This dissertation addresses the problem of modeling electromagnetic fields in and about the brain-skull-scalp system that are generated by active neural populations. Specifically, frequency dependence of Maxwell's fields is explored for the case of a dipole-like current source embedded in a spherical conductor surrounded by a vacuum. Frequency dependence was found to be small. Loosely, the difference between frequency dependent and frequency independent fields reached approximately 1% at 103 Hz and reached up to 16% at 104 Hz. Frequency dependence was found to be highly dependent on conductivity, the size of the conductor, and on the phase of generated fields. These findings indicate that the degree to which the magnetic field is coupled to the electric field depends on interference patterns occurring within the conductor. Several highly distinguishable exceptions to general trends in the data were found to be consistent with this view.
", "doi": "10.7907/TNQ6-P071", "publication_date": "1986", "thesis_type": "phd", "thesis_year": "1986" }, { "id": "thesis:1015", "collection": "thesis", "collection_id": "1015", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-03192008-094428", "primary_object_url": { "basename": "Naughton_mj_1986.pdf", "content": "final", "filesize": 4089116, "license": "other", "mime_type": "application/pdf", "url": "/1015/1/Naughton_mj_1986.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "On Numerical Boundary Conditions for the Navier-Stokes Equations", "author": [ { "family_name": "Naughton", "given_name": "Michael John", "clpid": "Naughton-Michael-John" } ], "thesis_advisor": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" } ], "thesis_committee": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Saffman", "given_name": "Philip G.", "clpid": "Saffman-P-G" }, { "family_name": "Lorenz", "given_name": "Jens", "clpid": "Lorenz-Jens" }, { "family_name": "Wu", "given_name": "Theodore Yao-tsu", "clpid": "Wu-T-Y-T" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Part I:
\r\n\r\nWe consider the numerical solution of the Navier-Stokes equations governing the unsteady flow of a viscous incompressible fluid. The analysis of numerical approximations to smooth nonlinear problems reduces to the examination of related linearized problems. The well posedness of the linear Navier-Stokes equations and the stability of finite difference approximations are studied by making energy estimates for the initial boundary value problems. Flows with open boundaries (i.e., inflow and outflow) and with solid walls are considered. We analyse boundary conditions of several types involving the velocity components or a combination of the velocity components and the pressure. The properties of these different types of boundary conditions are compared with emphasis on the suppression of undesirable numerical boundary layers for high Reynolds number calculations. The formulation of the Navier-Stokes equations which uses an elliptic equation for the pressure in lieu of the divergence equation for the velocity is shown to be equivalent to the usual formulation if the boundary conditions are treated correctly. The stability of numerical methods which use this formulation is demonstrated.
\r\n\r\nPart II:
\r\n\r\nWe consider the numerical solution of the stream function vorticity formulation of the two dimensional incompressible Navier-Stokes equations for unsteady flows on a domain with rigid walls. The no-slip boundary conditions on the velocity components at the rigid walls are prescribed. In the stream function vorticity formulation these become two boundary conditions on the stream function and there is no explicit boundary condition on the vorticity. The accuracy of the numerical approximations to the stream function and the vorticity is investigated.The common approach in calculations is to employ second order accurate finite difference approximations for all the space derivatives and the boundary conditions together with a time marching procedure involving iteration at each time step to satisfy the boundary conditions. With such schemes the vorticity may be only first order accurate. Higher order approximations to the no-slip boundary conditions have frequently been used to overcome this problem. A one dimensional initial boundary value problem containing the salient features is proposed and analysed. With the use of this model, the behaviour observed in calculations is explained.
", "doi": "10.7907/3QDE-CB55", "publication_date": "1986", "thesis_type": "phd", "thesis_year": "1986" }, { "id": "thesis:1159", "collection": "thesis", "collection_id": "1159", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-03262008-112516", "primary_object_url": { "basename": "Henderson_me_1985.pdf", "content": "final", "filesize": 2248741, "license": "other", "mime_type": "application/pdf", "url": "/1159/1/Henderson_me_1985.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Complex Bifurcation", "author": [ { "family_name": "Henderson", "given_name": "Michael Edwin", "orcid": "0009-0007-2206-0011", "clpid": "Henderson-Michael-Edwin" } ], "thesis_advisor": [ { "family_name": "Keller", "given_name": "Herbert Bishop", "clpid": "Keller-H-B" } ], "thesis_committee": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Saffman", "given_name": "Philip G.", "clpid": "Saffman-P-G" }, { "family_name": "Cohen", "given_name": "Donald S.", "clpid": "Cohen-D-S" }, { "family_name": "Keller", "given_name": "Herbert Bishop", "clpid": "Keller-H-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Real equations of the form g(x,\u03bb) = 0 are shown to have a complex extension G(u,\u03bb) = 0, defined on the complex Banach space 𝔹 \u2295 i𝔹. At a singular point of the real equation this extension has solution branches corresponding to both the real and imaginary roots of the Algebraic Bifurcation Equations (ABE's).
