Alessandra Jannelli

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

Abstract 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 non-standard finite difference 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 that 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.

Mathematical and numerical modeling for a bio-chemical aquarium

Based on bio-chemical ground we derive an aquarium mathematical model useful for predicting dangerous situations as well as for the startup cycle. This model is a basic step toward a more complex advection–diffusion–reaction model in 3D space variables: it defines the reaction part of the more complex partial differential equations model. For the numerical solution of our aquarium model we apply a low complexity second order method combined with a simple adaptive step-size selection procedure. The low accuracy and complexity of the resulting numerical algorithm are motivated because of the high complexity of the final 3D model. The reported numerical results, and comparisons with the know-how available in literature, show the validity of the proposed model.

A 3D mathematical model for the prediction of mucilage dynamics

Abstract We illustrate a three-dimensional mathematical model for the prediction of biological processes that typically occur in a sea region with minor water exchange. The model accounts for particle transport due to water motion, turbulent diffusion and reaction processes and we use a fractional-step approach for discretizing the related different terms.

Ill and Well-Posed One-Dimensional Models of Liquid Dynamics in a Horizontal Capillary

In this paper, we report a mathematical and numerical study of liquid dynamics models in a horizontal capillary. In particular, we prove that the classical model is illposed at initial time, and we present two different approaches in order to overcome this ill-posedness. By numerical viewpoint, we apply an adaptive strategy based on an onestep one-method approach, and we compare the obtained numerical approximations with suitable asymptotic solutions.

One-dimensional mathematical and numerical modeling of liquid dynamics in a horizontal capillary

This paper is concerned with a mathematical and numerical study of liquid dynamics in a horizontal capillary. We derive a two-liquids model for the prediction of capillary dynamics. This model takes into account the effects of real phenomena: like the outside flow action, or the entrapped gas inside a closed-end capillary. Moreover, the limitations of the one-dimensional model are clearly indicated. Finally, we report on several tests of interest: an academic test case that can be used to check available numerical methods, a test for decreasing values of the capillary radius, a simulation concerning a closed-end capillary, and two test cases for two liquids flow. In order to study the introduced mathematical model, our main tool, is a reliable one-step adaptive numerical approach based on a one-step one-method strategy.

Analytical and numerical solutions of fractional type advection-diffusion equation

In this paper, the case of an equation involving fractional derivatives with respect to a single independent variable has been analyzed. Our aim is to determine its Lie’s symmetry, and by using them, obtain analytical and numerical solutions.

Second Order Numerical Operator Splitting for 3D Advection–Diffusion-Reaction Models

In this paper, we present a numerical operator splitting for time integration of 3D advection-diffusion-reaction problems. In this approach, three distinct methods of second order accuracy are proposed for solving, separately, each term involved in the model. Numerical results, obtained for advection – reported in [Fazio and Jannelli, IAENG Int. J. Appl. Math., 39, 25–35, 2009] –, diffusion, and reaction terms, show the efficiency of proposed schemes.

Liquid Dynamics in a Horizontal Capillary: Extended Similarity Analysis

The topic of this study is an extended similarity analysis for a one-dimensional model of liquid dynamics in a horizontal capillary. The bulk liquid is assumed to be initially at rest and is put into motion by capillarity, that is the only driving force acting on it. Besides the smaller is the capillary radius the steeper becomes the initial transitory of the meniscus location derivative, and as a consequence the numerical solution to a prescribed accuracy becomes harder to achieve. Here, we show how an extended scaling invariance can be used to define a family of solutions from a computed one. The similarity transformation involves both geometric and physical feature of the model. As a result, density, surface tension, viscosity, and capillary radius are modified within the required invariance. Within our approach a target problem of practical interest can be solved numerically by solving a simpler transformed test case. The reference solution should be as accurate as possible, and therefore we suggest to use for it an adaptive numerical method. This study may be seen as a complement to the adaptive numerical solution of the considered initial value problems.

On shock solutions to balance equations for slow and fast chemical reaction

This paper deals with shock propagation features in a gas mixture undergoing reversible bimolecular reactions, governed by suitable closures at Euler level of Boltzmann-type equations. Slow and fast chemical processes are considered. At macroscopic level, the slow case is described by a set of balance laws, whereas the fast one yields a set of conservation equations. Within the framework of hierarchies of hyperbolic systems, it is possible to prove that the system governing fast reactions is an equilibrium subsystem of the one describing slow reactions, and then to show how the solutions of the slow system converge to those of the fast system, in case of steady shock problems as well as of Riemann problems.

Front fixing finite difference schemes for American put options model

In this paper, we present front-fixing finite difference schemes for numerical approximation of American put options model formulated as free boundary problem.