Adams-like techniques for zero-finder methods

Two families of zero-finding iterative methods for nonlinear equations are presented. We derive them solving an initial value problem using Adams-like multistep techniques. Namely, Adams methods have been used to solve the problem that consists in a differential equation in what appears the inverse function of the one which zero will be computed and the condition given by the value attained by it at the initial approximation. Performing this procedure several methods of different local orders of convergence have been obtained.     

Zero-finder methods derived using Runge-Kutta techniques

In this paper some families of zero-finding iterative methods for nonlinear equations are presented. The key idea to derive them is to solve an initial value problem applying Runge-Kutta techniques. More explicitly, these methods are used to solve the problem that consists in a differential equation in what appears the inverse function of the one which zero will be computed and the condition given by the value attained by it at the initial approximation. Carrying out this procedure several families of different orders of local convergence are obtained. Furthermore, the efficiency of these families are computed and two new families using like-Newton’s methods that improve the most efficient one are also given.     

Newton-type methods of high order and domains of semilocal and global convergence

We present the geometric construction of some classical iterative methods that have global convergence and “infinite” speed of convergence when they are applied to solve certain nonlinear equations f(t)=0. In particular, for nonlinear equations with the degree of logarithmic convexity of f^′, L_f^′(t)=f^′(t)f^?(t)/f^″(t)², is constant, a family of Newton-type iterative methods of high orders of convergence is constructed. We see that this family of iterations includes the classical iterative methods. The convergence of the family is studied in the real line and the complex plane, and domains of semilocal and global convergence are located.     

Continuous block implicit hybrid one-step methods for ordinary and delay differential equations

A class of high order continuous block implicit hybrid one-step methods has been proposed to solve numerically initial value problems for ordinary and delay differential equations. The convergence and Aω-stability of the continuous block implicit hybrid methods for ordinary differential equations are studied. Alternative form of continuous extension is constructed such that the block implicit hybrid one-step methods can be used to solve delay differential equations and have same convergence order as for ordinary differential equations. Some numerical experiments are conducted to illustrate the efficiency of the continuous methods.     

Symmetric multistep Obrechkoff methods with zero phase-lag for periodic initial value problems of second order differential equations

In this paper, symmetric multistep Obrechkoff methods of orders 8 and 12, involving a parameter p to solve a special class of second order initial value problems in which the first order derivative does not appear explicitly, are discussed. It is shown that the methods have zero phase-lag when p is chosen as 2π times the frequency of the given initial value problem.     

On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers

The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For R/r exceeding about 30, where r and R are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large R/r) than the convergence rate of its differential version. For this reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have revealed that this modification has the same convergence rate as its differential counterpart for R/r of up to 5 × 10³. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from R/r ～ 30 and is considerably more efficient for large values of R/r. It is also established that the convergence rates of both methods depend little on the stretching coefficient η of circularly rectangular mesh cells in a range of η that is well sufficient for effective use of the multigrid method for arbitrary values of R/r smaller than ～ 5 × 10³.     

Approximate Factorization in Shallow Water Applications

We consider the numerical integration of problems modelling phenomena in shallow water in 3 spatial dimensions. If the governing partial differential equations for such problems are spatially discretized, then the right-hand side of the resulting system of ordinary differential equations can be split into terms f ₁, f ₂, f ₃ and f ₄, respectively representing the spatial derivative terms with respect to the x, y and z directions, and the interaction terms. It is typical for shallow water applications that the interaction term f ₄ is nonstiff and that the function f ₃ corresponding with the vertical spatial direction is much more stiff than the functions f ₁ and f ₂ corresponding with the horizontal spatial directions. The reason is that in shallow seas the gridsize in the vertical direction is several orders of magnitude smaller than in the horizontal directions. In order to solve the initial value problem (IVP) for these systems numerically, we need a stiff IVP solver, which is necessarily implicit, requiring the iterative solution of large systems of implicit relations. The aim of this paper is the design of an efficient iteration process based on approximate factorization. Stability properties of the resulting integration method are compared with those of a number of integration methods from the literature. Finally, a performance test on a shallow water transport problem is reported.     

