Dissertations / Theses: 'Euler Equation - Numerical Solution'

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Latypov, Azat. "Numerical solution of Euler equations on streamline-aligned meshes." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape11/PQDD_0010/NQ52411.pdf.

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Sykes, L. A. "On the numerical solution of compressible flows containing shock discontinuities." Thesis, University of Hertfordshire, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.234329.

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Ungun, Yigit. "Numerical Solution Of One Dimensional Detonation Tube With Reactive Euler Equations Using High Resolution Method." Master's thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614128/index.pdf.

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In this thesis, numerical simulation of one dimensional detonation tube problem is solved with finite rate chemistry. For the numerical simulation, Euler equations have been used. Since detonation tube phenomena occurs in high speed flows, viscosity eects and gravity forces are negligible. In this thesis, Godunov type methods have been studied and afterwards high resolution method is used for the numerical solution of the detonation tube problem. To solve the chemistry aspect of the problem ZND theory have been used. For the numerical solution, a FORTRAN code is written and the numerical solution of the problems compared with the exact ZND solutions.

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Onur, Omer. "Effect Of Jacobian Evaluation On Direct Solutions Of The Euler Equations." Master's thesis, METU, 2003. http://etd.lib.metu.edu.tr/upload/2/1098268/index.pdf.

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A direct method is developed for solving the 2-D planar/axisymmetric Euler equations. The Euler equations are discretized using a finite-volume method with upwind flux splitting schemes, and the resulting nonlinear system of equations are solved using Newton&
#8217
s Method. Both analytical and numerical methods are used for Jacobian calculations. Numerical method has the advantage of keeping the Jacobian consistent with the numerical flux vector without extremely complex or impractical analytical differentiations. However, numerical method may have accuracy problem and may need longer execution time. In order to improve the accuracy of numerical method detailed error analyses were performed. It was demonstrated that the finite-difference perturbation magnitude and computer precision are the most important parameters that affect the accuracy of numerical Jacobians. A relation was developed for optimum perturbation magnitude that can minimize the error in numerical Jacobians. Results show that very accurate numerical Jacobians can be calculated with optimum perturbation magnitude. The effects of the accuracy of numerical Jacobians on the convergence of flow solver are also investigated. In order to reduce the execution time for numerical Jacobian evaluation, flux vectors with perturbed flow variables are calculated for only related cells. A sparse matrix solver based on LU factorization is used for the solution, and to improve the Jacobian matrix solution some strategies are considered. Effects of different flux splitting methods, higher-order discretizations and several parameters on the performance of the solver are analyzed.

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Morrelll, J. M. "A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations." Thesis, University of Reading, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.440088.

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Hu, Guanghui. "Numerical simulations of the steady Euler equations on unstructured grids." HKBU Institutional Repository, 2009. http://repository.hkbu.edu.hk/etd_ra/1106.

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7

SHELLEY, MICHAEL JOHN. "THE APPLICATION OF BOUNDARY INTEGRAL TECHNIQUES TO MULTIPLY CONNECTED DOMAINS (VORTEX METHODS, EULER EQUATIONS, FLUID MECHANICS)." Diss., The University of Arizona, 1985. http://hdl.handle.net/10150/188100.

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Very accurate methods, based on boundary integral techniques, are developed for the study of multiple, interacting fluid interfaces in an Eulerian fluid. These methods are applied to the evolution of a thin, periodic layer of constant vorticity embedded in irrotational fluid. Numerical regularity experiments are conducted and suggest that the interfaces of the layer develop a curvature singularity in infinite time. This is to be contrasted with the more singular vorticity distribution of a vortex sheet developing such a singularity in a finite time.

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8

Bright, Theresa Ann. "New solutions to the euler equations using lie group analysis and high order numerical techniques." Diss., Georgia Institute of Technology, 1993. http://hdl.handle.net/1853/29990.

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9

Williams, Rhys L. "Exact, asymptotic and numerical solutions to certain steady, axisymmetric, ideal fluid flow problems in IRÂ³." Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299262.

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10

Paillere, Henri J. "Multidimensional upwind residual distribution schemes for the Euler and Navier-Stokes equations on unstructured grids." Doctoral thesis, Universite Libre de Bruxelles, 1995. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/212553.

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Une approche multidimensionelle pour la résolution numérique des équations d'Euler et de Navier-Stokes sur maillages non-structurés est proposée. Dans une première partie, un exposé complet des schémas de distribution, dits de "fluctuation-splitting" ,est décrit, comprenant une étude comparative des schémas décentrés, positifs et de 2ème ordre, pour résoudre l'équation de convection à coefficients constants, ainsi qu'une étude théorique et numérique de la précision des schémas sur maillages réguliers et distordus. L'extension à des lois de conservation non-linéaires est aussi abordée, et une attention particulière est portée au problème de la linéarisation conservative. Dans une deuxième partie, diverses discrétisations des termes visqueux pour l'équation de convection-diffusion sont développées, avec pour but de déterminer l'approche qui offre le meilleur compromis entre précision et coût. L'extension de la méthode aux systèmes des lois de conservation, et en particulier à celui des équations d'Euler de la dynamique des gaz, représente le noyau principal de la thèse, et est abordée dans la troisième partie. Contrairement aux schémas de distribution classiques, qui reposent sur une extension formelle du cas scalaire, l'approche développée ici repose sur une décomposition du résidu par élément en équations scalaires, modélisant le transport de variables caracteristiques. La difficulté vient du fait que les équations d'Euler instationnaires ne se diagonalisent pas, et admettent une infinité de solutions élémentaires (ondes simples) se propageant dans toutes les directions d'espace. En régime stationnaire, en revanche, les équations se diagonalisent complètement dans le cas des écoulements supersoniques, et partiellement dans le cas des écoulements subsoniques. Ainsi, les équations sous forme conservative peuvent être remplacées par un système équivalent comprenant deux équations totalement découplées, exprimant l'invariance de l'entropie et de l'enthalpie totale le long des lignes de courant, et deux autres équations, modélisant les effets purement acoustiques. En régime supersonique, celles-ci se découplent aussi, et expriment la convection le long des lignes de Mach d'invariants de Riemann généralisés. La discrétisation de ces équations par des schémas scalaires décentrés permet de simuler des écoulements continus et discontinus avec une grande précision et sans oscillations. Finalement, dans une dernière partie, l'extension aux équations de Navier-Stokes est abordée, et la discrétisation des termes visqueux par une approche éléments finis est proposée. Les résultats numériques confirment la précision et la robustesse de la méthode.

Doctorat en sciences appliquées
info:eu-repo/semantics/nonPublished

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Kurniawan, Budi. "Numerical solution of Prandtl's lifting-line equation /." Title page, contents and summary only, 1992. http://web4.library.adelaide.edu.au/theses/09SM/09smk78.pdf.