\r\n\r\nWe solve the ABE's at simple quadratic folds, quadratic bifurcation points, and cubic bifurcation points, and show that these are complex bifurcation points. We also show that at a Hopf bifurcation point of the real equation there are two families of complex periodic orbits, parametrized by three real parameters.
\r\n\r\nBy taking sections of solutions of complex equations with two real parameters, we show that complex branches may connect disjoint solution branches of the real equation. These complex branches provide a simple and practical means of locating disjoint branches of real solutions.
\r\n\r\nFinally, we show how algorithms for computing real solutions may be modified to compute complex solutions. We use such an algorithm to find solutions of several example problems, and locate two sets of disjoint real branches.
", "doi": "10.7907/JF82-1T64", "publication_date": "1985", "thesis_type": "phd", "thesis_year": "1985" }, { "id": "thesis:1227", "collection": "thesis", "collection_id": "1227", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-03312008-100117", "primary_object_url": { "basename": "Henshaw_wd_1985.pdf", "content": "final", "filesize": 6861028, "license": "other", "mime_type": "application/pdf", "url": "/1227/1/Henshaw_wd_1985.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Part I. The Numerical Solution of Hyperbolic Systems of Conservation Laws. Part II. Composite Overlapping Grid Techniques", "author": [ { "family_name": "Henshaw", "given_name": "William Douglas", "orcid": "0009-0008-9088-7229", "clpid": "Henshaw-William-Douglas" } ], "thesis_advisor": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" } ], "thesis_committee": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Keller", "given_name": "Herbert Bishop", "clpid": "Keller-H-B" }, { "family_name": "Saffman", "given_name": "Philip G.", "clpid": "Saffman-P-G" }, { "family_name": "List", "given_name": "E. John", "clpid": "List-E-J" }, { "family_name": "Lorenz", "given_name": "Jens", "clpid": "Lorenz-Jens" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Part I
\r\n\r\nA method is described for the numerical solution of hyperbolic systems of conservation laws in one space dimension. The basis of the scheme is to use finite differences where the solution is smooth and the method of characteristics where the solution is not smooth. The method can accurately represent shocks. Results are presented for the solution of the equations of gas dynamics. The examples illustrate the accuracy of the method when discontinuities are present and the code's performance on difficult problems of interacting shocks and shock formation.
\r\n\r\nPart II
\r\n\r\nTechniques for the numerical solution of partial differential equations on composite overlapping meshes are discussed. Methods for the solution of time dependent and elliptic problems are illustrated, including a discussion of implicit time stepping and using the multigrid algorithm for the iterative solution of Poisson's equation. Two model problems are analyzed. The first gives insight into the accuracy of the solution to elliptic equations on overlapping meshes. The second deals with the numerical approximation of boundary conditions for vorticity stream function formulations. Computational results are presented.
", "doi": "10.7907/kz0y-2j77", "publication_date": "1985", "thesis_type": "phd", "thesis_year": "1985" }, { "id": "thesis:7555", "collection": "thesis", "collection_id": "7555", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:03262013-085556300", "primary_object_url": { "basename": "Stanley_ea_1985.pdf", "content": "final", "filesize": 17795444, "license": "other", "mime_type": "application/pdf", "url": "/7555/1/Stanley_ea_1985.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Diffusion in Glassy Polymers", "author": [ { "family_name": "Stanley", "given_name": "Elizabeth Ann", "clpid": "Stanley-Elizabeth-Ann" } ], "thesis_advisor": [ { "family_name": "Cohen", "given_name": "Donald S.", "clpid": "Cohen-D-S" } ], "thesis_committee": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Saffman", "given_name": "Philip G.", "clpid": "Saffman-P-G" }, { "family_name": "Knauss", "given_name": "Wolfgang Gustav", "clpid": "Knauss-W-G" }, { "family_name": "Cohen", "given_name": "Donald S.", "clpid": "Cohen-D-S" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Fluid diffusion in glassy polymers proceeds in ways that are not explained by the standard diffusion model. Although the reasons for the anomalous effects are not known, much of the observed behavior is attributed to the long times that polymers below their glass transition temperature take to adjust to changes in their condition. The slow internal relaxations of the polymer chains ensure that the material properties are history-dependent, and also allow both local inhomogeneities and differential swelling to occur. Two models are developed in this thesis with the intent of accounting for these effects in the diffusion process.
\r\n\r\nIn Part I, a model is developed to account for both the history dependence of the glassy polymer, and the dual sorption which occurs when gas molecules are immobilized by the local heterogeneities. A preliminary study of a special case of this model is conducted, showing the existence of travelling wave solutions and using perturbation techniques to investigate the effect of generalized diffusion mechanisms on their form. An integral averaging method is used to estimate the penetrant front position.
