The direct discretization method capitalizes the method of formation of generating function where fractional order operator s?m is expressed as a function of Z in the shift operator parameterization and continued fraction expansion (CFE) method is then utilized to get the corresponding discrete domain rational transfer function. There is an inherent problem with this discretization method using shift operator parameterization (discrete Z-domain). At fast sampling time, the discretized version of the continuous time operator or system should resemble that of the continuous time counterpart if the sampling theorem is satisfied. At very high sampling rate, the shift operator parameterized system fails to provide meaningful information due to its numerical ill conditioning. To overcome this problem, Delta operator parameterization for discretization is considered in this paper, where at fast sampling rate, the continuous time results can be obtained from the discrete time experiments and therefore a unified framework can be developed to get the discrete time results and continuous time results hand to hand. In this paper a new generating function is proposed to discretize the fractional order operator using the Gauss-Legendre 2-point quadrature rule. Additionally, the function has been expanded using the CFE in order to obtain rational approximation of the fractional order operator. The detailed mathematical formulations along with the simulation results in MATLAB, with different fractional order systems are considered, in order to prove the newness of this formulation for discretization of the FOS in complex Delta domain.

The spectral collocation method for solving a fractional integro-differential equation

<abstract><p>In this paper, we propose a high-precision numerical algorithm for a fractional integro-differential equation based on the shifted Legendre polynomials and the idea of Gauss-Legendre quadrature rule and spectral collocation method. The error analysis of this method is also given in detail. Some numerical examples are give to illustrate the exponential convergence of our method.</p></abstract>

Exponential convergence of the hp-version of the composite Gauss-Legendre quadrature for integrals with endpoint singularities

In this paper we propose and analyze an hp-version of the composite Gauss-Legendre quadrature rule which includes local subdivision of the integration domain and local variation of the number of quadrature nodes employed on each subinterval. We derive an a priori error estimate that is fully explicit in the local length of each subinterval, in the local number of quadrature nodes employed on each subinterval, and in the local regularity of the integrand. In particular, for integrands with singularities at one or both endpoints, we prove that exponential convergence can be achieved by using geometric partition of the integration domain and linearly increasing of the number of quadrature nodes in successive subintervals. Numerical experiments are provided to illustrate the theoretical results.

Numerical solution of a system of Volterra integral equations in application to avian-human influenza epidemic model

In this paper, we propose an efficient multistage method for solving the system of linear and nonlinear Volterra integral equations of the second kind. This numerical method is based on the Gauss-Legendre quadrature rule that obtains several values of unknown function at each step and it will be shown that the order of the convergence is $O(M^{−4})$, where M is the number of the nodes in the time discretization. The method has also the advantages of simplicity of application, less computational time, and useful performance for large intervals. In order to show the efficiency of the method, numerical results for the avian human influenza epidemic model are obtained that are comparable with the fourth order Runge-Kutta method.

A Rational Approximation Scheme for Computing Mittag-Leffler Function with Discrete Elliptic Operator as Input

In this work, we propose a new scheme based on numerical quadrature to calculate the two-parameter Mittag-Leffler function with discrete elliptic operator $$-{\mathcal {L}}_h$$ as input. Except pure mathematical interest from approximation theory, our consideration also arises from solving sub-diffusion equations numerically with time-independent diffusion coefficient. We obtain the scheme by applying Gauss-Legendre quadrature rule for the integral representation of the Mittag-Leffler function. Rigorous error analysis is carried out which shows that the scheme converges exponentially with the increase of quadrature nodes. The computational cost of the algorithm is solving K sparse linear systems with K the number of quadrature nodes. It is worth to point out that the scheme is completely parallel which can save much time if the dimension of $${\mathcal {L}}_h$$ is very large. Some numerical tests are provided to verify the efficiency and robustness of our scheme.

Finite difference/generalized Hermite spectral method for the distributed-order time-fractional reaction-diffusion equation on multi-dimensional unbounded domains

Distributed-order fractional differential equations, where the differential order is distributed over a range of values rather than being just a fixed value as it is in the classical differential equations, offer a powerful tool to describe multi-physics phenomena. In this article, we develop and analyze an efficient finite difference/generalized Hermite spectral method for the distributed-order time-fractional reaction-diffusion equation on one-, two-, and three-dimensional unbounded domains. Considering the Gauss-Legendre quadrature rule for the distributed integral term in temporal direction, we first approximate the original distributed-order time-fractional problem by the multi-term time-fractional differential equation. Then, we apply the L2-1σ formula for the discretization of the multi-term Caputo fractional derivatives. Moreover, we employ the generalized Hermite functions with scaling factor for the spectral approximation in space. The detailed implementations of the method are presented for one-, two-, and three-dimensional cases of the fractional problem. The stability and convergence of the method are strictly established, which shows that the proposed method is unconditionally stable and convergent with second-order accuracy in time. In addition, the optimal error estimate is derived for the space approximation. Finally, we perform numerical examples to support the theoretical claims.

