Riccardo Fazio Research Web-Page

Mathematical and Numerical Modeling

A 3D mathematical model for the prediction of mucilage dynamics , Alessandra Jannelli, Riccardo Fazio and Davide Ambrosi, Comput. & Fluids 32 (2003) 47--57

Abstract: We illustrate a 3D 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.

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Adaptive stiff solvers at low accuracy and complexity , Alessandra Jannelli and Riccardo Fazio, J. Comp. Appl. Math. 191 (2006) 246--258

Abstract: This paper is concerned with adaptive stiff solvers at low accuracy and complexity for systems of ordinary differential equations. The considered stiff solvers are: two second order Rosenbrock methods with low complexity, and the BDF method of the same order. For the adaptive algorithm we propose to use a monitoring function defined by comparing a measure of the local variability of the solution times the used step size and the order of magnitude of the solution instead of the classical approach based on some local error estimation. This simple step-size selection procedure is implemented in order to control the behavior of the numerical solution. It is easily used to automatically adjust the step size, as the calculation progresses, until user specified tolerance bounds for the introduced monitoring function are fulfilled. This leads to important advantages in accuracy, efficiency and general ease-of-use. At the end of the paper we present two numerical tests which show the performance of the implementation of the stiff solvers, with the proposed adaptive procedure.

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Mathematical and numerical modeling for a bio-chemical aquarium , Riccardo Fazio and Alessandra Jannelli, Appl. Math. Comp. 174 (2006) 1370--1383

Abstract: Based on bio-chemical ground we derive a 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.

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A two immiscible liquids penetration model for surface-driven capillary flows, Riccardo Fazio, Salvatore Iacono, Alessandra Jannelli, Giovanni Cavaccini, and Vittoria Pianese, PAMM (Proc. Appl. Math. Mech.) 7, (2007) 2150003--2150004 / DOI 10.1002/pamm.200700151

Abstract: This is a mathematical and numerical study of liquid dynamics in a horizontal capillary. We present a two-liquids model which takes into account the effects of real phenomena like the outside flow dynamics. Moreover, we report on results obtained by an adaptive numerical method.

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One-dimensional mathematical and numerical modeling of liquid dynamics in a horizontal capillary, Giovanni Cavaccini, Vittoria Pianese, Alessandra Jannelli, Salvatore Iacono, and Riccardo Fazio, J. Comput. Meth. Sci. Eng., 9 (2009) 3--16

Abstract: 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.

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Extended scaling invariance of one-dimensional models of liquid dynamics in a horizontal capillary, Riccardo Fazio, Salvatore Iacono, Alessandra Jannelli, Giovanni Cavaccini, and Vittoria Pianese, Math. Meth. Appl. Sci., 35 (2012) 935--942

Abstract: In this paper, we consider the adaptive numerical solution of one-dimensional models of liquid dynamics in a horizontal capillary. The bulk liquid is assumed to be initially at rest and is put into motion by capillarity: 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. Therefore, in order to solve a capillary problem effectively, it would be advisable to apply an adaptive numerical method. Here, we show how an extended scaling invariance that can be used to define a family of solutions from a computed one. In the viscous case, the similarity transformation involves solutions of liquids with different density, surface tension, viscosity, and capillary radii, whereas in the inviscid case, we can generate a family of solutions for the same liquid and capillaries with different radii. With our study, we are able to prove that the monitor function, used in the adaptive algorithm, is invariant with respect to the considered scaling group. It follows, from this particular results, that all the solutions within the generated family verify the adaptive criteria used for the computed one. Moreover, all the solutions have the same order of accuracy even if the maximum value of the step size varies under the action of the scaling group.

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