On the moving boundary formulation for parabolic problems on unbounded domains

The aim of this paper is to propose an original numerical approach for parabolic problems whose governing equations are defined on unbounded domains. We are interested in studying the class of problems admitting invariance property to Lie group of scalings. Thanks to similarity analysis the parabolic problem can be transformed into an equivalent boundary value problem governed by an ordinary differential equation and defined on an infinite interval. A free boundary formulation and a convergence theorem for this kind of transformed problems are available in [R. Fazio, A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM J. Numer. Anal. 33 (1996), pp. 1473–1483]. Depending on its scaling invariance properties, the free boundary problem is then solved numerically using either a noniterative, or an iterative method. Finally, the solution of the parabolic problem is retrieved by applying the inverse map of similarity.     

First vorticity–velocity–pressure numerical scheme for the Stokes problem

We consider the bidimensional Stokes problem for incompressible fluids and recall the vorticity, velocity and pressure variational formulation, which was previously proposed by one of the authors, and allows very general boundary conditions. We develop a natural implementation of this numerical method and we describe in this paper the numerical results we obtain. Moreover, we prove that the low degree numerical scheme we use is stable for Dirichlet boundary conditions on the vorticity. Numerical results are in accordance with the theoretical ones. In the general case of unstructured meshes, a stability problem is present for Dirichlet boundary conditions on the velocity, exactly as in the stream function-vorticity formulation. Finally, we show on some examples that we observe numerical convergence for regular meshes or embedded ones for Dirichlet boundary conditions on the velocity.     

A new numerical approach to a non-steady filtration problem

A free-boundary transient problem of seepage flow is studied from a numerical standpoint. From a suitable formulation of the problem in terms of variational inequality we introduce a new numerical approach of the implicit type and based on the finite element method. In this approach the problem is solved on a fixed region and the position of the free boundary is automatically found as part of the solution of the problem; so it is not necessary to solve a succession of problems with different positions of the free boundary. We prove stability and convergence for the approximate solution and we give several numerical results. Work supported by C. N. R. of Italy through the Laboratorio di Analisi Numerica of Pavia.     

A new numerical method for pricing fixed-rate mortgages with prepayment and default options

In this paper we consider the valuation of fixed-rate mortgages including prepayment and default options, where the underlying stochastic factors are the house price and the interest rate. The mathematical model to obtain the value of the contract is posed as a free boundary problem associated to a partial differential equation (PDE) model. The equilibrium contract rate is determined by using an iterative process. Moreover, appropriate numerical methods based on a Lagrange–Galerkin discretization of the PDE, an augmented Lagrangian active set method and a Newton iteration scheme are proposed. Finally, some numerical results to illustrate the performance of the numerical schemes, as well as the qualitative and quantitative behaviour of solution and the optimal prepayment boundary are presented.     

A Solution Method for a Certain Nonlinear Interface Problem in Unbounded Domains

In this paper, we consider a kind of nonlinear interface problem in unbounded domains. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive the optimal error estimate of finite element approximation to the coupled FEM-BEM problem. Then we introduce a preconditioning steepest descent method for solving the discrete system by constructing a cheap domain decomposition preconditioner. Moreover, we give a complete analysis to the convergence speed of this iterative method. Received March 30, 2000; revised November 29, 2000     

Element free method for static and free vibration analysis of spatial thin shell structures

The implementation of the element free Galerkin method (EFG) for spatial thin shell structures is presented in this paper. Both static deformation and free vibration analyses are considered. The formulation of the discrete system equations starts from the governing equations of stress resultant geometrically exact theory of shear flexible shells. Moving least squares approximation is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometry. Discrete system equations are obtained by incorporating these interpolations into the Galerkin weak form. The formulation is verified through numerical examples of static stress analysis and frequency analysis of spatial thin shell structures. For static load analysis, essential boundary conditions are enforced through penalty method and Lagrange multipliers while boundary conditions for frequency analysis are imposed through a weak form using orthogonal transformation techniques. The EFG results compare favorably with closed-form solutions and that of finite element analyses.     

Tangential decomposition

In this paper, we present the tangential block decomposition for block-tridiagonal matrices which is in many aspects similar to the frequency filtering method by Wittum [8] and also to the tangential frequency filtering decomposition by Wagner [6]–[7]. In opposite to the methods of Wittum and Wagner, for the class of model problems our approach does not use any test vectors for its implementation. Similar to many iterative methods, it needs only bounds for extremal eigenvalues. Theoretical properties of our scheme are similar to those for the ADI-method. The practical convergence of the presented method is illustrated by numerical examples.     

Accelerated monotone iterative methods for a boundary value problem of second-order discrete equations

An accelerated monotone iterative method for a boundary value problem of second-order discrete equations is presented. This method leads to an existence-comparison theorem as well as a computational algorithm for the solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Some numerical results are presented to illustrate the monotone convergence of the iterative sequences and the rate of convergence of the iterations.     

Spectral projected subgradient with a momentum term for the Lagrangean dual approach

The Lagrangean dual problem, with a non-differentiable convex objective function, is usually solved by using the subgradient method, whose convergence is guaranteed if the optimal value of the dual objective function is known. In practice, this optimal value is approximated by a previously computed bound. In this work, we combine the subgradient method with a different choice of steplength, based on the recently developed spectral projected gradient method, that does not require either exact or approximated estimates of the optimal value. We also add a momentum term to the subgradient direction that accelerates the convergence process towards global solutions. To illustrate the behavior of our new algorithm we solve Lagrangean dual problems associated with integer programming problems. In particular, we present and discuss encouraging numerical results for set covering problems and generalized assignment problems.     

An iterative algorithm based on level set method for the shape recovery problem

In this paper, we propose an iterative algorithm based on the level set method for the shape recovery problem. We use a suitable preconditioner for the artificial time-dependent system for the level set formulation and propose an iterative algorithm of the level set function. We prove the convergence of our algorithm under some hypothesis. Numerical experiments show the efficiency of the algorithm.     

Method of weighted residuals as applied to higher order,nonlinear, two-point boundary value problems

A new numerical method for solving a class of higher order nonlinear two-point boundary value problems is presented. The present paper is an extension of an earlier work where only second order problems were addressed. This iterative technique first linearizes the problem by an initial guess for the nonlinear terms. The linearized boundary value problem is transformed into an initial value problem by using a weighted residuals technique. The resulting initial value problem is then solved by utilizing a fourth order Runge-Kutta scheme. The new solution generated is used as an improved estimate and the process iterated until a desired level of convergence is attained. Numerical solutions for third and fourth order problems are included.     

A fast boundary element method for the two-dimensional Helmholtz Equation

In this paper, we present a fast method for solving boundary integral equations arising from the exterior Dirichlet problem for the two-dimensional Helmholtz equation. This method combines a quadrature method for discretizing the boundary integral equations with a preconditioned iterative method for solving the resulting dense, nonsymmetric linear systems. Using this method, a polynomial rate of convergence can be obtained by performing a finite number of iterations, which yields high computational efficiency. Various numerical examples are presented.     

Coupling of optimal variation of parameters method with Adomian’s polynomials for nonlinear equations representing fluid flow in different geometries

In this study, we describe a modified analytical algorithm for the resolution of nonlinear differential equations by the variation of parameters method (VPM). Our approach, including auxiliary parameter and auxiliary linear differential operator, provides a computational advantage for the convergence of approximate solutions for nonlinear boundary value problems. We consume all of the boundary conditions to establish an integral equation before constructing an iterative algorithm to compute the solution components for an approximate solution. Thus, we establish a modified iterative algorithm for computing successive solution components that does not contain undetermined coefficients, whereas most previous iterative algorithm does incorporate undetermined coefficients. The present algorithm also avoid to compute the multiple roots of nonlinear algebraic equations for undetermined coefficients, whereas VPM required to complete calculation of solution by computing roots of undetermined coefficients. Furthermore, a simple way is considered for obtaining an optimal value of an auxiliary parameter via minimizing the residual error over the domain of problem. Graphical and numerical results reconfirm the accuracy and efficiency of developed algorithm.

