Trapezoidal Quadrature Rule Research Articles

Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods

The pursuit of more efficient and reliable numerical methods to solve nonlinear systems of equations has long intrigued many researchers. Among these, the Broyden method has stood out since its introduction, serving as a foundational technique from which various derivative methods have evolved. These derivative methods, commonly referred to as Broyden-like iterative methods, often surpass the traditional Broyden method in terms of both the number of iterations required and the computational time needed. This study aimed to develop new Broyden-like methods by incorporating weighted combinations of different quadrature rules. Specifically, the research focused on leveraging the Composite Trapezoidal rule with n=3n=3, and comparing it against the Midpoint, Trapezoidal, and Simpson quadrature rules. By integrating these approaches, three novel methods were formulated. The findings revealed that several of these new methods demonstrated enhanced efficiency and robustness compared to their established counterparts. In a detailed comparative analysis with the classical Broyden method and other improved versions, the Midpoint–Composite Trapezoidal (M<i>T_3</i>) method emerged as the top performer. This method consistently provided superior numerical outcomes across all benchmark problems examined in the study. The results highlight the potential of these new methods to significantly advance the field of numerical analysis, offering more powerful tools for researchers and practitioners dealing with complex nonlinear systems of equations. Through this innovative approach, the study not only broadens the understanding of Broyden-like methods but also sets the stage for further advancements in the development of efficient numerical solutions.

Estimation of quadrature errors for layer potentials evaluated near surfaces with spherical topology

Numerical simulations with rigid particles, drops, or vesicles constitute some examples that involve 3D objects with spherical topology. When the numerical method is based on boundary integral equations, the error in using a regular quadrature rule to approximate the layer potentials that appear in the formulation will increase rapidly as the evaluation point approaches the surface and the integrand becomes sharply peaked. To determine when the accuracy becomes insufficient, and a more costly special quadrature method should be used, error estimates are needed. In this paper, we present quadrature error estimates for layer potentials evaluated near surfaces of genus 0, parametrized using a polar and an azimuthal angle, discretized by a combination of the Gauss-Legendre and the trapezoidal quadrature rules. The error estimates involve no unknown coefficients, but complex-valued roots of a specified distance function. The evaluation of the error estimates in general requires a one-dimensional local root-finding procedure, but for specific geometries, we obtain analytical results. Based on these explicit solutions, we derive simplified error estimates for layer potentials evaluated near spheres; these simple formulas depend only on the distance from the surface, the radius of the sphere, and the number of discretization points. The usefulness of these error estimates is illustrated with numerical examples.

Multi-Point Flux MFE Decoupled Method for Compressible Miscible Displacement Problem

In this paper, a multi-point flux mixed-finite-element decoupled method was considered for the compressible miscible displacement problem. For this compressible problem, a fully discrete backward Euler scheme was proposed, in which the velocity and pressure equations were decoupled by a multi-point flux MFE method using BDM1 elements combined with a trapezoidal quadrature rule. The concentration equation was handled by a standard FE method. The error analysis for velocity, pressure, and concentration were rigorously derived. Numerical experiments to verify the convergence rates and simulate the miscible displacement problem of a water–oil system were presented.

Tosoh reports its consolidated results for fiscal 2022: Speciality Group

Density tracking by quadrature (DTQ) is a numerical procedure for computing solutions to Fokker-Planck equations that describe probability densities for stochastic differential equations (SDEs). In this paper, we extend upon existing trapezoidal quadrature rule DTQ procedures by utilizing a flexible quadrature rule that allows for unstructured, adaptive meshes. We describe the procedure for N-dimensions, and demonstrate that the resulting adaptive procedure can be significantly more efficient than the trapezoidal DTQ method. We show examples of our procedure for problems ranging from one to five dimensions.

Adaptive density tracking by quadrature for stochastic differential equations

Density tracking by quadrature (DTQ) is a numerical procedure for computing solutions to Fokker-Planck equations that describe probability densities for stochastic differential equations (SDEs). In this paper, we extend upon existing trapezoidal quadrature rule DTQ procedures by utilizing a flexible quadrature rule that allows for unstructured, adaptive meshes. We describe the procedure for N-dimensions, and demonstrate that the resulting adaptive procedure can be significantly more efficient than the trapezoidal DTQ method. We show examples of our procedure for problems ranging from one to five dimensions.