\r\n\r\nIn Part II, a model is developed for particle diffusion along with displacements in isotropic viscoelastic materials. The nonlinear dependence of the materials on the fluid concentration is taken into account, while pure displacements are assumed to remain in the range of linear viscoelasticity. A fairly general model is obtained for three-dimensional irrotational movements, with the development of the model being based on the assumptions of irreversible thermodynamics. With the help of some dimensional analysis, this model is simplified to a version which is proposed to be studied for Case II behavior.
", "doi": "10.7907/pjzx-hb67", "publication_date": "1985", "thesis_type": "phd", "thesis_year": "1985" }, { "id": "thesis:339", "collection": "thesis", "collection_id": "339", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-01252007-143608", "primary_object_url": { "basename": "Smyth_nf_1984.pdf", "content": "final", "filesize": 3644668, "license": "other", "mime_type": "application/pdf", "url": "/339/1/Smyth_nf_1984.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Part I: Soliton on a Beach and Related Problems. Part II: Modulated Capillary Waves", "author": [ { "family_name": "Smyth", "given_name": "Noel Frederick", "orcid": "0000-0002-8787-3175", "clpid": "Smyth-Noel-Frederick" } ], "thesis_advisor": [ { "family_name": "Whitham", "given_name": "Gerald Beresford", "clpid": "Whitham-G-B" }, { "family_name": "Cohen", "given_name": "Donald S.", "clpid": "Cohen-D-S" } ], "thesis_committee": [ { "family_name": "Cohen", "given_name": "Donald S.", "clpid": "Cohen-D-S" }, { "family_name": "Kath", "given_name": "William L.", "clpid": "Kath-William-L" }, { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Raichlen", "given_name": "Fredric", "clpid": "Raichlen-F" }, { "family_name": "Whitham", "given_name": "Gerald Beresford", "clpid": "Whitham-G-B" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "I. Soliton on a Sloping Beach and Related Problems
\r\n\r\nThe problem of the behaviour of a soliton on a slowly varying beach is considered. It is shown that for a correct description, the full Boussinesq equations rather than a Korteweg-de Vries type approximation must be used. Using both energy conservation and two-timing expansions, the behaviour of the soliton is analysed. The slowly varying soliton is found not to conserve mass and momentum and it has been suggested that to conserve these quantities, both forward and reflected waves must be added behind the soliton, these waves being solutions of the linear shallow water equations. It is shown that to the order of approximation of the Boussinesq equations, only a forward wave (or tail) behind the soliton is necessary to fulfill mass and momentum conservation.
\r\n\r\nA perturbed Korteweg-de Vries equation for which the perturbation adds energy to the soliton is considered. It is found that a tail is formed behind the soliton. The development of this tail into new solitons is analysed.
\r\n\r\nII. Modulated Capillary Waves
\r\n\r\nAn exact hodograph solution for symmetric and antisymmetric capillary waves on a fluid sheet (of possibly infinite thickness) has been previously found. Using this solution, an exact averaged Lagrangian for slowly varying capillary waves is calculated. Modulation equations can be found from this averaged Lagrangian, but due to the algebraic complexity of the equations, the limit of waves on a thin fluid sheet is considered. From the modulation equations, the stability of symmetric and antisymmetric capillary waves on a thin fluid sheet is found. The modulation equations for antisymmetric waves form a hyperbolic system and the simple wave solutions for this system are calculated. These simple wave solutions are interpreted physically.
", "doi": "10.7907/VXDD-1M21", "publication_date": "1984", "thesis_type": "phd", "thesis_year": "1984" }, { "id": "thesis:3335", "collection": "thesis", "collection_id": "3335", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-09052006-153951", "primary_object_url": { "basename": "Yang_v_1984.pdf", "content": "final", "filesize": 7566442, "license": "other", "mime_type": "application/pdf", "url": "/3335/1/Yang_v_1984.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Pressure Oscillations in Liquid-Fueled Ramjet Engines", "author": [ { "family_name": "Yang", "given_name": "Vigor", "clpid": "Yang-Vigor" } ], "thesis_advisor": [ { "family_name": "Culick", "given_name": "Fred E. C.", "clpid": "Culick-F-E-C" } ], "thesis_committee": [ { "family_name": "Caughey", "given_name": "Thomas Kirk", "clpid": "Caughey-T-K" }, { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Sabersky", "given_name": "Rolf H.", "clpid": "Sabersky-R-H" }, { "family_name": "Zukoski", "given_name": "Edward E.", "clpid": "Zukoski-E-E" }, { "family_name": "Culick", "given_name": "Fred E. C.", "clpid": "Culick-F-E-C" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Pressure oscillations in liquid-fueled ramjet engines have been studied both analytically and numerically within the low frequency range. We examine first the linear unsteady motions in coaxial-dump configurations. The flowfield in the dump combustor is approximated by division into three parts: a flow of reactants, a region containing combustion products, and a recirculation zone, separated by two infinitesimally thin sheets: the flame and the vortex sheets. The three zones are matched at these sheets by taking into account kinematic and conservation relations. The oscillatory field in the inlet is coupled to the field in the combustor at the dump plane to determine the complex frequencies characterizing the linear stability of the engine. Favorable comparison with the experimental data obtained at the California Institute of Technology has been obtained.