     

The dynamic stresses induced by moving interfacial cracks

Elastodynamic problems involving moving mixed boundary conditions are considered. In particular, uniform and nonuniform propagation in Mode I, II and III types of motion of semi-infinite cracks along the interface of two dissimilar half-spaces are treated. The equations of motion are transformed to a new coordinate system in which the moving tip of the crack appears always at the origin of the coordinates. An implicit three-level numerical method of solution is given which is proved to be more efficient than a previous explicit one. Furthermore, an implicit method for the numerical formulation of the boundary conditions is presented and is shown to yield better results than a previous formulation. The stability analysis of the proposed finite difference approximation is given, and stability criteria are presented as well as a proof of the convergence of the iterative process involved in the numerical formulation of the boundary and interface conditions. The reliability of the present method of solution is examined in several situations where analytical results are known.     

A new directional stability transformation method of chaos control for first order reliability analysis

The HL-RF iterative algorithm of the first order reliability method (FORM) is popularly applied to evaluate reliability index in structural reliability analysis and reliability-based design optimization. However, it sometimes suffers from non-convergence problems, such as bifurcation, periodic oscillation, and chaos for nonlinear limit state functions. This paper derives the formulation of the Lyapunov exponents for the HL-RF iterative algorithm in order to identify these complicated numerical instability phenomena of discrete chaotic dynamic systems. Moreover, the essential cause of low efficiency for the stability transform method (STM) of convergence control of FORM is revealed. Then, a novel method, directional stability transformation method (DSTM), is proposed to reduce the number of function evaluations of original STM as a chaos feedback control approach. The efficiency and convergence of different reliability evaluation methods, including the HL-RF algorithm, STM and DSTM, are analyzed and compared by several numerical examples. It is indicated that the proposed DSTM method is versatile, efficient and robust, and the bifurcation, periodic oscillation, and chaos of FORM is controlled effectively.     

Computation of Self-similar Solution Profiles for the Nonlinear Schrödinger Equation

We discuss the numerical computation of self-similar blow-up solutions of the classical nonlinear Schrödinger equation in three space dimensions. These solutions become unbounded in finite time at a single point at which there is a growing and increasingly narrow peak. The problem of the computation of this self-similar solution profile reduces to a nonlinear, ordinary differential equation on an unbounded domain. We show that a transformation of the independent variable to the interval [0,1] yields a well-posed boundary value problem with an essential singularity. This can be stably solved by polynomial collocation. Moreover, a Matlab solver developed by two of the authors can be applied to solve the problem efficiently and provides a reliable estimate of the global error of the collocation solution. This is possible because the boundary conditions for the transformed problem serve to eliminate undesired, rapidly oscillating solution modes and essentially reduce the problem of the computation of the physical solution of the problem to a boundary value problem with a singularity of the first kind. Furthermore, this last observation implies that our proposed solution approach is theoretically justified for the present problem.     

The shooting method for the solution of ordinary differential equations: A control-theoretical perspective

A discrete space state representation is used to provide a feedback control-theoretical formulation for the iterative shooting method used to solve two-point boundary value problems in ordinary differential equations. Preconditioning matrices, error analysis and convergence conditions for these iterative methods are put into a feedback control perspective as well.     

Optimizing the trajectory of spatial movement of a flying object as a solid body

We consider the problem of optimal control of a flying object (FO) by the Pontryagin maximum principle with minimizing the control expenses. The dynamics of an FO in space is defined by Euler and Poisson equations. A two-point boundary value problem is solved with the Newton’s method. We give results of numerical modeling and recommendations to provide for convergence of the iterative procedure.     

A free boundary approach and keller's box scheme for BVPs on infinite intervals

A free boundary approach for the numerical solution of boundary value problems (BVPs) governed by a third-order differential equation and defined on infinite intervals was proposed recently [SIAM J. Numer. Anal., 33 (1996), pp. 1473–1483]. In that approach, the free boundary (that can be considered as the truncated boundary) is unknown and has to be found as part of the solution. This eliminates the uncertainty related to the choice of the truncated boundary in the classical treatment of BVPs defined on infinite intervals. In this article, we investigate some open questions related to the free boundary approach. We recall the extension of that approach to problems governed by a system of first-order differential equations, and for the solution of the related free boundary problem we consider now the reliable Keller's box difference scheme. Moreover, by solving a challenging test problem of interest in foundation engineering, we verify that the proposed approach is applicable to problems where none of the solution components is a monotone function.