The determination of the aggregate loss distribution through the numerical inverse of the characteristic function using the trapezoidal quadrature rule

Aggregate loss is the total loss suffered by an insured in a certain period. The aggregate loss depends on the claim frequency and the amount of the claim each time the insured makes a claim. The distribution of aggregate losses must be known to calculate motor vehicle insurance premiums. In general, there are two methods that can be used in determining the distribution of aggregate losses, namely exact and numerical. When an exact solution is difficult to find, numerical methods such as Monte Carlo, Panjer Recursion, and Fast Fourier Transform can be used. This research will discuss the determination of the distribution of aggregate losses through the numerical inverse of the characteristic function using the trapezoidal quadrature rule, on the data of motor vehicle insurance category 7 in Indonesia. The estimated cumulative distribution function for the largest aggregate loss is 0.999993. When x=0, it means that if someone does not file a claim, the estimated value of the cumulative distribution function is 0.9293. This value is close to the percentage of the number of insured, which is 0.9241.

Corrected trapezoidal rules for boundary integral equations in three dimensions

The manuscript describes a quadrature rule that is designed for the high order discretization of boundary integral equations (BIEs) using the Nyström method. The technique is designed for surfaces that can naturally be parameterized using a uniform grid on a rectangle, such as deformed tori, or channels with periodic boundary conditions. When a BIE on such a geometry is discretized using the Nyström method based on the Trapezoidal quadrature rule, the resulting scheme tends to converge only slowly, due to the singularity in the kernel function. The key finding of the manuscript is that the convergence order can be greatly improved by modifying only a very small number of elements in the coefficient matrix. Specifically, it is demonstrated that by correcting only the diagonal entries in the coefficient matrix, \(O(h^{3})\) convergence can be attained for the single and double layer potentials associated with both the Laplace and the Helmholtz kernels. A nine-point correction stencil leads to an \(O(h^5)\) scheme. The method proposed can be viewed as a generalization of the quadrature rule of Duan and Rokhlin, which was designed for the 2D Lippmann–Schwinger equation in the plane. The techniques proposed are supported by a rigorous error analysis that relies on Wigner-type limits involving the Epstein zeta function and its parametric derivatives.

A randomized trapezoidal quadrature

A randomized trapezoidal quadrature rule is proposed for continuous functions which enjoy less regularity than commonly required. Indeed, we consider functions in some fractional Sobolev space. Various error bounds for the randomized rule are established while an error bound for classical trapezoidal quadrature is obtained for comparison. The randomized trapezoidal quadrature rule is shown to improve the order of convergence by half.

Zeta correction: a new approach to constructing corrected trapezoidal quadrature rules for singular integral operators

A high-order accurate quadrature rule for the discretization of boundary integral equations (BIEs) on closed smooth contours in the plane is introduced. This quadrature can be viewed as a hybrid of the spectral quadrature of Kress (Math. Comput. Model. 15(3-5), 229–243 1991) and the locally corrected trapezoidal quadrature of Kapur and Rokhlin (SIAM J. Numer. Anal. 34(4), 1331–1356, 1997). The new technique combines the strengths of both methods, and attains high-order convergence, numerical stability, ease of implementation, and compatibility with the “fast” algorithms (such as the Fast Multipole Method or Fast Direct Solvers). Important connections between the punctured trapezoidal rule and the Riemann zeta function are introduced, which enable a complete convergence analysis and lead to remarkably simple procedures for constructing the quadrature corrections. The paper reports a detailed comparison between the new method and the methods of Kress, of Kapur and Rokhlin, and of Alpert (SIAM J. Sci. Comput. 20(5), 1551–1584, 1999).

A new family of generalized quadrature methods for solving nonlinear equations

Weerakoon and Fernando [A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93] were resorted on a trapezoidal quadrature rule to derive an arithmetic mean Newton method with third-order convergence of the iterative scheme to solve nonlinear equations. Different quadrature methods have been developed, which form a special class of third-order iterative schemes requiring three evaluations of functions on [Formula: see text] per iteration, where [Formula: see text] is generated from the first Newton step. As an extension of these methods, we derive a new family of iterative schemes by using a new weight function [Formula: see text] to generalize the quadrature methods, of which [Formula: see text] signifies an approximate area under the curve [Formula: see text] between [Formula: see text] and [Formula: see text]. Then, a generalization of the midpoint Newton method is obtained by using another weight function, which is based on three evaluations of functions on [Formula: see text] per iteration. The sufficient conditions of these two weight functions are derived, which guarantee that the convergence order of the proposed iterative schemes is three.