\r\n\r\nNumerical analysis has been applied to investigate the nonlinear behavior of the shock wave in the inlet diffuser. Both viscous effects and the influences of injecting fuel/air mixture are accounted for. The response of a shock wave to various disturbances, including finite and large amplitude oscillations, has been studied in detail. The results obtained serve as a basis for analyzing the stability characteristics of the inlet flow.
\r\n\r\nNumerical calculations have also been conducted for the pressure oscillations in side-dump ramjet engines. The flowfields have been constructed in two regions: the inlet section, including a region of fuel injection, and a dump combustor. Each region is treated separately and matched with the other at the dump plane. Following the calculation of the mean flowfield, the oscillatory characteristics of the engine are determined by its response to a disturbance imposed on the mean flow. Results for the frequencies and mode shapes have shown good agreement with the experimental data reported by the Naval Weapons Center, China Lake.
\r\n", "doi": "10.7907/rfpg-es59", "publication_date": "1984", "thesis_type": "phd", "thesis_year": "1984" }, { "id": "thesis:4338", "collection": "thesis", "collection_id": "4338", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-10312005-133116", "primary_object_url": { "basename": "Tatoian_jz_1983.pdf", "content": "final", "filesize": 2397489, "license": "other", "mime_type": "application/pdf", "url": "/4338/1/Tatoian_jz_1983.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "An Analytical Study of Electromagnetic Vector Field Propagation in a Nonlinear Electron Plasma", "author": [ { "family_name": "Tatoian", "given_name": "James Zareh", "clpid": "Tatoian-James-Zareh" } ], "thesis_advisor": [ { "family_name": "Papas", "given_name": "Charles Herach", "clpid": "Papas-C-H" } ], "thesis_committee": [ { "family_name": "Keller", "given_name": "Herbert Bishop", "clpid": "Keller-H-B" }, { "family_name": "Franklin", "given_name": "Joel N.", "clpid": "Franklin-J-N" }, { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Fornberg", "given_name": "Bengt", "clpid": "Fornberg-Bengt" }, { "family_name": "Papas", "given_name": "Charles Herach", "clpid": "Papas-C-H" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "From the equations of hydrodynamics and electrodynamics, a system of a coupled nonlinear equations governing the propagation of plane electromagnetic waves in a collisionless electron plasma is obtained. It is shown that solitary wave solutions exist for both the longitudinal and transverse components of the electromagnetic field. It is found that the velocity of the electromagnetic vector solitary wave depends on the amplitudes of all components of the field linearly. The relations among the longitudinal and transverse components that support the solitary waves are determined for different values of plasma temperature. It is shown that while transverse solitary waves cannot exist, except when they are supported by longitudinal waves, the latter can exist by themselves. The dynamics of the plasma electrons during the passage of a longitudinal wave is analyzed and the interaction of such waves with each other is studied. An upper bound on the amplitudes of these waves is obtained. The uniqueness and stability of the longitudinal waves are demonstrated. A Lagrangian density function and two conservation laws for the longitudinal wave equation are found. Frequency spectra of the solitary waves are calculated and their low frequency content is emphasized.
\r\n", "doi": "10.7907/hzt3-s585", "publication_date": "1983", "thesis_type": "phd", "thesis_year": "1983" }, { "id": "thesis:2683", "collection": "thesis", "collection_id": "2683", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-06222005-104752", "primary_object_url": { "basename": "Reyna_lgm_1983.pdf", "content": "final", "filesize": 4705367, "license": "other", "mime_type": "application/pdf", "url": "/2683/1/Reyna_lgm_1983.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "I. Stability of Tchebyshev Collocation. II. Interpolation for Surfaces with 1-D Discontinuities. III. On Composite Meshes", "author": [ { "family_name": "Reyna", "given_name": "Luis Guillermo Maria", "clpid": "Reyna-Luis-Guillermo-Maria" } ], "thesis_advisor": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" } ], "thesis_committee": [ { "family_name": "Keller", "given_name": "Herbert Bishop", "clpid": "Keller-H-B" }, { "family_name": "List", "given_name": "E. John", "clpid": "List-E-J" }, { "family_name": "Cohen", "given_name": "Donald S.", "clpid": "Cohen-D-S" }, { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "I. Stability of Tchebyshev Collocation
\r\n\r\nWe describe Tchebyshev collocation when applied to hyperbolic equations in one space dimension. We discuss previous stability results valid for scalar equations and study a procedure that when applied to a strictly hyperbolic system of equations leads to a stable numerical approximation in the L2-norm. The method consists of using orthogonal projections in the L2-norm to apply the boundary conditions and smooth the higher modes.
