Numerical simulation of stochastic ordinary differential equations in biomathematical modelling

In this work we discuss the effects of white and coloured noise perturbations on the parameters of a mathematical model of bacteriophage infection introduced by Beretta and Kuang in [Math. Biosc. 149 (1998) 57]. We numerically simulate the strong solutions of the resulting systems of stochastic ordinary differential equations (SDEs), with respect to the global error, by means of numerical methods of both Euler-Taylor expansion and stochastic Runge-Kutta type. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.

Numerical solutions of the sediment conservation law: A review and improved formulation for coastal morphological modelling

Numerical solutions of the sediment conservation law are reviewed in terms of their application to bed update schemes in coastal morphological models. It is demonstrated that inadequately formulated numerical techniques lead to the introduction of diffusion, dispersion and the bed elevation oscillations previously reported in the literature. Four different bed update schemes are then reviewed and tested against benchmark analytical solutions. These include a first order upwind scheme, two Lax-Wendroff schemes and a non-oscillating centred scheme (NOCS) recently applied to morphological modelling by Saint-Cast [Saint-Cast, F., 2002. Modelisation de la morphodynamique des corps sableux en milieu littoral (Modelling of coastal sand banks morphodynamics), University Bordeaux 1, Bordeaux, 245 pp.]. It is shown that NOCS limits and controls numerical errors while including all the sediment flux gradients that control morphological change. Further, no post solution filtering is required, which avoids difficulties with selecting filter strength. Finally, NOCS is compared to a recent Lax-Wendroff scheme with post-solution filtering for a longer term simulation of the morphological evolution around a trained river entrance. (C) 2006 Elsevier B.V. All rights reserved.

Numerical models used for the modelling of the transport of fibrous insulation debris

Mineral wool insulation material applied to the primary cooling circuit of a nuclear reactor maybe damaged in the course of a loss of coolant accident (LOCA). The insulation material released by the leak may compromise the operation of the emergency core cooling system (ECCS), as it maybe transported together with the coolant in the form of mineral wool fiber agglomerates (MWFA) suspensions to the containment sump strainers, which are mounted at the inlet of the ECCS to keep any debris away from the emergency cooling pumps. In the further course of the LOCA, the MWFA may block or penetrate the strainers. In addition to the impact of MWFA on the pressure drop across the strainers, corrosion products formed over time may also accumulate in the fiber cakes on the strainers, which can lead to a significant increase in the strainer pressure drop and result in cavitation in the ECCS. Therefore, it is essential to understand the transport characteristics of the insulation materials in order to determine the long-term operability of nuclear reactors, which undergo LOCA. An experimental and theoretical study performed by the Helmholtz-Zentrum Dresden-Rossendorf and the Hochschule Zittau/Görlitz1 is investigating the phenomena that maybe observed in the containment vessel during a primary circuit coolant leak. The study entails the generation of fiber agglomerates, the determination of their transport properties in single and multi-effect experiments and the long-term effects that particles formed due to corrosion of metallic containment internals by the coolant medium have on the strainer pressure drop. The focus of this presentation is on the numerical models that are used to predict the transport of MWFA by CFD simulations. A number of pseudo-continuous dispersed phases of spherical wetted agglomerates can represent the MWFA. The size, density, the relative viscosity of the fluid-fiber agglomerate mixture and the turbulent dispersion all affect how the fiber agglomerates are transported. In the cases described here, the size is kept constant while the density is modified. This definition affects both the terminal velocity and volume fraction of the dispersed phases. Only one of the single effect experimental scenarios is described here that are used in validation of the numerical models. The scenario examines the suspension and horizontal transport of the fiber agglomerates in a racetrack type channel. The corresponding experiments will be described in an accompanying presentation (see abstract of Seeliger et al.).

Solitons in Exciton-Polariton System: Reduced Modelling, Analysis, and Numerical Studies

In this dissertation, we study the behavior of exciton-polariton quasiparticles in semiconductor microcavities, under the sourceless and lossless conditions.

First, we simplify the original model by removing the photon dispersion term, thus effectively turn the PDEs system to an ODEs system,

and investigate the behavior of the resulting system, including the equilibrium points and the wave functions of the excitons and the photons.

Second, we add the dispersion term for the excitons to the original model and prove that the band of the discontinuous solitons now become dark solitons.

Third, we employ the Strang-splitting method to our sytem of PDEs and prove the first-order and second-order error bounds in the $H^1$ norm and the $L_2$ norm, respectively.

Using this numerical result, we analyze the stability of the steady state bright soliton solution. This solution revolves around the $x$-axis as time progresses

and the perturbed soliton also rotates around the $x$-axis and tracks closely in terms of amplitude but lags behind the exact one. Our numerical result shows orbital

stability but no $L_2$ stability.

