Multilevel augmentation methods for solving the Burgers' equation

In this article, multilevel augmentation method (MAM) for solving the Burgers' equation is developed. The Crank–Nicolson–Galerkin scheme of the Burgers' equation results in nonlinear algebraic systems at each time step, the computational cost for solving these nonlinear systems is huge. The MAM allows us to solve the nonlinear system at a fixed initial lower level and then compensate the error by solving a linear system at the higher level. We prove that the method has the same optimal convergence order as the projection method, while reducing the computational complexity greatly. Finally, numerical experiments are presented to confirm the theoretical analysis and illustrate the efficiency of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1665–1691, 2015     

Multilevel augmentation methods for differential equations

We develop multilevel augmentation methods for solving differential equations. We first establish a theoretical framework for convergence analysis of the boundary value problems of differential equations, and then construct multiscale orthonormal bases in H₀^m(0,1) spaces. Finally, the multilevel augmentation methods in conjunction with the multiscale orthonormal bases are applied to two-point boundary value problems of both second-order and fourth-order differential equations. Theoretical analysis and numerical tests show that these methods are computationally stable, efficient and accurate. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem. Mathematics subject classifications (2000) 65J15, 65R20. Zhongying Chen: Supported in part by the Natural Science Foundation of China under grants 10371137 and 10201034, the Foundation of Doctoral Program of National Higher Education of China under grant 20030558008, Guangdong Provincial Natural Science Foundation of China under grant 1011170 and the Foundation of Zhongshan University Advanced Research Center. Yuesheng Xu: Corresponding author. Supported in part by the US National Science Foundation under grants 9973427 and 0312113, by NASA under grant NCC5-399, by the Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under the program of “One Hundred Distinguished Young Scientists”.     

Lattice Boltzmann model for two-dimensional generalized sine-Gordon equation

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.     

A fast multiscale solver for modified Hammerstein equations

This paper presents a fast solver, called the multilevel augmentation method, for modified nonlinear Hammerstein equations. When we utilize the method to solve a large scale problem, most of components of the solution can be computed directly, and lower frequency components can be obtained by solving a fixed low-order algebraic nonlinear system. The advantage of using the algorithm to modified equations is that it leads to reduce the cost of numerical integrations greatly. The optimal error estimate of the method is established and the nearly linear computational complexity is proved. Finally, numerical examples are presented to confirm the theoretical results and illustrate the efficiency of the method.     

Identification problem for damped sine-Gordon equation with point sources

We establish the existence and uniqueness of solutions for sine-Gordon equations in a multidimensional setting. The equations contain a point-like source. Furthermore, the continuity and the Gâteaux differentiability of the solution map is established. An identification problem for parameters governing the equations is set, and is shown to have a solution. The objective function is proved to be Fréchet differentiable with respect to the parameters. An expression for the Fréchet derivative in terms of the solutions of the direct and the adjoint systems is presented. A criterion for optimal parameters is formulated as a bang-bang control principle. An application of these results to the one-dimensional sine-Gordon equation is considered.     

Domain decomposition methods for solving the burgers equation

This article presents some results of numerical tests of solving the two-dimensional non-linear unsteady viscous Burgers equation. We have compared the known convergence and parallel performance properties of the additive Schwarz domain decomposition method with or without a coarse grid for the model Poisson problem with those obtained by experiments for the Burgers problem.     

A note on King's fifth-order family of methods for solving equations

King's two-step fifth-order methods for solving equations require two function and two derivative evaluations per iteration. Using exactly the same information and composing two-point methods of order 2.732 with themselves, other two-step methods can be obtained with asymptotic convergence rates 7.464 which is better than the 5 of King.     

New eighth-order iterative methods for solving nonlinear equations

In this paper, three new families of eighth-order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods. Per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative. This implies that the efficiency index of the developed methods is 1.682, which is optimal according to Kung and Traub’s conjecture [7] for four function evaluations per iteration. Notice that Bi et al.’s method in [2] and [3] are special cases of the developed families of methods. In this study, several new examples of eighth-order methods with efficiency index 1.682 are provided after the development of each family of methods. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

多维区域上椭圆型方程Dirichlet问题的勒让德多小波多层扩充算法

用小波伽辽金方法求解多维区域上椭圆型方程齐次Dirichlet问题,构造了近似解空间的两个等价的勒让德多小波基,使得快速求解离散后的线性方程组的多层扩充算法得以实现.数值算例表明该算法是有效的.     