\r\n\r\nII. On 2-D Interpolation for Surfaces with 1-D Discontinuities
\r\n\r\nThis problem arises in the context of shock calculations in two space dimensions. Given the set of parabolic equations describing the shock phenomena the method proceeds by discretising in time and then solving the resulting elliptic equation by splitting. The specific problem is to reconstruct a two dimensional function which is fully resolved along a few parallel horizontal lines. The interpolation proceeds by determining the position of any discontinuity and then interpolating parallel to it.
\r\n\r\nIII. On Composite Meshes
\r\n\r\nWe collect several numerical experiments designed to determine possible numerical artifacts produced by the overlapping region of composite meshes. We also study the numerical stability of the method when applied to hyperbolic equations. Finally we apply it to a model of a wind driven ocean circulation model in a circular basin. We use stretching in the angular and radial directions which allow the necessary resolution to be obtained along the boundary.
", "doi": "10.7907/AAG6-MW97", "publication_date": "1983", "thesis_type": "phd", "thesis_year": "1983" }, { "id": "thesis:3619", "collection": "thesis", "collection_id": "3619", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-09182006-134307", "type": "thesis", "title": "Solution Adaptive Mesh Procedures for the Numerical Solution of Singular Perturbation Problems", "author": [ { "family_name": "Brown", "given_name": "David Leslie", "clpid": "Brown-David-Leslie" } ], "thesis_advisor": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" } ], "thesis_committee": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Cohen", "given_name": "Donald S.", "clpid": "Cohen-D-S" }, { "family_name": "Keller", "given_name": "Herbert Bishop", "clpid": "Keller-H-B" }, { "family_name": "Leal", "given_name": "L. Gary", "clpid": "Leal-L-G" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The accurate numerical solution of singular perturbation problems by finite difference methods is considered. (For efficient computations of this type, refinement of the finite difference mesh is important. The technique of solution-adaptive mesh refinement, in which the mesh is refined iteratively by looking at the properties of a computed solution, can be the simplest method by which to implement a mesh refinement.) The theoretical justification of solution-adaptive mesh refinement for singularly perturbed systems of first order ordinary differential equations (ODEs) is discussed. It is shown that a posteriori error estimates can be found for weighted one-sided difference approximations to systems of ODEs without turning points and to systems of ODEs with turning points that can be transformed to a typical normal form. These error estimates essentially depend only on the local meshwidths and on lower order divided differences of the computed solution, and so can be used in the implementation of solution-adaptive mesh refinement. It is pointed out, however, that not all systems with turning points fall into these categories, and solution-adaptive mesh refinement can sometimes be inadequate for the accurate resolution of solutions of these systems.
\r\n\r\nNumerical examples are presented in which the solutions of some model equations of fluid dynamics are resolved by transforming the problems to singularly perturbed ODEs and applying weighted one-sided difference approximations with solution-adaptive mesh refinement. In particular, well-resolved steady and moving shock solutions to Burgers' equation and to the equations of one-dimensional isentropic gas dynamics are obtained numerically. The method is further extended to problems in two space dimensions by using the method of dimensional splitting together with careful interpolation. In particular, in this extension the mesh refinement is only used to resolve the one-dimensional problems which are solved within the splitting algorithm. Numerical examples are presented in which two-dimensional oblique shocks are resolved.
\r\n", "doi": "10.7907/4DVY-AH34", "publication_date": "1982", "thesis_type": "phd", "thesis_year": "1982" }, { "id": "thesis:3618", "collection": "thesis", "collection_id": "3618", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-09182006-090057", "type": "thesis", "title": "I. Interactions of Fast and Slow Waves in Problems with Two Time Scales. II. A Numerical Experiment on the Structure of Two-Dimensional Turbulent Flow", "author": [ { "family_name": "Barker", "given_name": "John Wilson", "clpid": "Barker-John-Wilson" } ], "thesis_advisor": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" } ], "thesis_committee": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Keller", "given_name": "Herbert Bishop", "clpid": "Keller-H-B" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" }, { "family_name": "Saffman", "given_name": "Philip G.", "clpid": "Saffman-P-G" }, { "family_name": "Tadmor", "given_name": "E.", "clpid": "Tadmor-E" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "I. Interaction of Fast and Slow Waves in Problems with Two Time Scales
\r\n\r\nWe consider certain symmetric, hyperbolic systems of nonlinear first-order partial differential equations whose solutions vary on two time scales, a 'slow' scale t and a 'fast' scale t/\u03b5. The large (0(\u03b5-1)) part of the spatial operator is assumed to have constant coefficients, but a nonlinear term multiplying the time derivatives (a 'symmetriser') is allowed.
