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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 08 Dec 2023 21:54:25 +0000Part I. Numerical experiments for the computation of invariant curves in dynamical systems. Part II. Numerical convergence results for a one-dimensional Stefan problem
https://resolver.caltech.edu/CaltechETD:etd-05092007-080826
Authors: Morlet, Anne Chantal
Year: 1990
DOI: 10.7907/nfr6-ca66
Part I
We derive a model equation for the linearized equation of an invariant curve for a Poincare map. We discretize the model equation with a second-order and third-order finite difference schemes, and with a cubic spline interpolation scheme. We also approximate the solution of the model equation with a truncated Fourier expansion. We derive error estimates for the second-order and third-order finite difference schemes and for the cubic spline interpolation scheme. We numerically implement the four schemes we consider and plot some error curves.
Part II
We show for a one-dimensional Stefan problem, that the numerical solution converges to the solution of the continuous equations in the limit of zero meshsize and timestep. We discretize the continuous equations with a second-order finite difference scheme in space and Crank-Nicholson scheme in time. We derive error equations and we use L2 estimates to bound the error in terms of the truncation errors of the finite difference scheme. We confirm the analysis with numerical computations. We numerically prove that we have fourth-order convergence in space if we discretize the partial differential equations with a fourth-order scheme in space.https://thesis.library.caltech.edu/id/eprint/1705Mathematical control theory for liquid chromatography
https://resolver.caltech.edu/CaltechTHESIS:09082011-113722649
Authors: Ammons, Richard Lewis Martin
Year: 1992
DOI: 10.7907/bneq-qh86
<p>A more comprehensive mathematical theory for liquid
chromatography is set forth, incorporating dynamical models for mixed solvents and solutes, and new mathematical models for adsorption, including adsorbent and exchange processes.
The equations for solvent and solute are shown to possess unique solutions, using so-called energy methods. The solvent modulation of local velocity is found theoretically, as is solvent control of solute adsorption, diffusivity, and dispersion. The theory for solvent control of solute adsorption is found to be very accurate against experiment, and offers a useful method of treating normal phase, reversed phase, ion exchange, and ion pair liquid chromatography in a unified mathematical framework, under the name catalyzed adsorption. The long-recognised problem
of solvent localization is modelled, and the model shown to be consistent with experiment. Another classical problem, solvent demixing, is explained in terms of the nonlinear multicomponent solvent model, wherein solvent gradients steepen according to the adsorption and shock formation. Perturbation theory, based on a small packing number d_p/L « 1 (where d_p is substrate particle diameter, L is column length), is applied to the solvent-controlled pulsed solute dynamical equations. When moment techniques are used in conjunction with perturbation theory, very useful and simplified system control equations are obtained. These control equations are used in some model problems to discuss HETP (Height Equivalent to a Theoretical Plate) variations with Peclet number, with relative solvent concentration, and between solutes. Finally, numerical methods for the solvent and solute equations are discussed.</p>
https://thesis.library.caltech.edu/id/eprint/6655