Numerical Initial Value Problems in Ordinary Differential Equations

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

A literature review on mechanisms and models for the coalescence process of fluid particles

In this paper, we review the recent development of phase-field models and their numerical methods for multi-component fluid flows with interfacial phenomena. The models consist of a Navier-Stokes system coupled with a multi-component Cahn-Hilliard system through a phase-field dependent surface tension force, variable density and viscosity, and the advection term. The classical infinitely thin boundary of separation between two immiscible fluids is replaced by a transition region of a small but finite width, across which the composition of the mixture changes continuously. A constant level set of the phase-field is used to capture the interface between two immiscible fluids. Phase-field methods are capable of computing topological changes such as splitting and merging, and thus have been applied successfully to multi-component fluid flows involving large interface deformations. Practical applications are provided to illustrate the usefulness of using a phase-field method. Computational results of various experiments show the accuracy and effectiveness of phase-field models.

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Phase-Field Models for Multi-Component Fluid Flows

SUMMARY An hp-adaptive pseudospectral method is presented for numerically solving optimal control problems The method presented in this paper iteratively determines the number of segments, the width of each segment, and the polynomial degree required in each segment in order to obtain a solution to a userspecified accuracy Starting with a global pseudospectral approximation for the state, on each iteration the method determines locations for the segment breaks and the polynomial degree in each segment for use on the next iteration The number of segments and the degree of the polynomial on each segment continue to be updated until a user-specified tolerance is met The terminology ‘hp’ is used because the segment widths (denoted h) and the polynomial degree (denoted p) in each segment are determined simultaneously It is found that the method developed in this paper leads to higher accuracy solutions with less computational effort and memory than is required in a global pseudospectral method Consequently, the method makes it possible to solve complex optimal control problems using pseudospectral methods in cases where a global pseudospectral method would be computationally intractable Finally, the utility of the method is demonstrated on a variety of problems of varying complexity Copyright 2010 John Wiley & Sons, Ltd

/pdf/an-hp-adaptive-pseudospectral-method-for-solving-optimal-yisse331s0.pdf

An hp‐adaptive pseudospectral method for solving optimal control problems

Bubble columns are widely used for carrying out gas−liquid and gas−liquid−solid reactions in a variety of industrial applications. The dispersion and interfacial heat- and mass-transfer fluxes, which often limit the overall chemical reaction rates, are closely related to the fluid dynamics of the system through the liquid−gas contact area and the turbulence properties of the flow. There is thus considerable interest, within both academia and industry, to improve the limited understanding of the complex multiphase flow phenomena involved, which is preventing optimal scale-up and design of these reactors. In this paper, the progress reported in the literature during the past decade regarding the use of averaged Eulerian multifluid models and computational fluid dynamics (CFD) to model vertical bubble-driven flows is reviewed. The limiting steps in the model derivation are the formulation of proper boundary conditions, closure laws determining turbulent effects, interfacial transfer fluxes, and the bubble co...

Carlos Alberto Dorao

Papers

Modeling of Bubble Column Reactors: Progress and Limitations

A review on heat exchanger thermal hydraulic models for cryogenic applications

A least squares method for the solution of population balance problems

On the conceptual design of pre-cooling stage of LNG plants using propane or an ethane/propane mixture

Simulation of transients in natural gas pipelines