Low Volatility Options and Numerical Diffusion of Finite Difference Schemes

2000 Mathematics Subject Classification: 65M06, 65M12.

A new wavelet-based method for the solution of the population balance equation

A new wavelet-based method for solving population balance equations with simultaneous nucleation, growth and agglomeration is proposed, which uses wavelets to express the functions. The technique is very general, powerful and overcomes the crucial problems of numerical diffusion and stability that often characterize previous techniques in this area. It is also applicable to an arbitrary grid to control resolution and computational efficiency. The proposed technique has been tested for pure agglomeration, simultaneous nucleation and growth, and simultaneous growth and agglomeration. In all cases, the predicted and analytical particle size distributions are in excellent agreement. The presence of moving sharp fronts can be addressed without the prior investigation of the characteristics of the processes. (C) 2001 Published by Elsevier Science Ltd.

Modelling of hydrogen stratification and gas mixing in containment using the APROS code

Hydrogen stratification and atmosphere mixing is a very important phenomenon in nuclear reactor containments when severe accidents are studied and simulated. Hydrogen generation, distribution and accumulation in certain parts of containment may pose a great risk to pressure increase induced by hydrogen combustion, and thus, challenge the integrity of NPP containment. The accurate prediction of hydrogen distribution is important with respect to the safety design of a NPP. Modelling methods typically used for containment analyses include both lumped parameter and field codes. The lumped parameter method is universally used in the containment codes, because its versatility, flexibility and simplicity. The lumped parameter method allows fast, full-scale simulations, where different containment geometries with relevant engineering safety features can be modelled. Lumped parameter gas stratification and mixing modelling methods are presented and discussed in this master’s thesis. Experimental research is widely used in containment analyses. The HM-2 experiment related to hydrogen stratification and mixing conducted at the THAI facility in Germany is calculated with the APROS lump parameter containment package and the APROS 6-equation thermal hydraulic model. The main purpose was to study, whether the convection term included in the momentum conservation equation of the 6-equation modelling gives some remarkable advantages compared to the simplified lumped parameter approach. Finally, a simple containment test case (high steam release to a narrow steam generator room inside a large dry containment) was calculated with both APROS models. In this case, the aim was to determine the extreme containment conditions, where the effect of convection term was supposed to be possibly high. Calculation results showed that both the APROS containment and the 6-equation model could model the hydrogen stratification in the THAI test well, if the vertical nodalisation was dense enough. However, in more complicated cases, the numerical diffusion may distort the results. Calculation of light gas stratification could be probably improved by applying the second order discretisation scheme for the modelling of gas flows. If the gas flows are relatively high, the convection term of the momentum equation is necessary to model the pressure differences between the adjacent nodes reasonably.

A Lagrangian model of air-mass photochemistry and mixing using a trajectory ensemble: the Cambridge Tropospheric Trajectory model of Chemistry And Transport (CiTTyCAT) version 4.2

A Lagrangian model of photochemistry and mixing is described (CiTTyCAT, stemming from the Cambridge Tropospheric Trajectory model of Chemistry And Transport), which is suitable for transport and chemistry studies throughout the troposphere. Over the last five years, the model has been developed in parallel at several different institutions and here those developments have been incorporated into one "community" model and documented for the first time. The key photochemical developments include a new scheme for biogenic volatile organic compounds and updated emissions schemes. The key physical development is to evolve composition following an ensemble of trajectories within neighbouring air-masses, including a simple scheme for mixing between them via an evolving "background profile", both within the boundary layer and free troposphere. The model runs along trajectories pre-calculated using winds and temperature from meteorological analyses. In addition, boundary layer height and precipitation rates, output from the analysis model, are interpolated to trajectory points and used as inputs to the mixing and wet deposition schemes. The model is most suitable in regimes when the effects of small-scale turbulent mixing are slow relative to advection by the resolved winds so that coherent air-masses form with distinct composition and strong gradients between them. Such air-masses can persist for many days while stretching, folding and thinning. Lagrangian models offer a useful framework for picking apart the processes of air-mass evolution over inter-continental distances, without being hindered by the numerical diffusion inherent to global Eulerian models. The model, including different box and trajectory modes, is described and some output for each of the modes is presented for evaluation. The model is available for download from a Subversion-controlled repository by contacting the corresponding authors.

Efeitos numéricos na simulação de escoamentos gás-sólido em leito fuidizado borbulhante utilizando a teoria cinética dos escoamentos granulares

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Simulations of physics and chemistry of polar stratospheric clouds with a general circulation model

A polar stratospheric cloud submodel has been developed and incorporated in a general circulation model including atmospheric chemistry (ECHAM5/MESSy). The formation and sedimentation of polar stratospheric cloud (PSC) particles can thus be simulated as well as heterogeneous chemical reactions that take place on the PSC particles. For solid PSC particle sedimentation, the need for a tailor-made algorithm has been elucidated. A sedimentation scheme based on first order approximations of vertical mixing ratio profiles has been developed. It produces relatively little numerical diffusion and can deal well with divergent or convergent sedimentation velocity fields. For the determination of solid PSC particle sizes, an efficient algorithm has been adapted. It assumes a monodisperse radii distribution and thermodynamic equilibrium between the gas phase and the solid particle phase. This scheme, though relatively simple, is shown to produce particle number densities and radii within the observed range. The combined effects of the representations of sedimentation and solid PSC particles on vertical H2O and HNO3 redistribution are investigated in a series of tests. The formation of solid PSC particles, especially of those consisting of nitric acid trihydrate, has been discussed extensively in recent years. Three particle formation schemes in accordance with the most widely used approaches have been identified and implemented. For the evaluation of PSC occurrence a new data set with unprecedented spatial and temporal coverage was available. A quantitative method for the comparison of simulation results and observations is developed and applied. It reveals that the relative PSC sighting frequency can be reproduced well with the PSC submodel whereas the detailed modelling of PSC events is beyond the scope of coarse global scale models. In addition to the development and evaluation of new PSC submodel components, parts of existing simulation programs have been improved, e.g. a method for the assimilation of meteorological analysis data in the general circulation model, the liquid PSC particle composition scheme, and the calculation of heterogeneous reaction rate coefficients. The interplay of these model components is demonstrated in a simulation of stratospheric chemistry with the coupled general circulation model. Tests against recent satellite data show that the model successfully reproduces the Antarctic ozone hole.

Implementation of monotonic higher order upwind scheme in KIVA 4

KIVA is a FORTRAN code developed by Los Alamos national lab to simulate complete engine cycle. KIVA is a flow solver code which is used to perform calculation of properties in a fluid flow field. It involves using various numerical schemes and methods to solve the Navier-Stokes equation. This project involves improving the accuracy of one such scheme by upgrading it to a higher order scheme. The numerical scheme to be modified is used in the critical final stage calculation called as rezoning phase. The primitive objective of this project is to implement a higher order numerical scheme, to validate and verify that the new scheme is better than the existing scheme. The latest version of the KIVA family (KIVA 4) is used for implementing the higher order scheme to support handling the unstructured mesh. The code is validated using the traditional shock tube problem and the results are verified to be more accurate than the existing schemes in reference with the analytical result. The convection test is performed to compare the computational accuracy on convective transfer; it is found that the new scheme has less numerical diffusion compared to the existing schemes. A four valve pentroof engine, an example case of KIVA package is used as application to ensure the stability of the scheme in practical application. The results are compared for the temperature profile. In spite of all the positive results, the numerical scheme implemented has a downside of consuming more CPU time for the computational analysis. The detailed comparison is provided. However, in an overview, the implementation of the higher order scheme in the latest code KIVA 4 is verified to be successful and it gives better results than the existing scheme which satisfies the objective of this project.

Isotope composition of dolomite, calcite and pore-waters of ODP Leg 210 samples

Early diagenetic dolomite beds were sampled during the Ocean Drilling Programme (ODP) Leg 201 at four reoccupied ODP Leg 112 sites on the Peru continental margin (Sites 1227/684, 1228/680, 1229/681 and 1230/685) and analysed for petrography, mineralogy, d13C, d18O and 87Sr/86Sr values. The results are compared with the chemistry, and d13C and 87Sr/86Sr values of the associated porewater. Petrographic relationships indicate that dolomite forms as a primary precipitate in porous diatom ooze and siliciclastic sediment and is not replacing the small amounts of precursor carbonate. Dolomite precipitation often pre-dates the formation of framboidal pyrite. Most dolomite layers show 87Sr/86Sr-ratios similar to the composition of Quaternary seawater and do not indicate a contribution from the hypersaline brine, which is present at a greater burial depth. Also, the d13C values of the dolomite are not in equilibrium with the d13C values of the dissolved inorganic carbon in the associated modern porewater. Both petrography and 87Sr/86Sr ratios suggest a shallow depth of dolomite formation in the uppermost sediment (<30 m below the seafloor). A significant depletion in the dissolved Mg and Ca in the porewater constrains the present site of dolomite precipitation, which co-occurs with a sharp increase in alkalinity and microbial cell concentration at the sulphate-methane interface. It has been hypothesized that microbial 'hot-spots', such as the sulphate-methane interface, may act as focused sites of dolomite precipitation. Varying d13C values from -15 per mil to +15 per mil for the dolomite are consistent with precipitation at a dynamic sulphate-methane interface, where d13C of the dissolved inorganic carbon would likewise be variable. A dynamic deep biosphere with upward and downward migration of the sulphate-methane interface can be simulated using a simple numerical diffusion model for sulphate concentration in a sedimentary sequence with variable input of organic matter. Thus, the study of dolomite layers in ancient organic carbon-rich sedimentary sequences can provide a useful window into the palaeo-dynamics of the deep biosphere.

