Fourth-order Convergence Research Articles

The composite Chebyshev integration method for radiative integral transfer equations in rectangular enclosures

Recently, the Chebyshev integration method was developed to solve the radiative integral transfer equations (RITEs) after singularity removing (IJTS, 149 (2020) 106,158), and shown fourth order convergence accuracy. This superior property makes the Chebyshev integration method attractive to produce results with quite high accuracy to be benchmark solutions. However, the Chebyshev integration method, which is a global method, is only suitable for problems with smooth parameters and temperature distribution. To overcome this drawback, in this paper, the composite Chebyshev integration method is proposed. The computational domain is divided into several subdomains, and then the Chebyshev quadrature is applied in each subdomain. Several benchmark problems in the rectangular medium with continuous/stepwise-change scattering albedo and emissions are solved. Increasing the grid number with fixed subdomains, one can observe that the composite Chebyshev integration method still has the high order convergence accuracy. Besides, the solutions rounded to seven significant digits are given in tabular form for convenience. They are also used as benchmark solutions to assess the collocation spectral method (CSM) and its modified version for radiative transfer equation (RTE). The results indicate that the CSM suffers from severe ray effect when the discontinuities appear in the scattering albedo and temperature distribution. Though the modified CSM can avoid the ray effect due to stepwise-change emissions, it requires heavier computational load but produces less accurate results than the composite Chebyshev integration method for RITEs under the same spatial grid system. Compared with the CSM or modified CSM to solve the RTE, the composite Chebyshev integration method for RITEs can achieve much higher accuracy with acceptable computational time, thus could also be a good alternative for thermal radiation calculations in simple geometry.

Modifikasi Metode Householder Tiga Parameter yang Bebas Turunan Kedua dengan Orde Konvergensi Optimal

The Householder’s method is one of the iterative methods with a third-order convergence that used to solve a nonlinear equation. In this paper, the authors modified the iterative method using the expansion of second order Taylor’s series and approximated its second derivative using equality of two the third-order iterative methods. Based on the results of the study, it was found that the new iterative method has a fourth-order of convergence and requires three evaluations of function with an efficiency index of 1,587401. Numerical simulation is given by using several functions to compare the performance between the new method with other iterative methods. The results of numerical simulation show that the performance of the new method is better than other iterative methods.

Towards pseudo-spectral incompressible smoothed particle hydrodynamics (ISPH)

In this paper a pseudo-spectral incompressible smoothed particle hydrodynamics (FFT-ISPH) solver is presented. While the solution of the linear system arising from the pressure Poisson equation in physical space using iterative solvers in incompressible SPH is a viable solution, it is widely accepted that the computational cost of solving the pressure Poisson equation by iterative solvers is excessive and affects negatively on the efficiency of the solver. The proposed scheme is an intermediate between a fully spectral and a standard incompressible SPH solver. Herein, the solution of the pressure Poisson equation is performed in spectral space whereas the discretisation and time integration are performed in physical space. This results in a second and higher-order scheme with the classical first-order projection and a third-order Runge-Kutta time integration scheme, respectively. A detailed performance analysis shows gains in computational cost of two orders of magnitude. Further, it is demonstrated that the solution of the pressure Poisson equation in spectral space is independent of the number of neighbouring nodes, in contrast to the iterative solver whose cost increases by a factor of three with smoothing length to particle size ratio. Periodic, bounded, and mixed boundary conditions test cases have been used to demonstrate the applicability, accuracy and robustness of the scheme with second and fourth-order convergence rates. The scheme opens an avenue for simulations of high-order incompressible flows in SPH where filtering operations in the frequency domain can be performed straightforwardly.

A higher-order accurate difference approximation of singularly perturbed reaction-diffusion problem using grid equidistribution

This paper proposes a higher-order numerical approximation scheme to solve singularly perturbed reaction–diffusion boundary value problems. The proposed scheme is a combination of a fourth-order numerical difference method and classical central difference method applied on a non-equidistant grid. The non-equidistant grid is generated by equi-distributing a non-negative monitor function. The theoretical and empirical error analysis demonstrate that the proposed scheme has fourth-order uniform convergence with respect to the perturbation parameter.

