arXiv subjects

Riccardo Fazio

Publications and source records attributed to Riccardo Fazio.

26 records · Page 2Linked to original sources

arXiv2014

The Iterative Transformation Method for the Sakiadis Problem

In a transformation method the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. This paper is concerned with the application of the iterative transformation method to the Sakiadis problem. This method is an extension of the Toepfer's non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. As shown by this author [Appl. Anal., 66 (1997) pp. 89-100] the method provides a simple numerical test for the existence and uniqueness of solutions. Here we show how the method can be applied to problems with a homogeneous boundary conditions at infinity and in particular we solve the Sakiadis problem of boundary layer theory. Moreover, we show how to couple our method with Newton's root-finder. The obtained numerical results compare well with those available in literature. The main aim here is that any method developed for the Blasius, or the Sakiadis, problem might be extended to more challenging or interesting problems. In this context, the iterative transformation method has been recently applied to compute the normal and reverse flow solutions of Stewartson for the Falkner-Skan model [Comput. & Fluids, 73 (2013) pp. 202-209].

math.NA↗

arXiv2014

Free Boundary Formulation for BVPs on a Semi-Infinite Interval and Non-Iterative Transformation Methods

This paper is concerned with two examples on the application of the free boundary formulation to BVPs on a semi-infinite interval. In both cases we are able to provide the exact solution of both the BVP and its free boundary formulation. Therefore, these problems can be used as benchmarks for the numerical methods applied to BVPs on a semi-infinite interval and to free BVPs. Moreover, we emphasize how for two classes of free BVPs, we can define non-iterative initial value methods, whereas BVPs are usually solved iteratively. These non-iterative methods can be deduced within Lie's group invariance theory. Then, we show how to apply the non-iterative methods to the two introduced free boundary formulations in order to obtain meaningful numerical results. Finally, we indicate several problems from the literature where our non-iterative transformation methods can be applied.

math.NA↗

arXiv2014

Finite difference schemes on quasi-uniform grids for Bvps on infinite intervals

The classical numerical treatment of boundary value problems defined on infinite intervals is to replace the boundary conditions at infinity by suitable boundary conditions at a finite point, the so-called truncated boundary. A truncated boundary allowing for a satisfactory accuracy of the numerical solution has to be determined by trial and errors and this seems to be the weakest point of the classical approach. On the other hand, the free boundary approach overcomes the need for a priori definition of the truncated boundary. In fact, in a free boundary formulation the unknown free boundary can be identified with a truncated boundary and being unknown it has to be found as part of the solution. In this paper we consider a different way to overcome the introduction of a truncated boundary, namely finite differences schemes defined on quasi-uniform grids. A quasi-uniform grid allows us to describe the infinite domain by a finite number of intervals. The last node of such grid is placed on infinity so right boundary conditions are taken into account exactly. We apply the proposed approach to the Falkner-Skan model and to a problem of interest in foundation engineering. The obtained numerical results are found in good agreement with those available in literature. Moreover, we provide a simple way to improve the accuracy of the numerical results using Richardson's extrapolation. Finally, we indicate a possible way to extend the proposed approach to boundary value problems defined on the whole real line.

math.NA↗

arXiv2013

A Non-Iterative Transformation Method for Newton's Free Boundary Problem

In book II of Newton's "Principia Mathematica" of 1687 several applicative problems are introduced and solved. There, we can find the formulation of the first calculus of variations problem that leads to the first free boundary problem of history. The general calculus of variations problem is concerned with the optimal shape design for the motion of projectiles subject to air resistance. Here, for Newton's optimal nose cone free boundary problem, we define a non-iterative initial value method which is referred in literature as transformation method. To define this method we apply invariance properties of Newton's free boundary problem under a scaling group of point transformations. Finally, we compare our non-iterative numerical results with those available in literature and obtained via an iterative shooting method. We emphasize that our non-iterative method is faster than shooting or collocation methods and does not need any preliminary computation to test the target function as the iterative method or even provide any initial iterate. Moreover, applying Buckingham Pi-Theorem we get the functional relation between the unknown free boundary and the nose cone radius and height.

math.NA↗

arXiv2013

An Analytical and Numerical Study of Liquid Dynamics in a 1D Capillary under Entrapped Gas Action

Capillary dynamics has been and is yet an important field of research, because of its very relevant role played as the core mechanism at the base of many applications. In this context, we are particularly interested in the liquid penetration inspection technique. Due to the obviously needed level of reliability involved with such a non-destructive test, this paper is devoted to study how the presence of an entrapped gas in a close-end capillary may affect the inspection outcome. This study is carried out through a 1D ordinary differential model that despite its simplicity is able to point out quite well the capillary dynamics under the effect of an entrapped gas. The paper is divided into two main parts; the first starts from an introductory historical review of capillary flows modeling, goes on presenting the 1D second order ordinary differential model, taking into account the presence of an entrapped gas and therefore ends by showing some numerical simulation results. The second part is devoted to the analytical study of the model by separating the initial transitory behavior from the stationary one. Besides, these solutions are compared with the numerical ones and finally an expression is deduced for the threshold radius switching from a fully damped transitory to an oscillatory one.

physics.flu-dyn↗

arXiv2013

Numerical Methods for a Nonlinear BVP Arising in Physical Oceanography

In this paper we report and compare the numerical results for an ocean circulation model obtained by the classical truncated boundary formulation, the free boundary approach and a quasi-uniform grid treatment of the problem. We apply a shooting method to the truncated boundary formulation and finite difference methods to both the free boundary approach and the quasi-uniform grid treatment. Using the shooting method, supplemented by the Newton's iterations, we show that the ocean circulation model cannot be considered as a simple test case. In fact, for this method we are forced to use as initial iterate a value close to the correct missing initial condition in order to be able to get a convergent numerical solution. The reported numerical results allow us to point out how the finite difference method with a quasi-uniform grid is the less demanding approach and that the free boundary approach provides a more reliable formulation than the classical truncated boundary formulation.

math.NA↗

arXiv2012

Blasius Problem and Falkner-Skan model: Töpfer's Algorithm and its Extension

In this paper, we review the so-called Töpfer algorithm that allows us to find a non-iterative numerical solution of the Blasius problem, by solving a related initial value problem and applying a scaling transformation. Moreover, we remark that the applicability of this algorithm can be extended to any given problem, provided that the governing equation and the initial conditions are invariant under a scaling group of point transformations and that the asymptotic boundary condition is non-homogeneous. Then, we describe an iterative extension of Töpfer's algorithm that can be applied to a general class of problems. Finally, we solve the Falkner-Skan model, for values of the parameter where multiple solutions are admitted, and report original numerical results, in particular data related to the famous reverse flow solutions by Stewartson. The numerical data obtained by the extended algorithm are in good agreement with those obtained in previous studies.

math.NA↗

arXiv2012

Scaling Invariance and the Iterative Transformation Method for a Class of Parabolic Moving Boundary Problems

In this paper we apply a scaling invariance analysis to reduce a class of parabolic moving boundary problems to free boundary problems governed by ordinary differential equations. As well known free boundary problems are always non-linear and, consequently, their numerical solution is often obtained iteratively. Among the numerical methods, developed for the numerical solution of this kind of problems, we focus on the iterative transformation method that has been defined within scaling invariance theory. Then, as illustrative examples, we solve two problems of interest in the applications. The obtained numerical results are found in good agreement with exact or approximate ones.

math.NA↗