Generalized multiscale approximation of a mixed finite element method with velocity elimination for Darcy flow in fractured porous media

In this paper, we propose a multiscale method for solving the Darcy flow of a single-phase fluid in two-dimensional fractured porous media. We consider a discrete fracture-matrix (DFM) model that treats fractures as one-dimensional objects, and flows in both the matrix and fractures respect the Darcy’s law. A multipoint flux mixed finite element (MFMFE) method with the broken Raviart–Thomas (RT12) element and the trapezoidal quadrature rule is employed to approximate the matrix velocity and pressure, which results in a block diagonal, symmetric and positive definite mass matrix for the matrix velocity on general quadrilateral grids; the one-dimensional RT0 mixed finite element method with the one-dimensional trapezoidal quadrature rule is exploited to approximate the fracture velocity and pressure, which leads to a diagonal and positive definite mass matrix for the fracture velocity in each single fracture. All these features of the obtained mass matrices allow for velocity elimination. Multiscale basis functions are constructed for the two-dimensional matrix pressure following the generalized multiscale finite element method (GMsFEM) framework to capture the fine-scale information of heterogeneous fractured porous media and effectively reduce the degrees of freedom for the matrix pressure, while fine-grid basis functions are utilized for the one-dimensional fracture pressure in fractures. Various numerical tests with the oversampling technique for different fracture distributions are performed to show that the proposed multiscale method is effective and able to provide good approximations for the fine-grid solution.

Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy–Forchheimer model

In this paper, we propose a multiscale method for the Darcy–Forchheimer model in highly heterogeneous porous media. The problem is solved in the framework of generalized multiscale finite element method (GMsFEM) combined with a multipoint flux mixed finite element (MFMFE) method. We consider a MFMFE method that utilizes the lowest order Brezzi–Douglas–Marini (BDM1) mixed finite element spaces for the approximation of velocity and pressure. The symmetric trapezoidal quadrature rule is employed for the integral of bilinear forms related to velocity variables so that the local velocity elimination is allowed which leads to a cell-centered system for pressure. We construct the multiscale space for pressure and solve the problem on the coarse grid following the GMsFEM framework. In the offline stage, we construct local snapshot spaces and perform spectral decompositions to get the offline space with a smaller dimension. In the online stage, we use Newton iterative algorithm to solve the nonlinear problem and obtain the offline solution, which reduces the number of iterations greatly compared to the standard Picard iterative algorithm. Based on the offline basis functions and the offline solution, we calculate online basis functions on each coarse element to enrich the multiscale space iteratively. The online basis functions contain the important global information and are effective to reduce relative errors substantially. Numerical examples are provided to highlight the performance of the proposed multiscale method.

A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations

The search for a more efficient and robust numerical method for solving problems have become an interesting area for many researchers as most problems resulting into nonlinear system of equations would require a very good numerical method for its computation. The introduction of the Broyden method has served as the foundation to developing several others, which are referred to as Broyden – like methods by some authors. These methods, in most cases, have proven to be superior to the original classical Broyden method in terms of the number of iterations needed and the CPU time required to reach a solution. This research sought to develop new Broyden – like methods using weighted combinations of quadrature rules (i.e., Simpson -1/3 and Simpson -3/8 rules against Midpoint, Trapezoidal, and Simpson quadrature rules). The weighted combination of the quadrature rules in the development of the new methods led to the discovery of several new methods. Some of which have proven to be more efficient and robust when compared with some existing methods. A comparison of these newly developed methods with the classical Broyden method together with some existing improved Broyden method revealed that, one of the newly developed methods namely, Midpoint–Simpson-3/8 (MS–3/8) method, outperformed all the others, with the MS–3/8 method giving the best of numerical results in all the benchmark problems considered in the study.

A multipoint stress mixed finite element method for elasticity on quadrilateral grids

AbstractWe develop a multipoint stress mixed finite element method for linear elasticity with weak stress symmetry on quadrilateral grids, which can be reduced to a symmetric and positive definite cell centered system. The method utilizes the lowest order Brezzi–Douglas–Marini finite element spaces for the stress and the trapezoidal quadrature rule in order to localize the interaction of degrees of freedom, which allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell‐centered system for displacement and rotation. The second uses a continuous piecewise bilinear rotation and trapezoidal quadrature rule for the asymmetry bilinear form. This allows for further elimination of the rotation, resulting in a cell‐centered system for the displacement only. Stability and error analysis is performed for both methods. First‐order convergence is established for all variables in their natural norms. A duality argument is employed to prove second order superconvergence of the displacement at the cell centers. Numerical results are presented in confirmation of the theory.