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Simmel, Martin. "Numerical solution of the stochastic collection equation." Universitätsbibliothek Leipzig, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-216496.

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The Linear Discrete Method (LDM; SIMMEL 2000; SIMMEL ET AL. 2000) is used to solve the Stochastic Collection Equation (SCE) numerically. Comparisons are made to the Method of Moments (MOM; TzIVION ET AL. 1999) which is suggested as a reference for numerical solutions of the SCE. Simulations for both methods are shown for the GoLOVIN kernel (for which an analytical solution is available) and the hydrodynamic kernel after LONG (1974) as it is used by TZIVION ET AL. (1999). Different bin resolutions are investigated and the simulation times are compared. In addition, LDM simulations using the hydrodynamic kernel after BÖHM (1992b) are presented. The results show that for the GoLOVIN kernel, LDM is slightly closer to the analytic solution than MOM. For the LONG kernel, the low resolution results of LDM and MOM are of similar quality compared to the reference solution. For the BÖHM kernel, only LDM simulations were carried out which show good correspondence between low and high resolution results
Die lineare diskrete Methode (LDM; SIMMEL 2000; SIMMEL ET AL. 2000) wird dazu benutzt, die Gleichung für stochastisches Einsammeln (stochastic collection equation, SCE) numerisch zu lösen. Dabei werden Vergleiche gezogen zur Methode der Momente (Method of Moments, MOM; TzIVION ET AL. 1999), die als Referenz für numerische Lösungen der SCE vorgeschlagen wurde. Simulationsrechnungen für beide Methoden werden für die Koaleszenzfunktion nach GoLOVIN (für die eine analytische Lösung existiert) und die hydrodynamische Koaleszenzfunktion nach LONG (1974) wie sie von TZIVION ET AL. (1999) verwendet wird, gezeigt. Verschiedene Klassenauflösungen werden untersucht und die Simulationszeiten verglichen. Zusätzlich werden LDM-Simulationen mit der hydrodynamischen Koaleszenzfunktion nach BÖHM (1992b) gezeigt. Die Ergebnisse für die Koaleszenzfunktion nach GoLOVIN zeigen, daß die LDM der analytischen Lösung etwas näher kommt als MOM. Für die Koaleszenzfunktion nach LONG sind die Ergebnisse von LDM und MOM mit niedriger Auflösung von ähnlicher Qualität verglichen mit der Referenzlösung. Für die Koaleszenzfunktion nach BÖHM wurden nur Simulationen mit der LDM durchgeführt, die eine gute Übereinstimmung der Ergebnisse mit niedriger und hoher Auflösung zeigen

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13

Bao, Xuezhong, and Xuezhong Bao. "Mesh design for numerical solution of Richards' Equation." Thesis, The University of Arizona, 1995. http://hdl.handle.net/10150/626921.

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Richards' equation is difficult to solve numerically because water flow in the unsaturated zone is complicated by the fact that the soil's permeability depends on its water saturation. The accuracy and computational efficiency of numerical solution are affected by the form of the governing equation, the estimation of the internodal hydraulic conductivity and the time-stepping scheme. In order to save computational time and improve efficiency of numerical techniques, two classical dimensionless numbers, Peclet and Courant numbers, and their combinations, Fourier and Advective Peclet numbers, are used as the criteria for estimating the spatial and temporal increments needed for the numerical solution of the linear Richards' equation (for example expo_nential hydraulic functions) and two new dimensi_onless numbers, Modified Peclet and Courant numbers for the non-linear Richards' equation (van Genuchten's hydraulic functions). In this way, we can get numerical solutions of Richards' equation not only with no oscillation and mass conservation, but also with adequate convergency speed and accuracy. These numbers are only related to the soil-hydraulic properties and grid size.

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14

Penzl, T. "Numerical solution of generalized Lyapunov equations." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800893.

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Two efficient methods for solving generalized Lyapunov equations and their implementations in FORTRAN 77 are presented. The first one is a generalization of the Bartels--Stewart method and the second is an extension of Hammarling's method to generalized Lyapunov equations. Our LAPACK based subroutines are implemented in a quite flexible way. They can handle the transposed equations and provide scaling to avoid overflow in the solution. Moreover, the Bartels--Stewart subroutine offers the optional estimation of the separation and the reciprocal condition number. A brief description of both algorithms is given. The performance of the software is demonstrated by numerical experiments.

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15

Voss, Alexander. "Exact Riemann solution for the Euler equations with nonconvex and nonsmooth equation of state." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=976791641.

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16

Nguyen, Vinh Q. "A Numerical Study of Burgers' Equation With Robin Boundary Conditions." Thesis, Virginia Tech, 2001. http://hdl.handle.net/10919/31285.

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This thesis examines the numerical solution to Burgers' equation on a finite spatial domain with various boundary conditions. We first conduct experiments to confirm the numerical solutions observed by other researchers for Neumann boundary conditions. Then we consider the case where the non-homogeneous Robin boundary conditions approach non-homogeneous Neumann conditions. Finally we numerically approximate the steady state solutions to Burgers' equation with both the homogeneous and non-homogeneous Robin boundary conditions.
Master of Science

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17

Knappett, Daniel. "Numerical solution of the stationary FPK equation using Shannon wavelets." Thesis, University of Nottingham, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.367109.

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18

Derakhshan, M. S. "Efficient algorithms for numerical solution of some Volterra type equations." Thesis, University of Manchester, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.377730.

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19

Meral, Gulnihal. "Numerical Solution Of Nonlinear Reaction-diffusion And Wave Equations." Phd thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/3/12610568/index.pdf.

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In this thesis, the two-dimensional initial and boundary value problems (IBVPs) and the one-dimensional Cauchy problems defined by the nonlinear reaction- diffusion and wave equations are numerically solved. The dual reciprocity boundary element method (DRBEM) is used to discretize the IBVPs defined by single and system of nonlinear reaction-diffusion equations and nonlinear wave equation, spatially. The advantage of DRBEM for the exterior regions is made use of for the latter problem. The differential quadrature method (DQM) is used for the spatial discretization of IBVPs and Cauchy problems defined by the nonlinear reaction-diffusion and wave equations. The DRBEM and DQM applications result in first and second order system of ordinary differential equations in time. These systems are solved with three different time integration methods, the finite difference method (FDM), the least squares method (LSM) and the finite element method (FEM) and comparisons among the methods are made. In the FDM a relaxation parameter is used to smooth the solution between the consecutive time levels. It is found that DRBEM+FEM procedure gives better accuracy for the IBVPs defined by nonlinear reaction-diffusion equation. The DRBEM+LSM procedure with exponential and rational radial basis functions is found suitable for exterior wave problem. The same result is also valid when DQM is used for space discretization instead of DRBEM for Cauchy and IBVPs defined by nonlinear reaction-diffusion and wave equations.