Numerical modelling of double diffusion driven reactive flow transport in deformable fluid-saturated porous media with particular consideration of temperature-dependent chemical reaction rates

Numerical methods ave used to solve double diffusion driven reactive flow transport problems in deformable fluid-saturated porous media. in particular, thp temperature dependent reaction rate in the non-equilibrium chemical reactions is considered. A general numerical solution method, which is a combination of the finite difference method in FLAG and the finite element method in FIDAP, to solve the fully coupled problem involving material deformation, pore-fluid flow, heat transfer and species transport/chemical reactions in deformable fluid-saturated porous media has been developed The coupled problem is divided into two subproblems which are solved interactively until the convergence requirement is met. Owing to the approximate nature of the numerical method, if is essential to justify the numerical solutions through some kind of theoretical analysis. This has been highlighted in this paper The related numerical results, which are justified by the theoretical analysis, have demonstrated that the proposed solution method is useful for and applicable to a wide range of fully coupled problems in the field of science and engineering.

Numerical simulation of double-diffusion driven convective flow and rock alteration in three-dimensional fluid-saturated geological fault zones

Numerical methods are used to simulate the double-diffusion driven convective pore-fluid flow and rock alteration in three-dimensional fluid-saturated geological fault zones. The double diffusion is caused by a combination of both the positive upward temperature gradient and the positive downward salinity concentration gradient within a three-dimensional fluid-saturated geological fault zone, which is assumed to be more permeable than its surrounding rocks. In order to ensure the physical meaningfulness of the obtained numerical solutions, the numerical method used in this study is validated by a benchmark problem, for which the analytical solution to the critical Rayleigh number of the system is available. The theoretical value of the critical Rayleigh number of a three-dimensional fluid-saturated geological fault zone system can be used to judge whether or not the double-diffusion driven convective pore-fluid flow can take place within the system. After the possibility of triggering the double-diffusion driven convective pore-fluid flow is theoretically validated for the numerical model of a three-dimensional fluid-saturated geological fault zone system, the corresponding numerical solutions for the convective flow and temperature are directly coupled with a geochemical system. Through the numerical simulation of the coupled system between the convective fluid flow, heat transfer, mass transport and chemical reactions, we have investigated the effect of the double-diffusion driven convective pore-fluid flow on the rock alteration, which is the direct consequence of mineral redistribution due to its dissolution, transportation and precipitation, within the three-dimensional fluid-saturated geological fault zone system. (c) 2005 Elsevier B.V. All rights reserved.

Numerical prediction of diffusion and electric field-induced iron nanoparticle transport

Zero valent iron nanoparticles (nZVI) are considered very promising for the remediation of contaminated soils and groundwaters. However, an important issue related to their limited mobility remains unsolved. Direct current can be used to enhance the nanoparticles transport, based on the same principles of electrokinetic remediation. In this work, a generalized physicochemical model was developed and solved numerically to describe the nZVI transport through porous media under electric field, and with different electrolytes (with different ionic strengths). The model consists of the Nernst–Planck coupled system of equations, which accounts for the mass balance of ionic species in a fluid medium, when both the diffusion and electromigration of the ions are considered. The diffusion and electrophoretic transport of the negatively charged nZVI particles were also considered in the system. The contribution of electroosmotic flow to the overall mass transport was included in the model for all cases. The nZVI effective mobility values in the porous medium are very low (10−7–10−4 cm2 V−1 s−1), due to the counterbalance between the positive electroosmotic flow and the electrophoretic transport of the negatively charged nanoparticles. The higher the nZVI concentration is in the matrix, the higher the aggregation; therefore, low concentration of nZVI suspensions must be used for successful field application.

Comparison of different numerical methods for the solution of the time-fractional reaction-diffusion equation with variable diffusion coefficient

In this work we perform a comparison of two different numerical schemes for the solution of the time-fractional diffusion equation with variable diffusion coefficient and a nonlinear source term. The two methods are the implicit numerical scheme presented in [M.L. Morgado, M. Rebelo, Numerical approximation of distributed order reaction- diffusion equations, Journal of Computational and Applied Mathematics 275 (2015) 216-227] that is adapted to our type of equation, and a colocation method where Chebyshev polynomials are used to reduce the fractional differential equation to a system of ordinary differential equations

Adaptive numerical simulation of reaction - diffusion systems

Fluidized beds, granulation, heat and mass transfer, calcium dynamics, stochastic process, finite element methods, Rosenbrock methods, multigrid methods, parallelization

Numerical schemes of diffusion asymptotics and moment closures for kinetic equations

We investigate different models that are intended to describe the small mean free path regime of a kinetic equation, a particular attention being paid to the moment closure by entropy minimization. We introduce a specific asymptotic-induced numerical strategy which is able to treat the stiff terms of the asymptotic diffusive regime. We evaluate on numerics the performances of the method and the abilities of the reduced models to capture the main features of the full kinetic equation.

A fully adaptive numerical approximation for a two-dimensional epidemic model with nonlinear cross-diffusion

An epidemic model is formulated by a reactionâeuro"diffusion system where the spatial pattern formation is driven by cross-diffusion. The reaction terms describe the local dynamics of susceptible and infected species, whereas the diffusion terms account for the spatial distribution dynamics. For both self-diffusion and cross-diffusion, nonlinear constitutive assumptions are suggested. To simulate the pattern formation two finite volume formulations are proposed, which employ a conservative and a non-conservative discretization, respectively. An efficient simulation is obtained by a fully adaptive multiresolution strategy. Numerical examples illustrate the impact of the cross-diffusion on the pattern formation.