Mathematical and Numerical Modeling

- A 3D mathematical model for the prediction of mucilage dynamics
- Adaptive stiff solvers at low accuracy and complexity
- Mathematical and numerical modeling for a bio-chemical aquarium
- A two immiscible liquids penetration model for surface-driven capillary flows
- Extended scaling invariance of one-dimensional models of liquid dynamics in a horizontal capillary

** 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.

** 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.

** 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.

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 *

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*

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 *

Address:

Department of Mathematics and Computer Science

University of Messina

V.le F. Stagno D'Alcontres 31,

98166 Messina, Italy

Phone: +39 090 ?

Fax: +39 090 ?

Email: * rfazio@unime.it*

By Riccardo Fazio> Last modified: December 18, 2012