On the order of deferred correction

New deferred correction methods for the numerical solution of initial value problems in ordinary differential equations have recently been introduced by Dutt, Greengard and Rokhlin. A convergence proof is presented for these methods, based on the abstract Stetter-Lindberg-Skeel framework and Spijker-type norms. It is shown that p corrections of an order-r one-step solver yield order-r(p+1) accuracy.     

The least squares spectral element method for the Cahn–Hilliard equation

The problem of numerically resolving an interface separating two different components is a common problem in several scientific and engineering applications. One alternative is to use phase field or diffuse interface methods such as the Cahn–Hilliard (C–H) equation, which introduce a continuous transition region between the two bulk phases. Different numerical schemes to solve the C–H equation have been suggested in the literature. In this work, the least squares spectral element method (LS-SEM) is used to solve the Cahn–Hilliard equation. The LS-SEM is combined with a time–space coupled formulation and a high order continuity approximation by employing C¹¹p-version hierarchical interpolation functions both in space and time. A one-dimensional case of the Cahn–Hilliard equation is solved and the convergence properties of the presented method analyzed. The obtained solution is in accordance with previous results from the literature and the basic properties of the C–H equation (i.e. mass conservation and energy dissipation) are maintained. By using the LS-SEM, a symmetric positive definite problem is always obtained, making it possible to use highly efficient solvers for this kind of problems. The use of dynamic adjustment of number of elements and order of approximation gives the possibility of a dynamic meshing procedure for a better resolution in the areas close to interfaces.     

Variational iteration technique for solving nonlinear equations

In this paper, we use the variational iteration technique to suggest some new iterative methods for solving nonlinear equations f(x)=0. We also discuss the convergence criteria of these new iterative methods. Comparison with other similar methods is also given. These new methods can be considered as an alternative to the Newton method. We also give several examples to illustrate the efficiency of these methods. This technique can be used to suggest a wide class of new iterative methods for solving system of nonlinear equations.     

Inverse problem of groundwater modeling by iteratively regularized Gauss-Newton method with a nonlinear regularization term

A nonlinear minimization problem ‖F(d)−u‖?min, ‖u−u_δ‖≤δ, is a typical mathematical model of various applied inverse problems. In order to solve this problem numerically in the lack of regularity, we introduce iteratively regularized Gauss-Newton procedure with a nonlinear regularization term (IRGN-NRT). The new algorithm combines two very powerful features: iterative regularization and the most general stabilizing term that can be updated at every step of the iterative process. The convergence analysis is carried out in the presence of noise in the data and in the modified source condition. Numerical simulations for a parameter identification ill-posed problem arising in groundwater modeling demonstrate the efficiency of the proposed method.     

Martingales and the Robbins-Monro procedure in D[0, 1]

The Robbins-Monro procedure for recursive estimation of a zero point of a regression function f is investigated for the case f defined on and with values in the space D[0, 1] of real-valued functions on [0, 1] that are right-continuous and have left-hand limits, endowed with Skorohod's J₁-topology. There are proved an a.s. convergence result and an invariance principle where the limit process is a Gaussian Markov process with paths in the space of continuous C[0, 1]-valued functions on [0, 1]. At first the case f(x) ≡ x, i.e., the case of a martingale in D[0, 1], is treated and by this then the general case. An application to an initial value problem with only empirically available function values is sketched.     