Novel operational matrices for solving 2-dim nonlinear variable order fractional optimal control problems via a new set of basis functions

This paper provides an effective method for a class of 2-dim nonlinear variable order fractional optimal control problems (2DNVOFOCP). The technique is based on the new class of basis functions namely the generalized shifted Legendre polynomials. The dynamic constraint is described by a nonlinear variable order fractional differential equation where the fractional derivative is in the sense of Caputo. The 2-dim Gauss-Legendre quadrature rule together with the Lagrange multipliers method are utilized to find the solutions of the given 2DNVOFOCP. The convergence analysis of the presented method is investigated. The examined numerical examples manifest highly accurate results.

The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applications

In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given. Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial

Simulation for non-homogeneous transport equation by Nyström method

In this work we solve numerically the one-dimensional transport equation with semi-reflective boundary conditions and non-homogeneous domain. The proposed methodology consists of applying the Nyström method in order to discretize the integral formulation of this problem which is an equation involving weakly singular integral operators. For this purpose, analytical and computational techniques were applied to deal with the singularities. The Nyström method is an integral method which approximates the integral operator by a numerical quadrature and turns the integral equation into a finite dimensional linear system. This formulation allows us to use any function to describe both scattering cross section and total cross section. The algorithm is implemented in C language with the use of routines of GNU scientific library and computational techniques for code optimization. The scalar flux was calculated for two numerical quadrature, namely Gauss-Legendre quadrature and Boole's rule. The numerical results were determined for transport problem with homogeneous and non-homogeneous domains. In order to validate the proposed method-ology, our numerical results were compared with those from the literature and presented with several correct significant digits.

Verified Numerical Multiple Integration Based on Composite Gauss-Legendre Quadrature and Product Cubature Rules

Verified Numerical Multiple Integration Based on Composite Gauss-Legendre Quadrature and Product Cubature Rules

A numerical method for solving variable order fractional optimal control problems

This study is devoted to introducing a computational technique based on Bernstein polynomials to solve variable order fractional optimal control problems (VO-FOCPs). This class of problems generated by dynamical systems describe with variable order fractional derivatives in the Caputo sense. In the proposed method, the Bernstein operational matrix of the fractional variable-order derivatives will be derived. Then, this matrix is used to obtain an approximate solution to mentioned problems. With the use of Gauss-Legendre quadrature rule and the mentioned operational matrix, the considered VO-FOCPs are reduced to a system of equations that are solved to get approximate solutions. The obtained results show the accuracy of the numerical technique.

Exact Evaluation of Time-Domain Physical Optics Integral on Quadratic Triangular Surfaces

This article presents the exact evaluation of the physical optics (PO) integral in time-domain. The surface of a perfect electrically conducting (PEC) scatterer is modeled by quadratic triangles. Using the Radon transform interpretation, the PO integral is reduced to a line integral, which is formed by the intersection of the quadratic surface and the plane defined by the Dirac delta function. It is shown that the resulting line is a quadratic curve, e.g., an ellipse, in barycentric coordinates of the quadratic triangle, since the incident wave is a plane wave and the scattered fields are observed at the far-field. To evaluate the PO integral: 1) the PO integral is represented in barycentric coordinates and the type of the intersecting curve is determined; 2) an appropriate coordinate transformation is applied, e.g., elliptic coordinates for the ellipse; and then 3) the Gauss–Legendre quadrature rule (GLQR) is applied. It is shown that a suitable GLQR order yields the PO integral exactly. The validity and accuracy of the proposed method are demonstrated via numerical examples.

Mathematical modelling of streamwise velocity profile in open channels using Tsallis entropy

This study derived the vertical distribution of streamwise velocity in wide-open channels by maximizing Tsallis entropy, in accordance with the maximum entropy principle, subject to the total probability rule, and the constraints based on the conservation of mass, momentum, and energy. Entropy maximizing leads to a highly nonlinear differential equation for velocity, which was transformed into a relatively weaker nonlinear equation and then solved analytically using a non-perturbation approach that yielded a series solution. The convergence of the series solution was proved using both theoretical and numerical procedures. For the velocity profile assessment, we calculated the Lagrange multipliers and the entropy index by solving a system of nonlinear equations using the Gauss-Newton method after approximating the constraint integrals using Gauss-Legendre quadrature rule. The derived velocity profile was validated for some selected sets of laboratory and field data and was compared with the existing velocity profiles based on Rényi, Tsallis, and Shannon (with additional constraints) entropies. We found that the incorporation of additional constraints and the effect of the entropy index improved the velocity profile compared to the existing Tsallis and Rényi entropy-based velocity equations. Further, it was observed that the proposed model and the Shannon entropy-based model with additional constraints behaved the same for most of the data sets considered, as the corresponding values of the entropy index were close to 1, which is in agreement with the theoretical consideration. The methodology reported in this study can also be employed for addressing other open channel flow problems, such as sediment concentration and shear stress distribution.