\r\n\r\nIn physical applications, it is often the case that the fast scale motion is of little interest, and it is desired to calculate only the slow scale motion accurately. It is known that solutions with arbitrarily small amounts of fast scale motion can be obtained by careful choice of the initial data, and that an error of amplitude 0(\u03b5p), where p = 2 for one space dimension or p = 3 for two or three space dimensions, in this choice is allowable, resulting in fast scale waves of amplitude 0(\u03b5p) in the solution.
\r\n\r\nWe investigate what happens when the initial data are not prepared correctly for the suppression of the fast scale motion, but contain errors of amplitude 0(\u03b5). We show that then the perturbation in the solution will also be of amplitude 0(\u03b5). Further, we show that if the large part of the spatial operator is nonsingular in the sense that the number of large eigenvalues of the symbol, P(i\u03c9), of the spatial operator is independent of \u03c9, then the error introduced in the slow scale motion will be of amplitude 0(\u03b52), even though fast scale waves of amplitude 0(\u03b5) will be present in the solution. If the symmetriser is a constant, this holds even if the spatial operator is singular, and further if an error 0(\u03b5\u03bc) is made in the initial conditions, for any \u00b5 > 0, the resulting error in the slow scale motion will be 0(\u03b52\u03bc).
\r\n\r\nOur proofs are based on energy estimates which show that spatial derivatives of the solutions are 0(1), even if time derivatives are not, and the development of the solutions in asymptotic expansions.
\r\n\r\nII. A Numerical Experiment on the Structure of Two-Dimensional Turbulent Flow
\r\n\r\nSome previous theories and numerical calculations pertaining to the problem of two-dimensional turburlence are reviewed, and a new numerical experiment is proposed. The purpose of the experiment is to test the hypothesis that narrow regions of concentrated vorticity are produced in two-dimensional flows by advection of vorticity towards dividing streamlines in regions where the local flow is convergent.
\r\n\r\nThe numerical method to be used is described in detail. It integrates the inviscid Euler equations using a Fourier (pseudo-spectral) method for the space derivatives, and a predictor-corrector method due to Hyman (1979) for time stepping. Dissipation is included, following Fornberg (1977), by a chopping of the amplitudes of the higher Fourier modes every few time-steps. This acts as a high-wavenumber energy sink, allowing very high Reynolds number flows to be simulated with relatively little computational effort.
", "doi": "10.7907/ynsy-nh46", "publication_date": "1982", "thesis_type": "phd", "thesis_year": "1982" }, { "id": "thesis:3385", "collection": "thesis", "collection_id": "3385", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-09082006-131345", "type": "thesis", "title": "I. Similarity Solutions of the Equations of Three Phase Flow through Porous Media. II. The Fingering Problem in a Hele-Shaw Cell", "author": [ { "family_name": "Romero", "given_name": "Louis Anthony", "clpid": "Romero-Louis-Anthony" } ], "thesis_advisor": [ { "family_name": "Saffman", "given_name": "Philip G.", "clpid": "Saffman-P-G" } ], "thesis_committee": [ { "family_name": "Saffman", "given_name": "Philip G.", "clpid": "Saffman-P-G" }, { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Whitham", "given_name": "Gerald Beresford", "clpid": "Whitham-G-B" }, { "family_name": "Keller", "given_name": "Herbert Bishop", "clpid": "Keller-H-B" }, { "family_name": "Leal", "given_name": "L. Gary", "clpid": "Leal-L-G" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "I
\r\n\r\nIn part I of this thesis similarity solutions to the equations of three phase flow through porous media are examined. The three phases are water, steam, and a noncondensing phase, most likely oil. The main purpose of analyzing such flows is to gain understanding of the steam flooding of oil fields.
\r\n\r\nProvided steam is being injected at a higher pressure than the initial field pressure, it is shown that there will always be at least two saturation shocks. As one increases the pressure of the injected steam several regimes are encountered; first the flow develops a region where all the steam is completely condensed, then the position of two of the shocks are interchanged, and finally one of the shocks grows weaker and is eventually replaced by an expansion fan.
\r\n\r\nIn sections 12 and 13 the stability of steadily moving condensation fronts in porous media is analyzed. For one special problem it is found that the sign of the jump in pressure gradient at the interface determines whether the interfaces are stable or unstable. This result is applied with some caution to the similarity solutions found in the earlier sections.
\r\n\r\nII
\r\n\r\nRecently McLean analyzed the shapes of fingers in a Hele-Shaw cell, including the effects of surface tension. His work resolved the question of the uniqueness of the shapes first brought up by Saffman and Taylor in their analysis that did not include surface tension. It is however felt that there are still unresolved problems.
\r\n\r\nIn determining the pressure jump across an interface there are two principal radii of curvature. McLean only took into account the effect of the larger of these, assuming that the other was constant along the outline of the finger. Unless the smaller radius is very nearly constant, it should in fact give a larger contribution to the jump in pressure. In this thesis the effect of this smaller radius of curvature is modelled by assuming that it is a function of the normal velocity of the mean two dimensional surface of the finger.