Cluster identification and characterization in the riser of a circulating fluidized bed from numerical simulation results

A methodology of identification and characterization of coherent structures mostly known as clusters is applied to hydrodynamic results of numerical simulation generated for the riser of a circulating fluidized bed. The numerical simulation is performed using the MICEFLOW code, which includes the two-fluids IIT`s hydrodynamic model B. The methodology for cluster characterization that is used is based in the determination of four characteristics, related to average life time, average volumetric fraction of solid, existing time fraction and frequency of occurrence. The identification of clusters is performed by applying a criterion related to the time average value of the volumetric solid fraction. A qualitative rather than quantitative analysis is performed mainly owing to the unavailability of operational data used in the considered experiments. Concerning qualitative analysis, the simulation results are in good agreement with literature. Some quantitative comparisons between predictions and experiment were also presented to emphasize the capability of the modeling procedure regarding the analysis of macroscopic scale coherent structures. (c) 2007 Elsevier Inc. All rights reserved.

Finite element modelling of temperature gradient driven rock alteration and mineralization in porous rock masses

We present finite element simulations of temperature gradient driven rock alteration and mineralization in fluid saturated porous rock masses. In particular, we explore the significance of production/annihilation terms in the mass balance equations and the dependence of the spatial patterns of rock alteration upon the ratio of the roll over time of large scale convection cells to the relaxation time of the chemical reactions. Special concepts such as the gradient reaction criterion or rock alteration index (RAI) are discussed in light of the present, more general theory. In order to validate the finite element simulation, we derive an analytical solution for the rock alteration index of a benchmark problem on a two-dimensional rectangular domain. Since the geometry and boundary conditions of the benchmark problem can be easily and exactly modelled, the analytical solution is also useful for validating other numerical methods, such as the finite difference method and the boundary element method, when they are used to dear with this kind of problem. Finally, the potential of the theory is illustrated by means of finite element studies related to coupled flow problems in materially homogeneous and inhomogeneous porous rock masses. (C) 1998 Elsevier Science S.A. All rights reserved.

Finite element modelling of reactive mass transport problems in fluid-saturated porous media

We use the finite element method to solve reactive mass transport problems in fluid-saturated porous media. In particular, we discuss the mathematical expression of the chemical reaction terms involved in the mass transport equations for an isothermal, non-equilibrium chemical reaction. It has turned out that the Arrhenius law in chemistry is a good mathematical expression for such non-equilibrium chemical reactions especially from the computational point of view. Using the finite element method and the Arrhenius law, we investigate the distributions of PH (i.e. the concentration of H+) and the relevant reactive species in a groundwater system. Although the main focus of this study is on the contaminant transport problems in groundwater systems, the related numerical techniques and principles are equally applicable to the orebody formation problems in the geosciences. Copyright (C) 1999 John Wiley & Sons, Ltd.

Numerical and experimental study of seepage in unconfined aquifers with a periodic boundary condition

The assessment of groundwater conditions within an unconfined aquifer with a periodic boundary condition is of interest in many hydrological and environmental problems. A two-dimensional numerical model for density dependent variably saturated groundwater flow, SUTRA (Voss, C.I., 1984. SUTRA: a finite element simulation model for saturated-unsaturated, fluid-density dependent ground-water flow with energy transport or chemically reactive single species solute transport. US Geological Survey, National Center, Reston, VA) is modified in order to be able to simulate the groundwater flow in unconfined aquifers affected by a periodic boundary condition. The basic flow equation is changed from pressure-form to mixed-form. The model is also adjusted to handle a seepage-face boundary condition. Experiments are conducted to provide data for the groundwater response to the periodic boundary condition for aquifers with both vertical and sloping faces. The performance of the numerical model is assessed using those data. The results of pressure- and mixed-form approximations are compared and the improvement achieved through the mixed-form of the equation is demonstrated. The ability of the numerical model to simulate the water table and seepage-face is tested by modelling some published experimental data. Finally the numerical model is successfully verified against present experimental results to confirm its ability to simulate complex boundary conditions like the periodic head and the seepage-face boundary condition on the sloping face. (C) 1999 Elsevier Science B.V. All rights reserved.

Finite element modelling of dissipative structures for nonequilibrium chemical reactions in fluid-saturated porous media

We use the finite element method to model and predict the dissipative structures of chemical species for a nonequilibrium chemical reaction system in a fluid-saturated porous medium. In particular, we explore the conditions under which dissipative structures of the species may exist in the Brusselator type of nonequilibrium chemical reaction. Since this is the first time the finite element method and related strategies have been used to study the chemical instability problems in a fluid-saturated porous medium, it is essential to validate the method and strategies before they are put into application. For this purpose, we have rigorously derived the analytical solutions for dissipative structures of chemical species in a benchmark problem, which geometrically is a square. Comparison of the numerical solutions with the analytical ones demonstrates that the proposed numerical method and strategy are robust enough to solve chemical instability problems in a fluid-saturated porous medium. Finally, the related numerical results from two application examples indicate that both the regime and the magnitude of pore-fluid flow have significant effects on the nature of the dissipative structures that developed for a nonequilibrium chemical reaction system in a fluid-saturated porous medium. The motivation for this study is that self-organization under conditions of pore-fluid flow in a porous medium is a potential mechanism of the orebody formation and mineralization in the upper crust of the Earth. (C) 2000 Elsevier Science S.A. All rights reserved.