New fourth-order convergent algorithm for analysis of trusses with material and geometric nonlinearities

This paper presents a new algorithm to solve the system of nonlinear equations that describes the static equilibrium of trusses with material and geometric nonlinearities, adapting a three-step method with fourth-order convergence found in the literature. The co-rotational formulation of the Finite Element Method is used in the discretization of structures. The nonlinear behavior of the material is characterized by an elastoplastic constitutive model. The equilibrium paths with limit points of load and displacement are obtained using the linearized Arc-Length path-following technique. The numerical results obtained with the free program Scilab show that the new algorithm converges faster than standard procedures and modified Newton-Raphson, since the approximate solution of the problem is obtained with a smaller number of accumulated iterations and less CPU time. The equilibrium paths show that the structures exhibit a completely different behavior when the material nonlinearity is considered in the analysis with large displacements.

Convergence and Stability of a Parametric Class of Iterative Schemes for Solving Nonlinear Systems

A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations is presented. This study gives us important information about the stability and reliability of the members of the family. The numerical results obtained by applying different elements of the family for solving the Hammerstein integral equation and the Fisher’s equation confirm the theoretical results.

DEVELOPMENT AND TESTING OF THE MODIFIED SIMPSON’S RULE FOR DISCRETE ORDINATES TRANSPORT APPLICATIONS

A new angular quadrature type termed Modified Simpson Trapezoidal (MST) is developed based on the conventional Simpson’s 1/3 rule where the angular pattern over polar levels has a trapezoid shape. An adaptive coefficient correction scheme is developed to enable our new quadrature to integrate the angular flux over subintervals separated by the interior jump irregularities. A two-dimensional test problem is employed to verify the angular discretization error in the uncollided SN scalar flux computed with our new quadrature sets, as well as conventional angular quadrature types. Numerical results show that the MST quadrature error in the point-wise scalar flux converges with second order against increasing number of discrete angles, while the error obtained with other conventional quadrature types converges slower than first order depending on the regularity of the exact point-wise uncollided angular flux. In order to reduce the number of discrete points needed, a variant of the MST quadrature, namely MSTP30, is developed by using the Quadruple Range [1] polar quadrature with fixed 30 polar angles and applying the MST quadrature to the azimuthal dependence in each polar level. The angular discretization error in the point-wise SN scalar flux obtained with MSTP30 sets converges with fourth order because the polar discretization error is sufficiently reduced that MSTP30 behaves like a one-dimensional quadrature. Furthermore, because MSTP30 computes the integral over subintervals that keep the true solution’s irregularity at the boundaries, this fourth order convergence rate is unaffected by such inevitable irregularities.

Basic Steffensen's Method of Higher-Order Convergence

In this paper, we introduce a new analog of a variant of Steffensen's method of fourth-order convergence for solving non-linear equations based on the q-deference operator.

Fractal quintic spline method for nonlinear boundary-value problems

In this article, numerical solutions of nonlinear boundary-value problems are obtained using fractal quintic spline. Convergence analysis of the proposed method is also established. Proposed method has fourth-order convergence. Numerical examples are provided to show practical usefulness of the method and numerical results are compared with the existing numerical methods.

High-order compact difference method for two-dimension elliptic and parabolic equations with mixed derivatives

In this article, firstly, based on Taylor series expansion and truncation error correction technology, combined with the fourth-order Padé schemes of the first-order derivatives, a new fourth-order compact difference (CD) scheme is constructed to solve the two-dimensional (2D) linear elliptic equation a mixed derivative. In this new scheme, unknown function and its first-order derivatives are regarded as the unknown variables in calculation. Then, the method is extended to solve the 2D parabolic equation with a mixed derivative. To match the spatial fourth-order accuracy, The backward differentiation formula (BDF) is employed to gain the fourth-order accuracy for the temporal discretization. Truncation error is analyzed to display that the present scheme is fourth-order accuracy in space. In order to solve the resulting large-scale linear equations, a multigrid method is employed to accelerate the convergence speed of the conventional relaxation methods. Finally, numerical results indicate that the present schemes obtain fourth-order convergence and are more accurate than those in the literature.