A Crank–Nicolson-type finite-difference scheme and its algorithm implementation for a nonlinear partial integro-differential equation arising from viscoelasticity

A Crank–Nicolson-type finite-difference scheme is proposed and analyzed for a nonlinear partial integro-differential equation arising from viscoelasticity. The time derivative is approximated by the Crank–Nicolson scheme and the Riemann–Liouville fractional integral term is treated by means of the trapezoidal convolution quadrature rule. To construct a fully discrete difference scheme, the standard centered difference formula is utilized to approximate the second-order spatial derivative and the Galerkin method based on piecewise linear test functions is used to discrete the nonlinear convection term. Theoretical results of stability and convergence are derived using the non-negative character of the real quadratic form associated with the convolution quadrature, and combining with the discrete Gronwall inequality. Besides, a fixed point iterative numerical algorithm, which fills the gap that the existed numerical schemes have only theoretical analysis but no numerical results, is presented. Numerical results show the efficiency and feasibility of our scheme, and the orders of convergence are in good agreement with the theoretical results.

The Crank-Nicolson-type Sinc-Galerkin method for the fourth-order partial integro-differential equation with a weakly singular kernel

In this work, a Sinc-Galerkin method is considered and analyzed for solving the fourth-order partial integro-differential equation with a weakly singular kernel. The time derivative and Riemann-Liouville fractional integral term are approximated via the Crank-Nicolson method and the trapezoidal convolution quadrature rule, respectively. Then a fully discrete scheme is formulated via using the Sinc-Galerkin approximation. The exponential convergence rate in space of proposed method are derived. In addition, some properties of the Toeplitz matrix generated by the composite Sinc function at the Sinc node are extended to the cases of arbitrary order in the preliminary knowledge. Finally, some numerical examples are calculated to verify the accuracy and effectiveness of our method.

On fractional Kakutani–Matsuuchi water wave model: Implementing a reliable implicit finite difference method

In this study, a reliable implicit finite difference method based on the modified trapezoidal quadrature rule, backward Euler differences, nonstandard central approximations, and the Hadamard finite‐part integral is being considered to solve a viscous asymptotical model named as fractional Kakutani–Matsuuchi water wave model. The fractional derivative is used in the Riemann–Liouville sense. Based on the properties of Brouwer's fixed‐point theorem, the existence, uniqueness, convergence, and stability of the proposed method are proved. Furthermore, we show that the global convergence order of the method in maximum norm is O(τ, hmin{β,3 − α}), where 0 < α ≤ 1 and β > 0 are the order of fractional derivative and the Lipschitz constant. Also, τ and h are the time step and space step, respectively. Finally, several examples are used to illustrate the accuracy and performance of the method.

Numerical solution of two-dimensional nonlinear fuzzy delay integral equations via iterative method and trapezoidal quadrature rule

In recent years, fuzzy delay integral equation has received a considerable amount of attention. From the viewpoint of engineering, time delays in controllers may induce complex dynamic behavior and will limit the performance of the active control. Therefore, time delays play an important role in the dynamic behavior of active control systems. In our paper, we introduce an iterative method using trapezoidal quadrature formula for solving two-dimensional nonlinear fuzzy delay integral equations. In addition, the existence and uniqueness of the solution of these equations are proved. Also, the convergence analysis and a criterion to stop of the presented method are investigated. Moreover, we prove numerical stability analysis of the method through choosing the perturbation in the first iteration. Eventually, some numerical examples are provided to demonstrate the applicability of this procedure.

Estimating the distribution of a stochastic sum of IID random variables

Abstract Suggested is a non-parametric method and algorithm for estimating the probability distribution of a stochastic sum of independent identically distributed continuous random variables, based on combining and numerically inverting the associated empirical characteristic function (CF) derived from the observed data. This is motivated by classical problems in financial risk management, actuarial science, and hydrological modelling. This approach can be naturally generalized to more complex semi-parametric modelling and estimating approaches, e.g., by incorporating the generalized Pareto distribution fit for modelling heavy tails of the considered continuous random variables, or by considering the weighted mixture of the parametric CFs (used to incorporate the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the observed or historical data). The suggested numerical approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the associated CF and the trapezoidal quadrature rule used for the required numerical integration. The presented non-parametric estimation method is related to the bootstrap estimation approach, and thus, it shares similar properties. Applicability of the proposed estimation procedure is illustrated by estimating the aggregate loss distribution from the well-known Danish fire losses data.

Numerical Computations of Nonlocal Schrödinger Equations on the Real Line

The numerical computation of nonlocal Schrodinger equations (SEs) on the whole real axis is considered. Based on the artificial boundary method, we first derive the exact artificial nonreflecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458–3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonreflecting boundary condition and leads us to reformulate the original infinite discrete system into an equivalent finite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are finally provided to demonstrate the effectiveness of our approach.