On the use of rational iterations and domain decomposition methods for the Helmholtz problem

An iterative algorithm for the numerical solution of the Helmholtz problem is considered. It is difficult to solve the problem numerically, in particular, when the imaginary part of the wave number is zero or small. We develop a parallel iterative algorithm based on a rational iteration and a nonoverlapping domain decomposition method for such a non-Hermitian, non-coercive problem. Algorithm parameters (artificial damping and relaxation) are introduced to accelerate the convergence speed of the iteration. Convergence analysis and effective strategies for finding efficient algorithm parameters are presented. Numerical results carried out on an nCUBE2 are given to show the efficiency of the algorithm. To reduce the boundary reflection, we employ a hybrid absorbing boundary condition (ABC) which combines the first-order ABC and the physical $Q$ ABC. Computational results comparing the hybrid ABC with non-hybrid ones are presented. Received May 19, 1994 / Revised version received March 25, 1997     

A note on some improvements of the simultaneous methods for determination of polynomial zeros

Applying Gauss-Seidel approach to the improvements of two simultaneous methods for finding polynomial zeros, presented in [9], two iterative methods with faster convergence are obtained. The lower bounds of the R-order of convergence for the accelerated methods are given. The improved methods and their accelerated modifications are discussed in view of the convergence order and the number of numerical operations. The considered methods are illustrated numerically in the example of an algebraic equation.     

Implementation of a restarted Krylov subspace method for the evaluation of matrix functions

A new implementation of restarted Krylov subspace methods for evaluating f(A)b for a function f, a matrix A and a vector b is proposed. In contrast to an implementation proposed previously, it requires constant work and constant storage space per restart cycle. The convergence behavior of this scheme is discussed and a new stopping criterion based on an error indicator is given. The performance of the implementation is illustrated for three parabolic initial value problems, requiring the evaluation of exp(A)b.     

Iterative and non-iterative methods for non-linear Volterra integro-differential equations

Iterative and non-iterative methods for the solution of nonlinear Volterra integro-differential equations are presented and their local convergence is proved. The iterative methods provide a sequence solution and make use of fixed-point theory, whereas the non-iterative ones result in series solutions and also make use of fixed-point principles. By means of integration by parts and use of certain integral identities, it is shown that the initial conditions that appear in the iterative methods presented here can be eliminated and the resulting iterative technique is identical to the variational iteration method which is derived here without making any use at all of Lagrange multipliers and constrained variations. It is also shown that the formulation presented here can be applied to initial-value problems in ordinary differential, Volterra’s integral and integro-differential, pantograph, and nonlinear and linear algebraic equations. A technique for improving/accelerating the convergence of the iterative methods presented here is also presented and results in a Lipschitz constant that may be varied as the iteration progresses. It is shown that this acceleration technique is related to preconditioning methods for the solution of linear algebraic equations. It is also argued that the non-iterative methods presented in this paper may not competitive with iterative ones because of possible cancellation errors, if implemented numerically. An analytical continuation procedure based on dividing the interval of integration into disjoint subintervals is also presented and its limitations are discussed.     

A new family of eighth-order iterative methods for solving nonlinear equations

A family of eighth-order iterative methods with four evaluations for the solution of nonlinear equations is presented. Kung and Traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2^n-1. The new family of eighth-order methods agrees with the conjecture of Kung-Traub for the case n=4. Therefore this family of methods has efficiency index equal to 1.682. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

Ordinary differential equations on closed subsets of locally convex spaces with applications to fixed point theorems

The construction and convergence of an approximate solution to the initial value problem x′ = f(t, x), x(0) = x₀, defined on closed subsets of locally convex spaces are given. Sufficient conditions that guarantee the existence of an approximate solution are analyzed in relation to the Nagumo boundary condition used in the Banach space case. It is also shown that the Nagumo boundary condition does not guarantee the existence of an approximate solution. Applications to fixed point theorems for weakly inward mappings are given.     

Determining the temperature from incomplete boundary data

An iterative procedure for determining temperature fields from Cauchy data given on a part of the boundary is presented. At each iteration step, a series of mixed well‐posed boundary value problems are solved for the heat operator and its adjoint. A convergence proof of this method in a weighted L²‐space is included, as well as a stopping criteria for the case of noisy data. Moreover, a solvability result in a weighted Sobolev space for a parabolic initial boundary value problem of second order with mixed boundary conditions is presented. Regularity of the solution is proved. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)