Modelling solute transport in structured soils: Performance evaluation of the ADR and TRM models

The movement of chemicals through the soil to the groundwater or discharged to surface waters represents a degradation of these resources. In many cases, serious human and stock health implications are associated with this form of pollution. The chemicals of interest include nutrients, pesticides, salts, and industrial wastes. Recent studies have shown that current models and methods do not adequately describe the leaching of nutrients through soil, often underestimating the risk of groundwater contamination by surface-applied chemicals, and overestimating the concentration of resident solutes. This inaccuracy results primarily from ignoring soil structure and nonequilibrium between soil constituents, water, and solutes. A multiple sample percolation system (MSPS), consisting of 25 individual collection wells, was constructed to study the effects of localized soil heterogeneities on the transport of nutrients (NO3-, Cl-, PO43-) in the vadose zone of an agricultural soil predominantly dominated by clay. Very significant variations in drainage patterns across a small spatial scale were observed tone-way ANOVA, p < 0.001) indicating considerable heterogeneity in water flow patterns and nutrient leaching. Using data collected from the multiple sample percolation experiments, this paper compares the performance of two mathematical models for predicting solute transport, the advective-dispersion model with a reaction term (ADR), and a two-region preferential flow model (TRM) suitable for modelling nonequilibrium transport. These results have implications for modelling solute transport and predicting nutrient loading on a larger scale. (C) 2001 Elsevier Science Ltd. All rights reserved.

Finite element modelling of rock alteration and metamorphic process in hydrothermal systems

We use the finite element method to simulate the rock alteration and metamorphic process in hydrothermal systems. In particular, we consider the fluid-rock interaction problems in pore-fluid saturated porous rocks. Since the fluid rock interaction takes place at the contact interface between the pore-fluid and solid minerals, it is governed by the chemical reaction which usually takes place very slowly at this contact interface, from the geochemical point of view. Due to the relative slowness of the rate of the chemical reaction to the velocity of the pore-fluid flow in the hydrothermal system to be considered, there exists a retardation zone, in which the conventional static theory in geochemistry does not hold true. Since this issue is often overlooked by some purely numerical modellers, it is emphasized in this paper. The related results from a typical rock alteration and metamorphic problem in a hydrothermal system have shown not only the detailed rock alteration and metamorphic process, but also the size of the retardation zone in the hydrothermal system. Copyright (C) 2001 John Wiley & Sons, Ltd.

Finite element modelling of heat transfer through permeable cracks in hydrothermal systems with upward throughflow

We use the finite element method to model the heat transfer phenomenon through permeable cracks in hydrothermal systems with upward throughflow. Since the finite element method is an approximate numerical method, the method must be validated before it is used to soh,e any new, kind of problem. However, the analytical solution, which can be used to validate the finite element method and other numerical methods, is rather limited in the literature, especially, for the problem considered here. Keeping this in mind, we have derived analytical solutions for the temperature distribution along the vertical axis of a crack in a fluid-saturated porous layer. After the finite element method is validated by comparing the numerical solution with the analytical solution for the same benchmark problem, it is used to investigate the pore-fluid flow and heat transfer in layered hydrothermal systems with vertical permeable cracks. The related analytical and numerical results have demonstrated that vertical cracks are effective and efficient members to transfer heat energy from the bottom section to the top section in hydrothermal systems with upward throughflow.

BeachWin: modelling groundwater effects on swash sediment transport and beach profile changes

Field and laboratory observations have shown that a relatively low beach groundwater table enhances beach accretion. These observations have led to the beach dewatering technique (artificially lowering the beach water table) for combating beach erosion. Here we present a process-based numerical model that simulates the interacting wave motion on the beach. coastal groundwater flow, swash sediment transport and beach profile changes. Results of model simulations demonstrate that the model replicates accretionary effects of a low beach water table on beach profile changes and has the potential to become a tool for assessing the effectiveness of beach dewatering systems. (C) 2002 Elsevier Science Ltd. All rights reserved.

Modelling of species in hood fires by conditional moment closure

Carbon monoxide, the chief killer in fires, and other species are modelled for a series of enclosure fires. The conditions emulate building fires where CO is formed in the rich, turbulent, nonpremixed flame and is transported frozen to lean mixtures by the ceiling jet which is cooled by radiation and dilution. Conditional moment closure modelling is used and computational domain minimisation criteria are developed which reduce the computational cost of this method. The predictions give good agreement for CO and other species in the lean, quenched-gas stream, holding promise that this method may provide a practical means of modelling real, three-dimensional fire situations. (c) 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved.