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Lui, Ho Man. "Runge-Kutta Discontinuous Galerkin method for the Boltzmann equation." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/39215.
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Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2006.Includes bibliographical references (p. 85-87).
In this thesis we investigate the ability of the Runge-Kutta Discontinuous Galerkin (RKDG) method to provide accurate and efficient solutions of the Boltzmann equation. Solutions of the Boltzmann equation are desirable in connection to small scale science and technology because when characteristic flow length scales become of the order of, or smaller than, the molecular mean free path, the Navier-Stokes description fails. The prevalent Boltzmann solution method is a stochastic particle simulation scheme known as Direct Simulation Monte Carlo (DSMC). Unfortunately, DSMC is not very effective in low speed flows (typical of small scale devices of interest) because of the high statistical uncertainty associated with the statistical sampling of macroscopic quantities employed by this method. This work complements the recent development of an efficient low noise method for calculating the collision integral of the Boltzmann equation, by providing a high-order discretization method for the advection operator balancing the collision integral in the Boltzmann equation. One of the most attractive features of the RKDG method is its ability to combine high-order accuracy, both in physical space and time, with the ability to capture discontinuous solutions.
(cont.) The validity of this claim is thoroughly investigated in this thesis. It is shown that, for a model collisionless Boltzmann equation, high-order accuracy can be achieved for continuous solutions; whereas for discontinuous solutions, the RKDG method, with or without the application of a slope limiter such as a viscosity limiter, displays high-order accuracy away from the vicinity of the discontinuity. Given these results, we developed a RKDG solution method for the Boltzmann equation by formulating the collision integral as a source term in the advection equation. Solutions of the Boltzmann equation, in the form of mean velocity and shear stress, are obtained for a number of characteristic flow length scales and compared to DSMC solutions. With a small number of elements and a low order of approximation in physical space, the RKDG method achieves similar results to the DSMC method. When the characteristic flow length scale is small compared to the mean free path (i.e. when the effect of collisions is small), oscillations are present in the mean velocity and shear stress profiles when a coarse velocity space discretization is used. With a finer velocity space discretization, the oscillations are reduced, but the method becomes approximately five times more computationally expensive.
(cont.) We show that these oscillations (due to the presence of propagating discontinuities in the distribution function) can be removed using a viscosity limiter at significantly smaller computational cost.
by Ho Man Lui.
S.M.
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Auffredic, Jérémy. "A second order Runge–Kutta method for the Gatheral model." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-49170.
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In this thesis, our research focus on a weak second order stochastic Runge–Kutta method applied to a system of stochastic differential equations known as the Gatheral Model. We approximate numerical solutions to this system and investigate the rate of convergence of our method. Both call and put options are priced using Monte-Carlo simulation to investigate the order of convergence. The numerical results show that our method is consistent with the theoretical order of convergence of the Monte-Carlo simulation. However, in terms of the Runge-Kutta method, we cannot accept the consistency of our method with the theoretical order of convergence without further research.
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Ramos, Manoel Wallace Alves. "Métodos de Euler e Runge-Kutta: uma análise utilizando o Geogebra." Universidade Federal da Paraíba, 2017. http://tede.biblioteca.ufpb.br:8080/handle/tede/9381.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
Is evident the importance of ordinary differential equations in modeling problems in several areas of science. Coupled with this, is increasing the use of numerical methods to solve such equations. Computers have become an extremely useful tool in the study of differential equations, since through them it is possible to execute algorithms that construct numerical approximations for solutions of these equati- ons. This work introduces the study of numerical methods for ordinary differential equations presenting the numerical Eulerºs method, improved Eulerºs method and the class of Runge-Kuttaºs methods. In addition, in order to collaborate with the teaching and learning of such methods, we propose and show the construction of an applet created from the use of Geogebm software tools. The applet provides approximate numerical solutions to an initial value problem, as well as displays the graphs of the solutions that are obtained from the numerical Eulerºs method, im- proved Eulerºs method, and fourth-order Runge-Kuttaºs method.
É evidente a importancia das equações diferenciais ordinarias na modelagem de problemas em diversas áreas da ciência, bem como o uso de métodos numéricos para resolver tais equações. Os computadores são uma ferramenta extremamente útil no estudo de equações diferenciais, uma vez que através deles é possível executar algo- ritmos que constroem aproximações numéricas para soluções destas equações. Este trabalho é uma introdução ao estudo de métodos numéricos para equações diferen- ciais ordinarias. Apresentamos os métodos numéricos de Euler, Euler melhorado e a classe de métodos de Runge-Kutta. Além disso, com o propósito de colaborar com o ensino e aprendizagem de tais métodos, propomos e mostramos a construção de um applet criado a partir do uso de ferramentas do software Geogebra. O applet fornece soluções numéricas aproximadas para um problema de valor inicial, bem como eXibe os graficos das soluções que são obtidas a partir dos métodos numéricos de Euler, Euler melhorado e Runge-Kutta de quarta ordem.
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Saleh, Ali, and Ahmad Al-Kadri. "Option pricing under Black-Scholes model using stochastic Runge-Kutta method." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-53783.
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The purpose of this paper is solving the European option pricing problem under the Black–Scholes model. Our approach is to use the so-called stochastic Runge–Kutta (SRK) numericalscheme to find the corresponding expectation of the functional to the stochastic differentialequation under the Black–Scholes model. Several numerical solutions were made to study howquickly the result converges to the theoretical value. Then, we study the order of convergenceof the SRK method with the help of MATLAB.
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Edgar, Christopher A. "An adaptive Runge-Kutta-Fehlberg method for time-dependent discrete ordinate transport." Diss., Georgia Institute of Technology, 2015. http://hdl.handle.net/1853/53935.
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This dissertation focuses on the development and implementation of a new method to solve the time-dependent form of the linear Boltzmann transport equation for reactor transients. This new method allows for a stable solution to the fully explicit form of the transport equation with delayed neutrons by employing an error-controlled, adaptive Runge-Kutta-Fehlberg (RKF) method to differentiate the time domain. Allowing for the time step size to vary adaptively and as needed to resolve the time-dependent behavior of the angular flux and neutron precursor concentrations. The RKF expansion of the time domain occurs at each point and is coupled with a Source Iteration to resolve the spatial behavior of the angular flux at the specified point in time. The decoupling of the space and time domains requires the application of a quasi-static iteration between solving the time domain using adaptive RKF with error control and resolving the space domain with a Source Iteration sweep. The research culminated with the development of the 1-D Adaptive Runge-Kutta Time-Dependent Transport code (ARKTRAN-TD), which successfully implemented the new method and applied it to a suite of reactor transient benchmarks.
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Booth, Andrew S. "Collocation methods for a class of second order initial value problems with oscillatory solutions." