Method For Solution Of Equations Research Articles

Hybrid approaches for accelerated convergence of block-Jacobi iterative methods for solution of the neutron transport equation

The Parallel Block Jacobi (PBJ) [1,2] scheme is an iterative method with application in neutron transport solution which requires asynchronous sub-domain interfaces, where asynchronous here means flux continuity is not imposed across sub-domain interfaces. In this work, we present the development, analysis, and testing of hybrid methods which utilize both PBJ-type methods and synchronous sweep-based methods in a single iterative step. These methods are developed to reduce the penalty to iterative performance of the former incurred in problems containing optically thin cells, while maintaining the PBJ method's applicability to massively parallel solution on unstructured grids. As a prelude to this ultimate goal, this study is conducted in serial operation on 2-D Cartesian grids to facilitate implementation of the various tested iterative methods, to enable the reported spectral analysis, and for shorter computational execution times allowing the large amounts of performance data to be collected. The analysis consists of two Fourier analyses, demonstrating the sequential execution of a PBJ method and SI to be iteratively robust with respect to optical thickness. The results of these analyses and preliminary computational experimentation demonstrate the effectiveness of methods that execute PBJ and SI only in optically thick and thin regions of a problem, respectively. Further experimentation, including a parametric study, then demonstrates this approach to be effective when the thin region is solved using Inexact Parallel Block Jacobi (IPBJ), with sub-domains of a size such that the applicability to massively parallel solution on unstructured grids is not diminished. Finally, testing on realistic test problems containing large void regions confirms the viability of this approach. Performance comparisons include the traditional iterative acceleration methods Diffusion Synthetic Acceleration, Partial-current Nonlinear Diffusion Acceleration, and Adjacent-cell Preconditioner. The results show that while the hybrid PBJ methods generally consume more iterations than these traditional methods, our hybrid approach maintains an iteration count on the same order of magnitude as these acceleration methods, which are not fully developed for massively parallel solution on unstructured grids.

A New Type of Conjugate Gradient Technique for Solving Fuzzy Nonlinear Algebraic Equations

In this paper, we suggest a new method for the numerical solution of fuzzy nonlinear equations in parametric form using a new Conjugate Gradient Technique. Table of the numerical solution is given to show the efficiency of the proposed method and which is compared with classical algorithms such as (Fletcher and Reeves (FR), Polak and Ribiere (PRP), and Fletcher (CD)) techniques.

Revisiting high-order Taylor methods for astrodynamics and celestial mechanics

ABSTRACT We present heyoka, a new, modern and general-purpose implementation of Taylor’s integration method for the numerical solution of ordinary differential equations. Detailed numerical tests focused on difficult high-precision gravitational problems in astrodynamics and celestial mechanics show how our general-purpose integrator is competitive with and often superior to state-of-the-art specialized symplectic and non-symplectic integrators in both speed and accuracy. In particular, we show how Taylor methods are capable of satisfying Brouwer’s law for the conservation of energy in long-term integrations of planetary systems over billions of dynamical time-scales. We also show how close encounters are modelled accurately during simulations of the formation of the Kirkwood gaps and of Apophis’ 2029 close encounter with the Earth (where heyoka surpasses the speed and accuracy of domain-specific methods). heyoka can be used from both C++ and python, and it is publicly available as an open-source project.

A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order.

Coaxiality of Stepped Shaft Measurement Using the Structured Light Vision

A method is proposed to measure the coaxiality of stepped shafts based on line structured light vision. In order to solve the repeated positioning error of the measured shaft, the light plane equation solution method is proposed using movement distance and initial light plane equation. In the coaxiality measurement model, the equation of the reference axis is obtained by the overall least square method through the center point coordinates of each intercept line on the reference axis. The coaxiality error of each shaft segment relative to the reference axis is solved based on the principle of minimum containment. In the experiment, the coaxiality measurement method is evaluated, and the factors that affect the measurement accuracy are analyzed.

An efficient high-order least square-based finite difference-finite volume method for solution of compressible Navier-Stokes equations on unstructured grids

This paper presents an efficient high-order finite volume method for solution of compressible Navier-Stokes equations on unstructured grids. In this method, a high-order polynomial which is based on Taylor series expansion is applied to approximate the solution function within each control cell. The derivatives in the Taylor series expansion are approximated by the functional values at the cell centers of the considered cell and its neighboring cells using the mesh-free least square-based finite difference (LSFD) scheme. Naturally, this least square-based finite difference-finite volume (LSFD-FV) method inherits appealing characteristics of the LSFD scheme, i.e., the simple algorithm and easy implementation. In addition, since the LSFD scheme is mesh-free in nature, the developed high-order LSFD-FV method is endowed with the flexibility to handle the multi-dimensional problems with complex geometries on arbitrary grids. Different from other high-order methods, this LSFD-FV method applies a novel and effective strategy, i.e., the discrete gas-kinetic flux solver (DGKFS), to compute numerical fluxes at the cell interface. In this way, the inviscid and viscous fluxes are simultaneously and effectively evaluated by local reconstruction of solutions for the Boltzmann equation. The efficient time marching strategy coupled with the high-order LSFD-FV method is adopted to solve the resultant ordinary differential equations with the matching accuracy. A series of numerical examples are presented to validate the accuracy, robustness and flexibility of the developed method on unstructured grids. Numerical results demonstrate the superior performance of the developed high-order method on simulating compressible viscous flows, in comparison with the low-order counterpart and the k-exact method of the same order of accuracy.

New Method for Numerical Solution of Z-Fractional Differential Equations

In this paper, at first, we introduce fractional differential equations with [Formula: see text]-valuation. Then, we propose a numerical method to approximate the solution. The proposed method is a hybrid method based on the corrected fractional Euler’s method and the probability distribution function. Moreover, the corrected fractional Euler’s method based on the generalized Taylor formula and the modified trapezoidal rule is proposed that this method can be used in the problems’ limitation section of the [Formula: see text]-fractional Initial value problem of order [Formula: see text] with the fuzzy Caputo fractional differential (fractional derivatives are defined on the basis of the Hukuhara differences and the generalized fuzzy derivatives). The probability function is based on exponential distribution function and used to represent the reliability of the problem limitation part. Finally, by two examples, we show that the proposed method can arbitrarily approximate the fractional differential equations with [Formula: see text]-valuation.

Improved bracketing parabolic method for numerical solution of nonlinear equations

An analysis of numerical methods for solving algebraic and transcendental equations in terms of speed is performed. In contrast to the reliable bisection method, the well-known parabolic methods by Müller, Brent, and Ridders have been determined to have higher speed, but do not always ensure the specified accuracy of the solution. Based on the data obtained, an improved parabolic method that guarantees accuracy confirmed by computational experiment is developed. This proposed method combines the quadratic speed of parabolic approaches and the ability to provide stable convergence to the solution for slowly varying functions inherent to the bisection method.

Four step hybrid block method for the direct solution of fourth order ordinary differential equations

‎This paper proposes a direct four-step implicit hybrid block method for directly solving general fourth-order initial value problems of ordinary differential equations‎. ‎In deriving this method‎, ‎the approximate solution in the form of power series is interpolated at four points‎, ‎i.e $ x_n,,‎, ‎x_{n+1},,x_{n+2},,x_{n+3} $ while its forth derivative is collocated at all grid points‎, ‎i.e $ x_n‎, ‎,,x_{n+frac{1}{4}},,‎, ‎x_{n+1}‎ , ‎,,x_{n+2}‎, ‎,,x_{n+frac{5}{2}}‎, ‎,,x_{n+3}‎, ‎,,x_{n+frac{7}{2}} $ and $ x_{n+4} $ to produce the main continuous schemes‎. ‎In order to verify the applicability of the new method‎, ‎the properties of the new method such as local truncation error‎, ‎zero stability‎, ‎order and convergence are also established‎. ‎The performance of the newly developed method is then compared with the existing methods in terms of error by solving the same test problems‎. ‎The numerical results reveal that the proposed method produces better accuracy than several existing methods when solving the same initial value problems (IVPs) of second order ODEs‎.

An optimal method for a staggered-grid finite-difference solution of elastic wave equations including rotational deformation

The staggered-grid finite-difference (SGFD) method is one of the most popular means used in seismic numerical modelling because of its computational efficiency and it is easy to implement. However, it will lead to serious numerical errors while using coarse grids or wavelet with high-frequency. In this work, we proposed an optimal spatial SGFD method based on a new improved particle swarm optimization (PSO) algorithm. Then the dispersion of numerical solution of the first-order spatial derivatives is analyzed. Meanwhile, numerical modelling of elastic wave equations and elastic wave equations including rotational deformation are performed with the optimal spatial SGFD method based on the improved PSO algorithm. Numerical dispersion analysis and numerical modeling results indicate that the optimal spatial SGFD method based on the improved PSO algorithm has high modeling accuracy and can efficiently suppress the numerical dispersion.

Optimization of one-step hybrid method for direct solution of fifth order ordinary differential equations of initial value problems

This paper focuses on the derivation, analysis and implementation of a hybrid method by optimizing the order of the method by introduction of six-hybrid points for direct solution of fifth order ordinary differential equations of initial value problems (IVPs). Power series was used as the basis function for the solution of the IVP. The basis function was interpolated at some selected hybrid points whereas the fifth derivative of the approximate solution was collocated at all the interval of integration of the method to generate a system of linear equations for the determination of the unknown parameters. The derived method was tested for consistency, zero stability, convergence and absolute stability. The method was tested with two linear test problems to confirm its accuracy and usability. The comparison of the results with some existing methods shows the superiority of the accuracy of the method.

Optimization of one-step hybrid method for direct solution of fifth order ordinary differential equations of initial value problems

Optimization of one-step hybrid method for direct solution of fifth order ordinary differential equations of initial value problems

Modified iteration method for numerical solution of nonlinear differential equations arising in science and engineering

In this paper, a modified iteration method (MIM) has been proposed to solve nonlinear second-order ODEs. Convergence analysis and error estimate of the proposed method are also discussed. Computational efficiency of this method is illustrated through numerical examples.

An efficient method for the numerical solution of the nonlinear Hammerstein fractional integral equations

In this paper, we present a numerical method for solving nonlinear Hammerstein fractional integral equations. The method approximates the solution by Picard iteration together with a numerical integration designed for weakly singular integrals. Error analysis of the proposed method is also investigated. Numerical examples approve its efficiency in terms of accuracy and computational cost.

Some number-theoretic methods for solving partial derivatives

In this paper, a new method is constructed for solving partial differential equations using a sequence of nested generalized parallelepiped grids. This method is a generalization and development of the V. S. Ryaben’kii and N. M. Korobov method for the approximate solution of partial differential equations for the case of using arbitrary generalized parallelepiped grids for integer lattices. The error of this method was also found. In the case of using an infinite sequence of nested generalized parallelepiped grids, a fairly fast convergence will take place. In addition, a variant of constructing optimal grids in the two-dimensional case is proposed. It is based on the integer approximation of algebraic lattices. In the two-dimensional case, the grids constructed in this way will always give generalized parallelepiped grids. Moreover, there are simple ways to assess the quality of the resulting meshes. One such method, based on the use of a hyperbolic parameter, is considered in this paper.

Operator splitting around Euler–Maruyama scheme and high order discretization of heat kernels

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker–Campbell–Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler–Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

Lattice Boltzmann Simulation of Plane Strain Problems

AbstractThe Lattice Boltzmann Method (LBM), e.g. in [3] and [4], can be interpreted as an alternative method for the numerical solution of partial differential equations. The LBM is usually applied to solve fluid flows. However, the interpretation of the LBM as a general numerical tool, allows to extend the LBM to solid mechanics as well. In this spirit, the LBM has been studied in recent years. First publications [5], [6] present a LBM scheme for the numerical solution of the dynamic behavior of a linear elastic solid under simplified deformation assumptions. For so‐called anti‐plane shear deformation, the only non‐zero displacement component is governed by the two‐dimensional wave equation. In this work, the existing LBM for the two‐dimensional wave equation is extended to more general plane strain problems. The algorithm reduces the plane strain problem to the solution of two separate wave equations for the volume dilatation and the non‐zero component of the rotation vector, respectively.

ЧИСЕЛЬНИЙ АНАЛІЗ КРУГЛИХ ПЛАСТИН НА ПРУЖНІЙ ОСНОВІ ЗІ ЗМІННИМ КОЕФІЦІЄНТОМ ПОСТЕЛІ

Abstract. The paper presents the results of a study of the stress-strain state of a circular plate of constant cylindrical stiffness, which lies on a variable elastic foundation and is under the influence of a continuously distributed transverse load. Twelve variants of calculation are considered ‒ six for a steel round plate and six more ‒ for a concrete round plate under two conditions of fixing and three different laws of variation of the bed coefficient. To solve the problem, the finite element method implemented in the LIRA-SAPR software package is used. It is noted that in the case when the bedding coefficient is a variable value depending on the coordinate in which the foundation settlement is determined, the analytical approach leads to the need to solve the corresponding differential equations with variable coefficients. Therefore, calculations of circular and annular plates lying on a variable elastic foundation by means of analytical solutions of differential equations are extremely rare in scientific periodicals and are of a private nature. An effective method for the analytical solution of differential equations with variable coefficients for a number of problems in mechanics was proposed by one of the authors of the article, however, the application of the method to the calculation of a circular plate on an elastic foundation with a variable bed coefficient requires verification, therefore, here we consider the features of the finite element analysis of such a plate under different boundary conditions and different laws of variation of the bed coefficient. In all versions, the results completely coincide with the known results of bending of slabs that do not have an elastic base and in the case when this base exists and its resistance is constant. The discrepancy here is very insignificant ‒ in the third significant digit after the decimal point for deflection when hinged and in the second for moments. In case of rigid clamping, the deflections and moments also differ from the corresponding values of the known solutions in the second significant digit after the decimal point.

Biorthogonal wavelet-based multigrid method for the numerical solution of elliptic partial differential equations

Wavelets have generated significant interest from both theoretical and applied researchers over the last few decades. We proposed biorthogonal wavelet-based multigrid method for the numerical solution of elliptic partial differential equations. The performance of the proposed scheme is better than the existing ones in terms of superconvergence with low computational time. Some of the illustrative examples demonstrate the applicability of the method.

Efficacious Analytical Technique Applied to Fractional Fornberg–Whitham Model and Two-Dimensional Fractional Population Model

This paper presents an efficacious analytical and numerical method for solution of fractional differential equations. This technique, here in named q-HATM (q-homotopy analysis transform method) is applied to a one-dimensional fractional Fornberg–Whitham model and a two-dimensional fractional population model emanating from biological sciences. The overwhelming agreement of our analytical solution by the q-HATM technique with the exact solution indeed establishes the efficacy of q-HATM to solve the fractional Fornberg–Whitham model and the two-dimensional fractional population model. Furthermore, comparisons by means of extensive analysis using numerics, graphs and error analysis are presented to affirm the preference of q-HATM technique over other methods. A variant of the q-HATM using symmetry can also be considered to solve these problems.