\r\n\r\nIt is found that if one only takes into account the smaller radius of curvature, the problem is not uniquely determined, as in the case with no surface tension at all. When both radii of curvature are taken into account, the effect of the smaller radius of curvature is to modify the finger shapes in a way that is qualitatively in agreement with experimental data. Also, McLean's results are checked by an independent numerical scheme, and the results are found to be in excellent agreement. Using both methods of solution a second solution branch other than that described by McLean was also found.
", "doi": "10.7907/MR6S-7C08", "publication_date": "1982", "thesis_type": "phd", "thesis_year": "1982" }, { "id": "thesis:1601", "collection": "thesis", "collection_id": "1601", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05042006-103859", "type": "thesis", "title": "The Accurate Numerical Solution of Highly Oscillatory Ordinary Differential Equations", "author": [ { "family_name": "Scheid", "given_name": "Robert Elmer, Jr", "clpid": "Scheid-Robert-Elmer" } ], "thesis_advisor": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" } ], "thesis_committee": [ { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Cohen", "given_name": "Donald S.", "clpid": "Cohen-D-S" }, { "family_name": "Keller", "given_name": "Herbert Bishop", "clpid": "Keller-H-B" }, { "family_name": "Tadmor", "given_name": "E.", "clpid": "Tadmor-E" }, { "family_name": "Caughey", "given_name": "Thomas Kirk", "clpid": "Caughey-T-K" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "We consider systems of ordinary differential equations with rapidly oscillating solutions. Conventional numerical methods require an excessively small time step (Δt = 0(εh), where h is the step size necessary for the resolution of a smooth function of t and 1/ε measures the size of the large eigenvalues of the system's Jacobian).
\r\n\r\nFor the linear problem with well-separated large eigenvalues we introduce smooth transformations which lead to the separation of the time scales and computation with a large time step (Δt = 0(h)). For more general problems, including systems with weak polynomial nonlinearities, we develop an asymptotic theory which leads to expansions whose terms are suitable for numerical approximation. Resonances can be detected and resolved often with a large time step (Δt = 0(h)). Passage through resonance in nonautonomous systems can be resolved by a moderate time step (Δt = 0(√εh)).
\r\n", "doi": "10.7907/4JVY-JB67", "publication_date": "1982", "thesis_type": "phd", "thesis_year": "1982" }, { "id": "thesis:3813", "collection": "thesis", "collection_id": "3813", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-09282006-094746", "type": "thesis", "title": "I. Propagating and Waiting Fronts in Nonlinear Diffusion. II. Sustained Reentry Roll Resonance", "author": [ { "family_name": "Kath", "given_name": "William Lawrence", "clpid": "Kath-William-Lawrence" } ], "thesis_advisor": [ { "family_name": "Cohen", "given_name": "Donald S.", "clpid": "Cohen-D-S" } ], "thesis_committee": [ { "family_name": "Cohen", "given_name": "Donald S.", "clpid": "Cohen-D-S" }, { "family_name": "Lagerstrom", "given_name": "Paco A.", "clpid": "Lagerstrom-P-A" }, { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Knowles", "given_name": "James K.", "clpid": "Knowles-J-K" }, { "family_name": "Saffman", "given_name": "Philip G.", "clpid": "Saffman-P-G" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Part I
\r\n\r\nWe examine a nonlinear diffusion equation that arises in the study of a number of physical problems, where the equation is nonlinear because the diffusion coefficient is proportional to a power of the concentration. Previous authors have proven using similarity solutions, that this dependence produces fronts (interfaces between regions of zero and nonzero concentration) which propagate with finite speed, as well as waiting-time behavior, where the fronts remain stationary for a finite amount of time before beginning to move. These similarity solutions provide limited information about the solution for general initial conditions, however.
\r\n\r\nTo alleviate this deficiency, we construct approximate solutions for the above nonlinear diffusion equation using singular perturbation theory. We do so by considering the equation in the limit of nearly linear diffusion, but the analysis reveals the basic qualitative behavior outside this limit as well.
\r\n\r\nThe basic behavior follows from the leading-order approximation of a transformed equation, and propagating and waiting fronts are due to the formation (in this approximation) of what we call corner shocks. This enables us to determine for which initial conditions waiting time behavior will occur.
\r\n\r\nThe transformed equation must be solved to first order to find the solution of the original equation to leading order, and when corner shocks occur at a point of nonzero concentration, this first order analysis shows that they become rounded (which we call a corner layer). When a corner shock occurs at a point of zero concentration, this rounding does not take place, and the corner shock remains sharp. This allows us to give a simple procedure for constructing approximate solutions of the nonlinear diffusion equation when corner shocks occur only at points of zero concentration.
\r\n\r\nPart II
\r\n\r\nWe study a model of reentry roll resonance, a situation encountered when an almost axially symmetric vehicle reenters the earth's atmosphere, using the method of near identity transformations. This method allows us to extend previous results for the case of sustained resonance, when roll buildup occurs.