Efficient technique for solving variable order fractional optimal control problems

We apply a novel computation approach to determine the numerical solution of variable-order fractional optimal control problems. The dynamic constraint of these problems is considered with variable-order (VO) fractional derivatives. The numerical procedure is based on the Genocchi polynomials. To carry out the suggested technique, the Genocchi operational matrix of the VO fractional integral is calculated. By utilizing the mentioned operational matrix and the Gauss–Legendre quadrature rule, a system of equations is made. Then, the produced system is simply solved. Some numerical examples are given. The obtained results show the applicability and usefulness of the numerical technique.

Semi-implicit Hermite–Galerkin Spectral Method for Distributed-Order Fractional-in-Space Nonlinear Reaction–Diffusion Equations in Multidimensional Unbounded Domains

In this paper, we construct an efficient Hermite–Galerkin spectral method for the nonlinear reaction–diffusion equations with distributed-order fractional Laplacian in multidimensional unbounded domains. By applying Gauss–Legendre quadrature rule for the distributed integral term, we first approximate the original distributed-order fractional problem by the multi-term fractional-in-space differential equation. Applying Hermite–Galerkin spectral method in space and backward difference method in time, we establish semi-implicit fully discrete scheme. For two- and three-dimensional cases of the original fractional problem, the linear systems are solved by the preconditioned conjugate gradients method. The main advantage of our method is that the original fractional problem is directly solved in the unbounded domains, thus avoiding the errors introduced by the domain truncations. The stability analysis is rigourously established, which shows that our scheme is unconditionally stable under suitable assumption on the nonlinear term. Several numerical examples are presented to validate both stability and accuracy of the numerical method. The numerical results of the fractional Allen–Cahn, Gray–Scott, and Belousov–Zhabotinskii models show that our semi-implicit methods produce good numerical solutions.

An integro-differential non-local model for cell migration and its efficient numerical solution

Cell migration is fundamental in a wide variety of physiological and pathological phenomena, being exploited in biomedical engineering as well. In this respect, we here present a hybrid non-local integro-differential model where a representative cell, reproduced by a point particle with an orientation, moves on a planar domain upon signals coming from environmental variables. From a numerical point of view, non-locality implies the need to evaluate integral terms which may present non-regular integrand functions because of heterogeneities in the environmental conditions and/or in cell sensing region. Having in mind multicellular applications, we here propose a robust computational method able to handle such non-regularities. The procedure is based on low order Runge–Kutta methods and on an ad hoc application of the Gauss–Legendre quadrature rule. The accuracy and efficiency of the resulting computational method is then tested by selected benchmark settings. In this context, the ad hoc application of the quadrature rule reveals to be crucial to obtain a high accuracy with a remarkably low number of quadrature nodes with respect to the standard Gauss–Legendre quadrature formula, and which thus results in a reduced overall computational cost. Finally, the proposed method is further coupled with the cubic spline interpolation scheme which allows to deal also with possible poor (i.e., point-wise defined) molecular spatial information. The performed simulations (which accounts also for different scenarios) show how the interpolation of the molecular variables affects the efficiency of the overall method and further justify the proposed procedure.

Close evaluation of layer potentials in three dimensions

We present a simple and effective method for evaluating double- and single-layer potentials for Laplace's equation in three dimensions close to the boundary. The close evaluation of these layer potentials is challenging because they are nearly singular integrals. The method we propose is based on writing these layer potentials in spherical coordinates where the point at which their kernels are peaked maps to the north pole. An N-point Gauss-Legendre quadrature rule is used for integration with respect to the polar angle rather than the cosine of the polar angle. A 2N-point periodic trapezoid rule is used to compute the integral with respect to the azimuthal angle which acts as a natural and effective averaging operation in this coordinate system. The numerical method resulting from combining these two quadrature rules in this rotated coordinate system yields results that are consistent with asymptotic behaviors of the double- and single-layer potentials at close evaluation distances. In particular, we show that the error in computing the double-layer potential, after applying a subtraction method, is quadratic with respect to the evaluation distance from the boundary, and the error is linear for the single-layer potential. We improve upon the single-layer potential by introducing an alternate approximation based on a perturbation expansion and obtain an error that is quadratic with respect to the evaluation distance from the boundary.

On nodal point sets for flux reconstruction

Nodal point sets, and associated collocation projections, play an important role in a range of high-order methods, including Flux Reconstruction (FR) schemes. Historically, efforts have focused on identifying nodal point sets that aim to minimise the L∞ error of an associated interpolating polynomial. The present work combines a comprehensive review of known approximation theory results, with new results, and numerical experiments, to motivate that in fact point sets for FR should aim to minimise the L2 error of an associated interpolating polynomial. New results include identification of a nodal point set that minimises the L2 norm of an interpolating polynomial, and a proof of the equivalence between such an interpolating polynomial and an L2 approximating polynomial with coefficients obtained using a Gauss–Legendre quadrature rule. Numerical experiments confirm that FR errors can be reduced by an order-of-magnitude by switching from popular point sets such as Chebyshev, Chebyshev–Lobatto and Legendre–Lobatto to Legendre point sets.