A high-order compact difference method on fitted meshes for Neumann problems of time-fractional reaction–diffusion equations with variable coefficients

This paper is concerned with numerical methods for a class of nonhomogeneous Neumann problems of time-fractional reaction–diffusion equations with variable coefficients. The solutions of this kind of problems often have weak singularity at the initial time. This makes the existing numerical methods with uniform time mesh often lose accuracy. In this paper, we propose and analyze a high-order compact finite difference method with nonuniform time mesh. The time-fractional derivative is approximated by Alikhanov’s high-order approximation on a class of fitted time meshes. For the spatial variable coefficient differential operator, a new fourth-order boundary discretization is developed under the nonhomogeneous Neumann boundary condition, and then a new fourth-order compact finite difference approximation on a space uniform mesh is obtained. Under the assumption of the weak initial singularity of solution, we prove that for the general case of the variable coefficients, the proposed method is unconditionally stable and the numerical solution converges to the solution of the problem under consideration. The convergence result also gives an optimal error estimate of the numerical solution in the discrete L2-norm, which shows that the method has the spatial fourth-order convergence, while it attains the temporal optimal second-order convergence provided a proper mesh grading parameter is employed. Numerical results that confirm the sharpness of the error analysis are presented.

A compact quadratic spline collocation method for the time-fractional Black–Scholes model

A compact quadratic spline collocation (QSC) method for the time-fractional Black–Scholes model governing European option pricing is presented. Firstly, after eliminating the convection term by an exponential transformation, the time-fractional Black–Scholes equation is transformed to a time-fractional sub-diffusion equation. Then applying $$L1 - 2$$ formula for the Caputo time-fractional derivative and using a collocation method based on quadratic B-spline basic functions for the space discretization, we establish a higher accuracy numerical scheme which yields $$3-\alpha $$ order convergence in time and fourth-order convergence in space. Furthermore, the uniqueness of the numerical solution and the convergence of the algorithm are investigated. Finally, numerical experiments are carried out to verify the theoretical order of accuracy and demonstrate the effectiveness of the new technique. Moreover, we also study the effect of different parameters on option price in time-fractional Black–Scholes model.

Fast and reliable high-accuracy computation of Gauss–Jacobi quadrature

Iterative methods with certified convergence for the computation of Gauss–Jacobi quadratures are described. The methods do not require a priori estimations of the nodes to guarantee its fourth-order convergence. They are shown to be generally faster than previous methods and without practical restrictions on the range of the parameters. The evaluation of the nodes and weights of the quadrature is exclusively based on convergent processes which, together with the fourth-order convergence of the fixed point method for computing the nodes, makes this an ideal approach for high-accuracy computations, so much so that computations of quadrature rules with even millions of nodes and thousands of digits are possible on a typical laptop.

The numerical study of advection–diffusion equations by the fourth-order cubic B-spline collocation method

A fourth-order numerical method based on cubic B-spline functions has been proposed to solve a class of advection–diffusion equations. The proposed method has several advantageous features such as high accuracy and fast results with very small CPU time. We have applied the Crank–Nicolson method to solve the advection–diffusion equation. The stability analysis is performed, and the method is shown to be unconditionally stable. Error analysis is carried out to show that the proposed method has fourth-order convergence. The efficiency of the proposed B-spline method has been checked by applying on ten important advection–diffusion problems of three types, having Dirichlet, Neumann and periodic boundary conditions. Considered examples prove the mentioned advantages of the method. The computed results are also compared with those available in the literature, and it is found that our method is giving better results.

A Sixth-Order Numerical Method Based on Shishkin Mesh for Singularly Perturbed Boundary Value Problems

Recently, Lodhi and Mishra (J Comput Appl Math 319:170–187, 2017) presented the standard B-spline method based on quintic B-spline basis functions to solve a type of singularly perturbed boundary value problems (SPBVP). We note that their method provides only fourth-order convergence approximation to the solution of such problem. In this paper, we present a novel optimal B-spline technique, based on same quintic B-spline basis function as used in Lodhi and Mishra (2017), for solving linear and nonlinear SPBVP. The advantage of the suggested method over the method in Lodhi and Mishra (2017) is that our method has sixth-order rate of convergence. To obtain higher order of convergence, a high-order perturbation of the SPBVP is generated. The method is tested for its efficiency by applying it on five test problems.

Energy-preserving time high-order AVF compact finite difference schemes for nonlinear wave equations with variable coefficients

In this article, we develop and analyze two energy-preserving high-order average vector field (AVF) compact finite difference schemes for solving variable coefficient nonlinear wave equations with periodic boundary conditions. Specifically, we first consider the variable coefficient nonlinear wave equation as an infinite-dimensional Hamiltonian system. Then the fourth-order compact finite difference and AVF techniques are applied to the resulting Hamiltonian system for the spatial discretization and time integration, respectively, which yield two energy-preserving high-order schemes, one is a time second-order AVF compact finite difference scheme (AVF(2)-CFD) and the other is a time fourth-order AVF compact finite difference scheme (AVF(4)-CFD). We theoretically prove that the proposed schemes satisfy energy conservations in the discrete forms and are uniquely solvable. Also, we prove the convergence of the proposed schemes and their error estimates, where the AVF(4)-CFD scheme is of fourth-order convergence in both time and space. Numerical experiments for the nonlinear wave equations with various nonlinearities are given to show their performances, which confirm the theoretical results.