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20

Palitta, Davide. "Preconditioning strategies for the numerical solution of convection-diffusion partial differential equations." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2014. http://amslaurea.unibo.it/7464/.

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Il trattamento numerico dell'equazione di convezione-diffusione con le relative condizioni al bordo, comporta la risoluzione di sistemi lineari algebrici di grandi dimensioni in cui la matrice dei coefficienti è non simmetrica. Risolutori iterativi basati sul sottospazio di Krylov sono ampiamente utilizzati per questi sistemi lineari la cui risoluzione risulta particolarmente impegnativa nel caso di convezione dominante. In questa tesi vengono analizzate alcune strategie di precondizionamento, atte ad accelerare la convergenza di questi metodi iterativi. Vengono confrontati sperimentalmente precondizionatori molto noti come ILU e iterazioni di tipo inner-outer flessibile. Nel caso in cui i coefficienti del termine di convezione siano a variabili separabili, proponiamo una nuova strategia di precondizionamento basata sull'approssimazione, mediante equazione matriciale, dell'operatore differenziale di convezione-diffusione. L'azione di questo nuovo precondizionatore sfrutta in modo opportuno recenti risolutori efficienti per equazioni matriciali lineari. Vengono riportati numerosi esperimenti numerici per studiare la dipendenza della performance dei diversi risolutori dalla scelta del termine di convezione, e dai parametri di discretizzazione.

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Rajasingam, Prasanthan. "Numerical Solution of the coupled algebraic Riccati equations." OpenSIUC, 2013. https://opensiuc.lib.siu.edu/theses/1323.

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In this paper we develop new and improved results in the numerical solution of the coupled algebraic Riccati equations. First we provide improved matrix upper bounds on the positive semidefinite solution of the unified coupled algebraic Riccati equations. Our approach is largely inspired by recent results established by Liu and Zhang. Our main results tighten the estimates of the relevant dominant eigenvalues. Also by relaxing the key restriction our upper bound applies to a larger number of situations. We also present an iterative algorithm to refine the new upper bounds and the lower bounds and numerically compute the solutions of the unified coupled algebraic Riccati equations. This construction follows the approach of Gao, Xue and Sun but we use different bounds. This leads to different analysis on convergence. Besides, we provide new matrix upper bounds for the positive semidefinite solution of the continuous coupled algebraic Riccati equations. By using an alternative primary assumption we present a new upper bound. We follow the idea of Davies, Shi and Wiltshire for the non-coupled equation and extend their results to the coupled case. We also present an iterative algorithm to improve our upper bounds. Finally we improve the classical Newton's method by the line search technique to compute the solutions of the continuous coupled algebraic Riccati equations. The Newton's method for couple Riccati equations is attributed to Salama and Gourishanar, but we construct the algorithm in a different way using the Fr\'echet derivative and we include line search too. Our algorithm leads to a faster convergence compared with the classical scheme. Numerical evidence is also provided to illustrate the performance of our algorithm.

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Simmel, Martin. "Two numerical solutions for the stochastic collection equation." Universitätsbibliothek Leipzig, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-215378.

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Two different methods are used to solve the stochastic collection equation (SCE) numerically. They are called linear discrete method (LDM) and bin shift method (BSM), respectively. Conceptually, both of them are similar to the well-known discrete method (DM) of Kovetz and Olund. For LDM and BSM, their concept is extended to two prognostic moments. Therefore, the \"splitting factors\" (which are constant in time for DM) become time-dependent for LDM and BSM. Simulations are shown for the Golovin kernel (for which an analytical solution is available) and the hydrodynamic kernel after Hall. Different bin resolutions and time steps are investigated. As expected, the results become better with increasing bin resolution. LDM and BSM do not show the anomalous dispersion which is a weakness of DM
Es werden zwei verschiedene Methoden zur numerischen Lösung der \"Gleichung für stochastisches Einsammeln\" (stochastic collection equation, SCE) vorgestellt. Sie werden als Lineare Diskrete Methode (LDM) bzw. Bin Shift Methode (BSM) bezeichnet. Konzeptuell sind beide der bekannten Diskreten Methode (DM) von Kovetz und Olund ähnlich. Für LDM und BSM wird deren Konzept auf zwei prognostische Momente erweitert. Für LDM und BSM werden die\" Aufteil-Faktoren\" (die für DM zeitlich konstant sind) dadurch zeitabhängig. Es werden Simulationsrechnungen für die Koaleszenzfunktion nach Golovin (für die eine analytische Lösung existiert) und die hydrodynamische Koaleszenzfunktion nach Hall gezeigt. Verschiedene Klassenauflösungen und Zeitschritte werden untersucht. Wie erwartet werden die Ergebnisse mit zunehmender Auflösung besser. LDM und BSM zeigen nicht die anomale Dispersion, die eine Schwäche der DM ist

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Andallah, Laek Sazzad. "A hexagonal collision model for the numerical solution of the Boltzmann equation." [S.l. : s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=974861626.

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Sjöberg, Paul. "Numerical solution of the Fokker-Planck approximation of the chemical master equation /." Uppsala : Department of Information Technology, Uppsala University, 2005. http://www.it.uu.se/research/publications/lic/2005-010/.

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Sjöberg, Paul. "Numerical solution of the Fokker–Planck approximation of the chemical master equation." Licentiate thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-86354.

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The chemical master equation (CME) describes the probability for the discrete molecular copy numbers that define the state of a chemical system. Each molecular species in the chemical model adds a dimension to the state space. The CME is a difference-differential equation which can be solved numerically if the state space is truncated at an upper limit of the copy number in each dimension. The size of the truncated CME suffers from an exponential growth for an increasing number of chemical species. In this thesis the chemical master equation is approximated by a continuous Fokker-Planck equation (FPE) which makes it possible to use sparser computational grids than for CME. FPE on conservative form is used to compute steady state solutions by computation of an extremal eigenvalue and the corresponding eigenvector as well as time-dependent solutions by an implicit time-stepping scheme. The performance of the numerical solution is compared to a standard Monte Carlo algorithm. The computational work for a solutions with the same estimated error is compared for the two methods. Depending on the problem, FPE or the Monte Carlo algorithm will be more efficient. FPE is well suited for problems in low dimensions, especially if high accuracy is desirable.

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Sugimoto, Rie. "Special wave finite and infinite elements for the solution of the Helmholtz equation." Thesis, Durham University, 2003. http://etheses.dur.ac.uk/3142/.

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The theory and the formulation of the special wave finite elements are discussed, and the special integration schemes for the elements are developed. Then the special wave infinite elements, a new concept of the mapped wave infinite elements with multiple wave directions, are developed. Computational models using these elements coupled together are tested by the applications of wave problems. In the special wave finite elements, the potential at each node is expanded in a discrete series of approximating plane waves propagating in different directions. Because of this a single element can contain many wavelengths, unlike the standard finite elements. This is a great advantage in the reduction of the degree of freedom of the problem, however the computational cost of the numerical integration over an element becomes high due to the oscillatory shape functions. Therefore the special semi-analytical integration schemes for the special wave finite elements are developed. The schemes are independent of wavenumber and efficient for short waves problems. In many cases of wave problems, it is practical to consider the domain as being infinite. However the finite element method can not deal with infinite domains. Infinite elements are an extension of the concept of finite elements in which the element has an infinite extent in one or more directions to address this limitation. In the special wave infinite element developed in this study multiple waves propagating in different directions are considered, in contrast to conventional infinite elements in which only a single wave propagating in the radial direction is considered. The shape functions of the special wave infinite elements contain trigonometric functions to describe multiple waves, and the amplitude decay factor to satisfy the radiation condition. The special wave infinite elements become a straightforward extension to the special wave finite elements for wave problems in an unbounded domain.

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Chittabathini, Kumaraswamy. "Investigation of the inverse cascade process in wall-bounded logarithmic flow as a solution of the Euler equation." Thesis, University of Edinburgh, 2008. http://hdl.handle.net/1842/27787.

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Wall-bounded shear flows (WBSF) can be regarded as turbulent, organized coherent structures and occur in many different circumstances. The self-similarity of statistical characteristics of turbulence at different heights in the log layer of WBSF might reflect coherent structures which are also self-similar. McNaughton suggested that these coherent structures are in the form of 'Theodorsen ejection amplifier' (TEA) patterns. The TEA model of the structure of turbulence may be responsible for the formation of the three-dimensional inverse cascade process in log layers over smooth walls. The inverse cascade can serve as an efficient mechanism of energy transfer from small to large scales and enables us to understand the dynamics of large-scale coherent structures in the near-wall region. As far as the author is aware, no previous research has been conducted into the existence of a 3-D inverse cascade in WBSF. The objective of the thesis is to investigate numerically an upscale cascade process that has been hypothesized as a basic element of WBSF, and to examine an inverse cascade of this kind as a valid solution of the Euler equations. Initially, the numerical experiments were performed using the commercial Computational Fluid Dynamics (CFD) FLUENT 6.0 software, to reproduce the 'ejection amplifier' cycle (TEA structure) found by McNaughton and Bluendell (2002). In the numerical experiments, fluid was injected from the wall into the base of an ideal, ffictionless logarithmic flow while an equal volume of fluid was removed by suction along two flanking slots. This arrangement is known to create hairpin vortices in physical experiments. The FLUENT simulation results followed the subsequent formation of a hairpin eddy which induced a second, larger ejection from within its arc. The limited computing resources did not allow the FLUENT simulations to be followed far enough to examine possibility of any subsequent hairpins and ejections, so the feasibility of the TEA cascade was not firmly established. Research-oriented Large-Eddy Simulation (LES) code has been used to examine the inverse cascade process, and to overcome the computational limitations of the FLUENT solver. Several numerical experiments have been done using the LES code. The flow velocity data obtained from the simulations have been used to study the formation and growth of hairpin vortices and ejections, and their regeneration into 'ejection amplifier' structures. A comparison has been made between the CFD FLUENT predictions and initial LES run results so as to validate the LES solver. The results of the initial LES experiment reproduced the formation of the primary hairpin vortex (PHV), but did not reproduce the formation of primary a 'ejection amplifier' cycle. This is because the injection parameters and the spatial resolution were influencing primary hairpin development. The possibility of an upscale cascade of 'ejection amplifier' structure formation has been investigated by changing the injection/suction velocity, size and location in both low and high resolution domains. Fifteen LES simulation runs have been done, in which sets of variables and parameters have been systematically varied. The results obtained from all the LES runs showed that the injected disturbance developed into a primary hairpin vortex. When the slot was near the inflow boundary of the simulation domain, the low resolution runs did not indicate the formation of a primary 'ejection amplifier' cycle from the primary hairpin vortex development. These results suggest that the frequency of hairpin generation decreases with decreasing injection velocity. When the disturbance was injected at the center of the low resolution domain, development of the primary hairpin vortex was not affected by the inflow boundary. However, because of the large injection velocity and large slot the primary hairpin vortex also did not evolve into a primary 'ejection amplifier' cycle. The low resolution simulations done using a small slot with large injection velocity showed that the primary hairpin vortex developed into a primary 'ejection amplifier' cycle, but its development was discontinued because of the small injection. All the high resolution runs that were done using a large slot and a high injection velocity showed the formation of a primary 'ejection amplifier' cycle. The high resolution runs that were done using different injection periods also showed the formation of primary 'ejection amplifier' cycles. However, none of the simulations developed fully into an inverse cascade of ejection amplifier structures. In general, these results suggested that the TEA structure formation depends on the injection parameters (injection velocity, injection size, injection duration and injection location) and resolution. The injected disturbances are able to generate TEA structures, but have not been able to generate upscale cascades of TEA structures in log flow. This suggests that the present LES is not able to establish the 3-D inverse cascade process in wall-bounded log flow.

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Coleman, Keenan L. "On the numerical solution of the integral equation formulation for transient structural synthesis." Thesis, Monterey, California: Naval Postgraduate School, 2014. http://hdl.handle.net/10945/43891.

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Approved for public release; distribution is unlimited
Structural synthesis is the analysis of the dynamic response of a system when either subsystems are combined (substructure coupling) or modifications are made to substructures (structural modification). The integral equation formulation for structural synthesis is a method that requires only the baseline transient response, the baseline modal parameters, and the impedance of the structural modification. The integral formulation results in a Volterra integral equation of the second-kind. An adaptive time-marching scheme is utilized to solve the integral equation formulation for structural synthesis. When structural modifications of large magnitude are made, the solution to the integral equation can become unstable. To overcome this conditional stability, the derivative of the synthesis equation is examined and demonstrated to be stable regardless of the magnitude of the structural modification.

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Sabonis, Cynthia Anne. "Numerical Scheme for the Solution to Laplace's Equation using Local Conformal Mapping Techniques." Digital WPI, 2014. https://digitalcommons.wpi.edu/etd-theses/782.

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This paper introduces a method to determine the pressure in a fixed thickness, smooth, periodic domain; namely a lead-over-pleat cartridge filter. Finding the pressure within the domain requires the numerical solution of Laplace's equation, the first step of which is approximating, by interpolation, the curved portions of the filter to a circle in the xy plane.A conformal map is then applied to the filter, transforming the region into a rectangle in the uv plane. A finite difference method is introduced to numerically solve Laplace's equation in the rectangular domain. There are currently methods in existence to solve partial differential equations on non- regular domains. In a method employed by Monchmeyer and Muller, a scheme is used to transform from cartesian to spherical polar coordinates. Monchmeyer and Muller stress that for non-linear domains, extrapolation of existing cartesian difference schemes may produce incorrect solutions, and therefore, a volume centered discretization is used. A difference scheme is then derived that relies on mean values. This method has second order accuracy.(Rosenfeld,Moshe, Kwak, Dochan, 1989) The method introduced in this paper is based on a 7-point stencil which takes into account the unequal spacing of the points. From all neighboring pairs, a linear system of equations is constructed, which takes into account the periodic domain.This method is solved by standard iterative methods. The solution is then mapped back to the original domain, with second order accuracy. The method is then tested to obtain a solution to a domain which satisfies $y=sin(x)$ at the center, a shape similar to that of a lead-over-pleat cartridge filter. As a result, a model for the pressure distribution within the filter is obtained.

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Al-Harbi, Saleh M. "Implicit Runge-Kutta methods for the numerical solution of stiff ordinary differential equation." Thesis, University of Manchester, 1999. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.488322.

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The primary aim of this thesis is to calculate the numerical solution of a given stiff system of ordinary differential equations. We deal with the implementation of the implicit Runge-Kutta methods, in particular for Radau IIA order 5 which is now a competitive method for solving stiff initial value problems. New software based on Radau IIA, called IRKMR5 written in MATLAB has been developed for fixed order (order 5) with variable stepsizes, which is quite efficient when it is used to solve stiff problems. The code is organized in a modular form so that it facilitates both the understanding of it and its modification whenever needed. The new software is not only more functional than its Fortran 77 Radau IIA counterpart but also more robust and better documented. When implicit methods are used to solve nonlinear problems it is necessary to solve systems of nonlinear algebraic equations. New investigations for a modified Newton iteration are undertaken. This new strategy manages the iterative solutions of nonlinear equations in the ODEs solver. It also involves when to re-evaluate the Jacobian and the iteration matrix. The strategy also significantly reduces the number of function evaluations and linear solves. We subsequently consider the mathematical analysis of the nonlinear algebraic equations that arise from using s-stage fully implicit Runge-Kutta methods. Results for uniqueness of solutions and an error bound was established. The termination criterion in the iterative solution of the nonlinear equations is also studied as well as two types of termination criterion (displacement and the residual test). The residual test has been compared with the displacement test on some test examples and the results are tabulated.

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Mohamed, Nurul Akmal. "Numerical solution and spectrum of boundary-domain integral equations." Thesis, Brunel University, 2013. http://bura.brunel.ac.uk/handle/2438/7592.

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A numerical implementation of the direct Boundary-Domain Integral Equation (BDIE)/ Boundary-Domain Integro-Differential Equations (BDIDEs) and Localized Boundary-Domain Integral Equation (LBDIE)/Localized Boundary-Domain Integro-Differential Equations (LBDIDEs) related to the Neumann and Dirichlet boundary value problem for a scalar elliptic PDE with variable coefficient is discussed in this thesis. The BDIE and LBDIE related to Neumann problem are reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretisation of the BDIE/BDIDEs and LBDIE/LBDIDEs with quadrilateral domain elements leads to systems of linear algebraic equations (discretised BDIE/BDIDEs/LBDIE/BDIDEs). Then the systems obtained from BDIE/BDIDE (discretised BDIE/BDIDE) are solved by the LU decomposition method and Neumann iterations. Convergence of the iterative method is analyzed in relation with the eigen-values of the corresponding discrete BDIE/BDIDE operators obtained numerically. The systems obtained from LBDIE/LBDIDE (discretised LBDIE/LBDIDE) are solved by the LU decomposition method as the Neumann iteration method diverges.

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Simmel, Martin. "Two numerical solutions for the stochastic collection equation." Wissenschaftliche Mitteilungen des Leipziger Instituts für Meteorologie ; 17 = Meteorologische Arbeiten aus Leipzig ; 5 (2000), S. 61-73, 2000. https://ul.qucosa.de/id/qucosa%3A15149.

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Two different methods are used to solve the stochastic collection equation (SCE) numerically. They are called linear discrete method (LDM) and bin shift method (BSM), respectively. Conceptually, both of them are similar to the well-known discrete method (DM) of Kovetz and Olund. For LDM and BSM, their concept is extended to two prognostic moments. Therefore, the \"splitting factors\" (which are constant in time for DM) become time-dependent for LDM and BSM. Simulations are shown for the Golovin kernel (for which an analytical solution is available) and the hydrodynamic kernel after Hall. Different bin resolutions and time steps are investigated. As expected, the results become better with increasing bin resolution. LDM and BSM do not show the anomalous dispersion which is a weakness of DM.
Es werden zwei verschiedene Methoden zur numerischen Lösung der \"Gleichung für stochastisches Einsammeln\" (stochastic collection equation, SCE) vorgestellt. Sie werden als Lineare Diskrete Methode (LDM) bzw. Bin Shift Methode (BSM) bezeichnet. Konzeptuell sind beide der bekannten Diskreten Methode (DM) von Kovetz und Olund ähnlich. Für LDM und BSM wird deren Konzept auf zwei prognostische Momente erweitert. Für LDM und BSM werden die\" Aufteil-Faktoren\" (die für DM zeitlich konstant sind) dadurch zeitabhängig. Es werden Simulationsrechnungen für die Koaleszenzfunktion nach Golovin (für die eine analytische Lösung existiert) und die hydrodynamische Koaleszenzfunktion nach Hall gezeigt. Verschiedene Klassenauflösungen und Zeitschritte werden untersucht. Wie erwartet werden die Ergebnisse mit zunehmender Auflösung besser. LDM und BSM zeigen nicht die anomale Dispersion, die eine Schwäche der DM ist.

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Höök, Lars Josef. "Variance reduction methods for numerical solution of plasma kinetic diffusion." Licentiate thesis, KTH, Fusionsplasmafysik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-91332.

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Performing detailed simulations of plasma kinetic diffusion is a challenging task and currently requires the largest computational facilities in the world. The reason for this is that, the physics in a confined heated plasma occur on a broad range of temporal and spatial scales. It is therefore of interest to improve the computational algorithms together with the development of more powerful computational resources. Kinetic diffusion processes in plasmas are commonly simulated with the Monte Carlo method, where a discrete set of particles are sampled from a distribution function and advanced in a Lagrangian frame according to a set of stochastic differential equations. The Monte Carlo method introduces computational error in the form of statistical random noise produced by a finite number of particles (or markers) N and the error scales as αN−β where β = 1/2 for the standard Monte Carlo method. This requires a large number of simulated particles in order to obtain a sufficiently low numerical noise level. Therefore it is essential to use techniques that reduce the numerical noise. Such methods are commonly called variance reduction methods. In this thesis, we have developed new variance reduction methods with application to plasma kinetic diffusion. The methods are suitable for simulation of RF-heating and transport, but are not limited to these types of problems. We have derived a novel variance reduction method that minimizes the number of required particles from an optimization model. This implicitly reduces the variance when calculating the expected value of the distribution, since for a fixed error the optimization model ensures that a minimal number of particles are needed. Techniques that reduce the noise by improving the order of convergence, have also been considered. Two different methods have been tested on a neutral beam injection scenario. The methods are the scrambled Brownian bridge method and a method here called the sorting and mixing method of L´ecot and Khettabi[1999]. Both methods converge faster than the standard Monte Carlo method for modest number of time steps, but fail to converge correctly for large number of time steps, a range required for detailed plasma kinetic simulations. Different techniques are discussed that have the potential of improving the convergence to this range of time steps.
QC 20120314

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34

Engblom, Stefan. "Numerical Solution Methods in Stochastic Chemical Kinetics." Doctoral thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-9342.

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This study is concerned with the numerical solution of certain stochastic models of chemical reactions. Such descriptions have been shown to be useful tools when studying biochemical processes inside living cells where classical deterministic rate equations fail to reproduce actual behavior. The main contribution of this thesis lies in its theoretical and practical investigation of different methods for obtaining numerical solutions to such descriptions. In a preliminary study, a simple but often quite effective approach to the moment closure problem is examined. A more advanced program is then developed for obtaining a consistent representation of the high dimensional probability density of the solution. The proposed method gains efficiency by utilizing a rapidly converging representation of certain functions defined over the semi-infinite integer lattice. Another contribution of this study, where the focus instead is on the spatially distributed case, is a suggestion for how to obtain a consistent stochastic reaction-diffusion model over an unstructured grid. Here it is also shown how to efficiently collect samples from the resulting model by making use of a hybrid method. In a final study, a time-parallel stochastic simulation algorithm is suggested and analyzed. Efficiency is here achieved by moving parts of the solution phase into the deterministic regime given that a parallel architecture is available. Necessary background material is developed in three chapters in this summary. An introductory chapter on an accessible level motivates the purpose of considering stochastic models in applied physics. In a second chapter the actual stochastic models considered are developed in a multi-faceted way. Finally, the current state-of-the-art in numerical solution methods is summarized and commented upon.

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Oliveira, Anabela Pacheco de Pacheco de Oliveira Anabela De Oliveira Anabela Pacheco. "A comparison of Eulerian-Lagrangian methods for the solution of the transport equation /." Full text open access at:, 1994. http://content.ohsu.edu/u?/etd,208.

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Kobayashi, Kenta. "A numerical verification method for the global uniqueness of a positive solution for Nekrasov's equation." 京都大学 (Kyoto University), 2003. http://hdl.handle.net/2433/149038.

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37

Boyanova, Petia. "On Numerical Solution Methods for Block-Structured Discrete Systems." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-173530.

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The development, analysis, and implementation of efficient methods to solve algebraic systems of equations are main research directions in the field of numerical simulation and are the focus of this thesis. Due to their lesser demands for computer resources, iterative solution methods are the choice to make, when very large scale simulations have to be performed. To improve their efficiency, iterative methods are combined with proper techniques to accelerate convergence. A general technique to do this is to use a so-called preconditioner. Constructing and analysing various preconditioning methods has been an active field of research already for decades. Special attention is devoted to the class of the so-called optimal order preconditioners, that possess both optimal convergence rate and optimal computational complexity. The preconditioning techniques, proposed and studied in this thesis, utilise the block structure of the underlying matrices, and lead to methods that are of optimal order. In the first part of the thesis, we construct an Algebraic MultiLevel Iteration (AMLI) method for systems arising from discretizations of parabolic problems, using Crouzeix-Raviart finite elements. The developed AMLI method is based on an approximated block factorization of the original system matrix, where the partitioning is associated with a sequence of nested discretization meshes. In the second part of the thesis we develop solution methods for the numerical simulation of multiphase flow problems, modelled by the Cahn-Hilliard (C-H) equation. We consider the discrete C-H problem, obtained via finite element discretization in space and implicit schemes in time. We propose techniques to precondition the Jacobian of the discrete nonlinear system, based on its natural two-by-two block structure. The preconditioners are used in the framework of inexact Newton methods. We develop two nonlinear solution algorithms for the Cahn-Hilliard problem. Both lead to efficient optimal order methods. One of the main advantages of the proposed methods is that they are implemented using available software toolboxes for both sequential and distributed execution. The theoretical analysis of the solution methods presented in this thesis is combined with numerical studies that confirm their efficiency.

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Kurus, Gulay. "Solution Of Helmholtz Type Equations By Differential Quadarature Method." Master's thesis, METU, 2000. http://etd.lib.metu.edu.tr/upload/2/12605383/index.pdf.

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This thesis presents the Differential Quadrature Method (DQM) for solving Helmholtz, modified Helmholtz and Helmholtz eigenvalue-eigenvector equations. The equations are discretized by using Polynomial-based and Fourier-based differential quadrature technique wich use basically polynomial interpolation for the solution of differential equation.

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39

Kumar, Chaman. "Explicit numerical schemes of SDEs driven by Lévy noise with super-linear coeffcients and their application to delay equations." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/15946.

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We investigate an explicit tamed Euler scheme of stochastic differential equation with random coefficients driven by Lévy noise, which has super-linear drift coefficient. The strong convergence property of the tamed Euler scheme is proved when drift coefficient satisfies one-sided local Lipschitz condition whereas diffusion and jump coefficients satisfy local Lipschitz conditions. A rate of convergence for the tamed Euler scheme is recovered when local Lipschitz conditions are replaced by global Lipschitz conditions and drift satisfies polynomial Lipschitz condition. These findings are consistent with those of the classical Euler scheme. New methodologies are developed to overcome challenges arising due to the jumps and the randomness of the coefficients. Moreover, as an application of these findings, a tamed Euler scheme is proposed for the stochastic delay differential equation driven by Lévy noise with drift coefficient that grows super-linearly in both delay and non-delay variables. The strong convergence property of the tamed Euler scheme for such SDDE driven by Lévy noise is studied and rate of convergence is shown to be consistent with that of the classical Euler scheme. Finally, an explicit tamed Milstein scheme with rate of convergence arbitrarily close to one is developed to approximate the stochastic differential equation driven by Lévy noise (without random coefficients) that has super-linearly growing drift coefficient.

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Al-Jawary, Majeed Ahmed Weli. "The radial integration boundary integral and integro-differential equation methods for numerical solution of problems with variable coefficients." Thesis, Brunel University, 2012. http://bura.brunel.ac.uk/handle/2438/6449.

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The boundary element method (BEM) has become a powerful method for the numerical solution of boundary-value problems (BVPs), due to its ability (at least for problems with constant coefficients) of reducing a BVP for a linear partial differential equation (PDE) defined in a domain to an integral equation defined on the boundary, leading to a simplified discretisation process with boundary elements only. On the other hand, the coefficients in the mathematical model of a physical problem typically correspond to the material parameters of the problem. In many physical problems, the governing equation is likely to involve variable coefficients. The application of the BEM to these equations is hampered by the difficulty of finding a fundamental solution. The first part of this thesis will focus on the derivation of the boundary integral equation (BIE) for the Laplace equation, and numerical results are presented for some examples using constant elements. Then, the formulations of the boundary-domain integral or integro-differential equation (BDIE or BDIDE) for heat conduction problems with variable coefficients are presented using a parametrix (Levi function), which is usually available. The second part of this thesis deals with the extension of the BDIE and BDIDE formulations to the treatment of the two-dimensional Helmholtz equation with variable coefficients. Four possible cases are investigated, first of all when both material parameters and wave number are constant, in which case the zero-order Bessel function of the second kind is used as fundamental solution. Moreover, when the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or a BDIDE. Finally, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. In the third part, the radial integration method (RIM) is introduced and discussed in detail. Modifications are introduced to the RIM, particularly the fact that the radial integral is calculated by using a pure boundary-only integral which relaxes the “star-shaped” requirement of the RIM. Then, the RIM is used to convert the domain integrals appearing in both BDIE and BDIDE for heat conduction and Helmholtz equations to equivalent boundary integrals. For domain integrals consisting of known functions the transformation is straightforward, while for domain integrals that include unknown variables the transformation is accomplished with the use of augmented radial basis functions (RBFs). The most attractive feature of the method is that the transformations are very simple and have similar forms for both 2D and 3D problems. Finally, the application of the RIM is discussed for the diffusion equation, in which the parabolic PDE is initially reformulated as a BDIE or a BDIDE and the RIM is used to convert the resulting domain integrals to equivalent boundary integrals. Three cases have been investigated, for homogenous, non-homogeneous and variable coefficient diffusion problems.

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Levesley, Jeremy. "A study of Chebyshev weighted approximations to the solution of Symm's integral equation for numerical conformal mapping." Thesis, Coventry University, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.304879.

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Sundqvist, Per. "Numerical Computations with Fundamental Solutions." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-5757.

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43

Holman, Benjamin Robert. "Analytical Study and Numerical Solution of the Inverse Source Problem Arising in Thermoacoustic Tomography." Diss., The University of Arizona, 2016. http://hdl.handle.net/10150/612954.

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In recent years, revolutionary "hybrid" or "multi-physics" methods of medical imaging have emerged. By combining two or three different types of waves these methods overcome limitations of classical tomography techniques and deliver otherwise unavailable, potentially life-saving diagnostic information. Thermoacoustic (and photoacoustic) tomography is the most developed multi-physics imaging modality. Thermo- and photo-acoustic tomography require reconstructing initial acoustic pressure in a body from time series of pressure measured on a surface surrounding the body. For the classical case of free space wave propagation, various reconstruction techniques are well known. However, some novel measurement schemes place the object of interest between reflecting walls that form a de facto resonant cavity. In this case, known methods cannot be used. In chapter 2 we present a fast iterative reconstruction algorithm for measurements made at the walls of a rectangular reverberant cavity with a constant speed of sound. We prove the convergence of the iterations under a certain sufficient condition, and demonstrate the effectiveness and efficiency of the algorithm in numerical simulations. In chapter 3 we consider the more general problem of an arbitrarily shaped resonant cavity with a non constant speed of sound and present the gradual time reversal method for computing solutions to the inverse source problem. It consists in solving back in time on the interval [0, T] the initial/boundary value problem for the wave equation, with the Dirichlet boundary data multiplied by a smooth cutoff function. If T is sufficiently large one obtains a good approximation to the initial pressure; in the limit of large T such an approximation converges (under certain conditions) to the exact solution.

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44

Perella, Andrew James. "A class of Petrov-Galerkin finite element methods for the numerical solution of the stationary convection-diffusion equation." Thesis, Durham University, 1996. http://etheses.dur.ac.uk/5381/.

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A class of Petrov-Galerkin finite element methods is proposed for the numerical solution of the n dimensional stationary convection-diffusion equation. After an initial review of the literature we describe this class of methods and present both asymptotic and nonasymptotic error analyses. Links are made with the classical Galerkin finite element method and the cell vertex finite volume method. We then present numerical results obtained for a selection of these methods applied to some standard test problems. We also describe extensions of these methods which enable us to solve accurately for derivative values of the solution.

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45

Simmel, Martin. "Numerical solution of the stochastic collection equation: comparison of the linear discrete method and the method of moments." Wissenschaftliche Mitteilungen des Leipziger Instituts für Meteorologie ; 17 = Meteorologische Arbeiten aus Leipzig ; 5 (2000), S. 26-34, 2001. https://ul.qucosa.de/id/qucosa%3A15197.

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The Linear Discrete Method (LDM; SIMMEL 2000; SIMMEL ET AL. 2000) is used to solve the Stochastic Collection Equation (SCE) numerically. Comparisons are made to the Method of Moments (MOM; TzIVION ET AL. 1999) which is suggested as a reference for numerical solutions of the SCE. Simulations for both methods are shown for the GoLOVIN kernel (for which an analytical solution is available) and the hydrodynamic kernel after LONG (1974) as it is used by TZIVION ET AL. (1999). Different bin resolutions are investigated and the simulation times are compared. In addition, LDM simulations using the hydrodynamic kernel after BÖHM (1992b) are presented. The results show that for the GoLOVIN kernel, LDM is slightly closer to the analytic solution than MOM. For the LONG kernel, the low resolution results of LDM and MOM are of similar quality compared to the reference solution. For the BÖHM kernel, only LDM simulations were carried out which show good correspondence between low and high resolution results.
Die lineare diskrete Methode (LDM; SIMMEL 2000; SIMMEL ET AL. 2000) wird dazu benutzt, die Gleichung für stochastisches Einsammeln (stochastic collection equation, SCE) numerisch zu lösen. Dabei werden Vergleiche gezogen zur Methode der Momente (Method of Moments, MOM; TzIVION ET AL. 1999), die als Referenz für numerische Lösungen der SCE vorgeschlagen wurde. Simulationsrechnungen für beide Methoden werden für die Koaleszenzfunktion nach GoLOVIN (für die eine analytische Lösung existiert) und die hydrodynamische Koaleszenzfunktion nach LONG (1974) wie sie von TZIVION ET AL. (1999) verwendet wird, gezeigt. Verschiedene Klassenauflösungen werden untersucht und die Simulationszeiten verglichen. Zusätzlich werden LDM-Simulationen mit der hydrodynamischen Koaleszenzfunktion nach BÖHM (1992b) gezeigt. Die Ergebnisse für die Koaleszenzfunktion nach GoLOVIN zeigen, daß die LDM der analytischen Lösung etwas näher kommt als MOM. Für die Koaleszenzfunktion nach LONG sind die Ergebnisse von LDM und MOM mit niedriger Auflösung von ähnlicher Qualität verglichen mit der Referenzlösung. Für die Koaleszenzfunktion nach BÖHM wurden nur Simulationen mit der LDM durchgeführt, die eine gute Übereinstimmung der Ergebnisse mit niedriger und hoher Auflösung zeigen.

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46

Piserchia, Andrea. "New integrated numerical approaches to the Smoluchowski equation for the interpretation of molecular properties in solution phase chemistry." Doctoral thesis, Scuola Normale Superiore, 2018. http://hdl.handle.net/11384/85814.

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The necessity of a proper modelization of molecular diffusion in solution phase chemistry in order to rationalize properties and observables of the same give rise to the question \which tools from the theoretical chemistry field can be used?". The most commonly used approaches in order to answer this question relies on the usage of stochastic models. Then, depending on the particular conformational dynamics of the studied systems and the time scale on which molecular relaxation phenomena occurs, different models and tools can fit the desired level of description. Among all these approaches I focused on the Smoluchowski equation since its wide application in the theoretical chemistry community and since it well adapts to many diffusion problems in solution phase chemistry. Smoluchowski equation is a partial derivative equation that describes how the probability density of a set of coordinates evolves along time. Once the equation is solved, the probability profiles enclose informations upon specific properties and observables. My research activity in this field focused on new and more general numerical approaches to the solution of Smoluchowski equation with the application to specific case studies of chemical interest. Initially I have studied a class of methods apt to solve partial derivative differential equations. In particular, the chosen methodology is known as Discrete Variable Representation (DVR). Successively I have used this methodology in order to solve the one-dimensional Smoluchowski equation, in the stochastic processes framework applied to molecular systems. Comparing this method with preexisting ones I have validated this novel approach. This method has been implemented on a FORTRAN code with the aim of having an integrated approach for the solution of the one-dimensional Smoluchowski equation. Then I focused my research activity on the enhancement of the several physico- chemical ingredients that enter into the one-dimensional Smoluchowski equation. In par- ticular I extended an earlier diffusion tensor model including the dependence of the same on a generalized coordinate. Then, applying DVR theory to one-dimensional Smoluchowski equation I gave a more complete and general theoretical scenario of the same. In the framework of stochastic processes applied to molecular systems I have implemented and validated this novel approach. At last, for what concerns the Smoluchowski equation solved with DVR, I extended the same formalism to coupled one-dimensional Smoluchowski equations along the same generalized coordinate where there is the possibility of reactive exchanges of population and/or sinking terms between different coupled states. This is of great interest for example in the context of photoexcitations, where one has a population evolving through a ground and an excited state, along a specific coordinate, e.g. a twisting coordinate. From the temporal evolution of the probability density of the excited state one can retrieve lifetimes and/or compute the time resolved spectra at different times, etc. This last implementation merge the use of DVR basis with product approximation and diffusion tensor calculation along a generalized coordinate. The result is an integrated computational \black-box" tool in the framework of Gaussian software that give access to the generic user the possibility to study these specific systems of interest.

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Jawahar, P. "A High-Resolution Procedure For Euler And Navier-Stokes Computations On Unstructured Grids." Thesis, Indian Institute of Science, 2000. http://hdl.handle.net/2005/226.

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A finite-volume procedure, comprising a gradient-reconstruction technique and a multidimensional limiter, has been proposed for upwind algorithms on unstructured grids. The high-resolution strategy, with its inherent dependence on a wide computational stencil, does not suffer from a catastrophic loss of accuracy on a grid with poor connectivity as reported recently is the case with many unstructured-grid limiting procedures. The continuously-differentiable limiter is shown to be effective for strong discontinuities, even on a grid which is composed of highly-distorted triangles, without adversely affecting convergence to steady state. Numerical experiments involving transient computations of two-dimensional scalar convection to steady-state solutions of Euler and Navier-Stokes equations demonstrate the capabilities of the new procedure.

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Saleemi, Asima Parveen. "Finite Difference Methods for the Black-Scholes Equation." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-48660.

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Financial engineering problems are of great importance in the academic community and BlackScholes equation is a revolutionary concept in the modern financial theory. Financial instruments such as stocks and derivatives can be evaluated using this model. Option evaluation, is extremely important to trade in the stocks. The numerical solutions of the Black-Scholes equation are used to simulate these options. In this thesis, the explicit and the implicit Euler methods are used for the approximation of Black-scholes partial differential equation and a second order finite difference scheme is used for the spatial derivatives. These temporal and spatial discretizations are used to gain an insight about the stability properties of the explicit and the implicit methods in general. The numerical results show that the explicit methods have some constraints on the stability, whereas, the implicit Euler method is unconditionally stable. It is also demostrated that both the explicit and the implicit Euler methods are only first order convergent in time and this implies too small step-sizes to achieve a good accuracy.

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Yevik, Andrei. "Numerical approximations to the stationary solutions of stochastic differential equations." Thesis, Loughborough University, 2011. https://dspace.lboro.ac.uk/2134/7777.

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This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler's numerical method, applied to the stochastic differential equation, is proved to generate a discrete random dynamical system. The existence of stationary solution is proved again using global attractor approach. At last we prove that the approximate stationary point converges in mean-square sense to the exact one as the time step of the numerical scheme diminishes.

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Goh, K. H. M. "Numerical solution of quadratically non-linear boundary value problems using integral equation techniques : with applications to nozzle and wall flows /." Title page, contents and summary only, 1986. http://web4.library.adelaide.edu.au/theses/09PH/09phg614.pdf.

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Dissertations / Theses on the topic 'Euler Equation - Numerical Solution'

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