A hybrid approximation method for solving Hutchinson’s equation

The hybrid function approximation method for solving Hutchinson’s equation which is a nonlinear delay partial differential equation, is investigated. The properties of hybrid of block-pulse functions and Lagrange interpolating polynomials based on Legendre-Gauss-type points are presented and are utilized to replace the system of nonlinear delay differential equations resulting from the application of Legendre pseudospectral method, by a system of nonlinear algebraic equations. The validity and applicability of the proposed method are demonstrated through two illustrative examples on Hutchinson’s equation.     

Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.     

On the similarity of some three-point methods for solving nonlinear equations

In this short note we discuss certain similarities between some three-point methods for solving nonlinear equations. In particular, we show that the recent three-point method published in [R. Thukral, A new eighth-order iterative method for solving nonlinear equations, Appl. Math. Comput. 217 (2010) 222-229] is a special case of the family of three-point methods proposed previously in [R. Thukral, M.S. Petkovi?, Family of three-point methods of optimal order for solving nonlinear equations, J. Comput. Appl. Math. 233 (2010) 2278-2284].     

The “Gauss-Seidelization” of iterative methods for solving nonlinear equations in the complex plane

In this paper we introduce a process we have called “Gauss-Seidelization” for solving nonlinear equations. We have used this name because the process is inspired by the well-known Gauss-Seidel method to numerically solve a system of linear equations. Together with some convergence results, we present several numerical experiments in order to emphasize how the Gauss-Seidelization process influences on the dynamical behavior of an iterative method for solving nonlinear equations.     

Finite Volume Multilevel Approximation of the Shallow Water Equations

The authors consider a simple transport equation in one-dimensional space and the linearized shallow water equations in two-dimensional space, and describe and implement a multilevel finite-volume discretization in the context of the utilization of the incremental unknowns. The numerical stability of the method is proved in both cases.     

Eighth-order methods with high efficiency index for solving nonlinear equations

In this paper, we derive a new family of eighth-order methods for solving simple roots of nonlinear equations by using weight function methods. Per iteration these methods require three evaluations of the function and one evaluation of its first derivative, which implies that the efficiency indexes are 1.682. Numerical comparisons are made to show the performance of the derived methods, as shown in the illustration examples.     

Derivative free two-point methods with and without memory for solving nonlinear equations

Two families of derivative free two-point iterative methods for solving nonlinear equations are constructed. These methods use a suitable parametric function and an arbitrary real parameter. It is proved that the first family has the convergence order four requiring only three function evaluations per iteration. In this way it is demonstrated that the proposed family without memory supports the Kung-Traub hypothesis (1974) on the upper bound 2ⁿ of the order of multipoint methods based on n + 1 function evaluations. Further acceleration of the convergence rate is attained by varying a free parameter from step to step using information available from the previous step. This approach leads to a family of two-step self-accelerating methods with memory whose order of convergence is at least and even in special cases. The increase of convergence order is attained without any additional calculations so that the family of methods with memory possesses a very high computational efficiency. Numerical examples are included to demonstrate exceptional convergence speed of the proposed methods using only few function evaluations.     

Various Newton-type iterative methods for solving nonlinear equations

The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.     

Incremental unknowns preconditioning for solving the Helmholtz equation

An efficient preconditioner is developed for solving the Helmholtz problem in both high and low frequency (wavenumber) regimes. The preconditioner is based on hierarchical unknowns on nested grids, known as incremental unknowns (IU). The motivation for the IU preconditioner is provided by an eigenvalue analysis of a simplified Helmholtz problem. The performance of our preconditioner is tested on the iterative solution of two‐dimensional electromagnetic scattering problems. When compared with other well‐known methods, our technique is shown to often provide a better numerical efficacy and, most importantly, to be more robust. Moreover, for the best performance, the number of IU levels used in the preconditioner should be designed for the coarsest grid to have roughly two points per linear wavelength. This result is consistent with the conventional sampling criteria for wave phenomena in contrast with existing IU applications for solving the Laplace/Poisson problem, where the coarsest grid comprises just one interior point. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007