An Optimal Fourth Order Derivative-Free Numerical Algorithm for Multiple Roots

A plethora of higher order iterative methods, involving derivatives in algorithms, are available in the literature for finding multiple roots. Contrary to this fact, the higher order methods without derivatives in the iteration are difficult to construct, and hence, such methods are almost non-existent. This motivated us to explore a derivative-free iterative scheme with optimal fourth order convergence. The applicability of the new scheme is shown by testing on different functions, which illustrates the excellent convergence. Moreover, the comparison of the performance shows that the new technique is a good competitor to existing optimal fourth order Newton-like techniques.

An efficient class of iterative methods for computing generalized outer inverse $${M_{T,S}^{(2)}}$$

In this paper, we propose a new matrix iteration scheme for computing the generalized outer inverse for a given complex matrix. The convergence analysis of the proposed scheme is established under certain necessary conditions, which indicates that the methods possess at least fourth-order convergence. The theoretical discussions show that the convergence order improves from 4 to 5 for a particular parameter choice. We prove that the sequence of approximations generated by the family satisfies the commutative property of matrices, provided the initial matrix commutes with the matrix under consideration. Some real-world and academic problems are chosen to validate our methods for solving the linear systems arising from statically determinate truss problems, steady-state analysis of a system of reactors, and elliptic partial differential equations. Moreover, we include a wide variety of large sparse test matrices obtained from the matrix market library. The performance measures used are the number of iterations, computational order of convergence, residual norm, efficiency index, and the computational time. The numerical results obtained are compared with some of the existing robust methods. It is demonstrated that our method gives improved results in terms of computational speed and efficiency.

A compact exponential difference method for multi-term time-fractional convection-reaction-diffusion problems with non-smooth solutions

This paper is concerned with a numerical method for a class of one-dimensional multi-term time-fractional convection-reaction-diffusion problems, where the differential equation contains a sum of the Caputo time-fractional derivatives of different orders between 0 and 1. In general the solutions of such problems typically exhibit a weak singularity at the initial time. A compact exponential finite difference method, using the well-known L1 formula for each time-fractional derivative and a fourth-order compact exponential difference approximation for the spatial discretization, is proposed on a mesh that is generally nonuniform in time and uniform in space. Taking into account the initial weak singularity of the solution, the stability and convergence of the method is proved and the optimal error estimate in the discrete L2-norm is obtained by developing a discrete energy analysis technique which enables us to overcome the difficulties caused by the nonsymmetric discretization matrices. The error estimate shows that the method has the spatial fourth-order convergence, and reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence. The extension of the method to two-dimensional problems is also discussed. Numerical results confirm the theoretical convergence result, and show the applicability of the method to convection dominated problems.

High-order time-stepping methods for two-dimensional Riesz fractional nonlinear reaction–diffusion equations

Anomalous diffusion and long-range spatial interactions in anisotropic media could be captured by considering the Riesz fractional derivatives rather than the classical Laplacian. While analytical solutions of the resulting fractional reaction–diffusion models may not be available, the numerical methods for approximating them are challenging. In this paper, fourth-order L-stable (ETDRK04) and A-stable (ETDRK22) linearly implicit predictor-corrector type time-stepping methods are presented. These methods are implemented on two-dimensional Riesz fractional nonlinear reaction–diffusion equations with smooth and non-smooth initial data. Three types of nonlinear reaction–diffusion models are considered: Allen–Cahn equation with cubic nonlinearity, Fisher’s equation with quadratic nonlinearity, and Enzyme Kinetics equation with rational nonlinearity. Fourth-order temporal convergence rate of the methods is proved analytically and computed numerically. Profiles of the numerical solutions corresponding to different orders and rates of diffusion are included. Computational efficiency of the ETDRK04 and ETDRK22 methods over the well known Cox–Matthews ETDRK4 is presented. The superiority of the provided methods, in terms of computational accuracy, efficiency, and reliability, is demonstrated through the numerical experiments.