\r\n\r\nIn particular, we give necessary conditions both for entrainmeat to sustained resonance and for sustained resonance to continue. These conditions imply that it is possible for sustained resonance to last for a finite time and then for unlocking of the resonance to occur. In addition, from the analysis we make a conjecture concerning sufficient conditions for sustained resonance.
", "doi": "10.7907/bpk3-rt60", "publication_date": "1981", "thesis_type": "phd", "thesis_year": "1981" }, { "id": "thesis:1609", "collection": "thesis", "collection_id": "1609", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05042006-134537", "type": "thesis", "title": "Mathematical Modeling of Photochemical Air Pollution", "author": [ { "family_name": "McRae", "given_name": "Gregory John", "clpid": "McRae-Gregory-John" } ], "thesis_advisor": [ { "family_name": "Seinfeld", "given_name": "John H.", "orcid": "0000-0003-1344-4068", "clpid": "Seinfeld-J-H" } ], "thesis_committee": [ { "family_name": "Brooks", "given_name": "Norman H.", "clpid": "Brooks-N-H" }, { "family_name": "Kreiss", "given_name": "Heinz-Otto", "clpid": "Kreiss-H-O" }, { "family_name": "Shair", "given_name": "Fredrick H.", "clpid": "Shair-F-H" }, { "family_name": "Holmes", "given_name": "John R.", "clpid": "Holmes-John-R" }, { "family_name": "Seinfeld", "given_name": "John H.", "orcid": "0000-0003-1344-4068", "clpid": "Seinfeld-J-H" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Air pollution is an environmental problem that is both pervasive and difficult to control. An important element of any rational control approach is a reliable means for evaluating the air quality impact of alternative abatement measures. This work presents such a capability, in the form of a mathematical description of the production and transport of photochemical oxidants within an urban airshed. The combined influences of advection, turbulent diffusion, chemical reaction, emissions and surface removal processes are all incorporated into a series of models that are based on the species continuity equations. A delineation of the essential assumptions underlying the formulation of a three-dimensional, a Lagrangian trajectory, a vertically integrated and single cell air quality model is presented. Since each model employs common components and input data the simpler forms can be used for rapid screening calculations and the more complex ones for detailed evaluations.
\r\n\r\nThe flow fields, needed for species transport, are constructed using inverse distance weighted polynomial interpolation techniques that map routine monitoring data onto a regular computational mesh. Variational analysis procedures are then employed to adjust the field so that mass is conserved. Initial concentration and mixing height distributions can be established with the same interpolation algorithms.
\r\n\r\nSubgrid scale turbulent transport is characterized by a gradient diffusion hypothesis. Similarity solutions are used to model the surface layer fluxes. Above this layer different treatments of turbulent diffusivity are required to account for variations in atmospheric stability. Convective velocity scaling is utilized to develop eddy diffusivities for unstable conditions. The predicted mixing times are in accord with results obtained during sulfur hexafluoride (SF6) tracer experiments. Conventional models are employed for neutral and stable conditions.
\r\n\r\nA new formulation for gaseous deposition fluxes is presented that provides a means for estimating removal rates as a function of atmospheric stability. The model satisfactorily reproduces measured deposition velocities for reactive materials. In addition it is shown how computational cell size influences the representation of surface removal.
\r\n\r\nChemical interactions between twenty nine chemical species are described by a 52 step kinetic mechanism. The atmospheric hydrocarbon chemistry is modeled by the reactions of six lumped classes: alkanes, ethylene, other olefins, aromatics, formaldehyde and other aldehydes; a grouping that enables representation of a wide range of smog chamber experiments and atmospheric conditions. Chemical lumping minimizes the number of species while maintaining a high degree of detail for the inorganic reactions. Variations in rate data, stoichiometric coefficients and initial conditions have been studied using the Fourier Amplitude Sensitivity Test.
\r\n\r\nThe wide variation in time scales, non-linearity of the chemistry and differences in transport processes complicates selection of numerical algorithms. Operator splitting techniques are used to decompose the governing equation into elemental steps of transport and chemistry. Each transport operator is further split into advective and diffusive components so that linear finite element and compact finite difference schemes can be applied to their best advantage. Because most of the computer time is consumed by the chemical kinetics those species that could be accurately described by pseudo-steady state approximations were identified reducing the number of species, described by differential equations, to 15.
\r\n\r\nWhile the mathematical formulation of the complete system contains no regional or area specific information, performance evaluation studies were carried out using data measured in the South Coast Air Basin of Southern California. Detailed emissions and meteorological information were assembled for the period 26-28 June 1974. A comparison between predictions and observed air quality, during multi-day periods, indicates that the model can satisfactorily describe urban scale atmospheric concentration dynamics.
", "doi": "10.7907/n8p7-f149", "publication_date": "1981", "thesis_type": "phd", "thesis_year": "1981" } ]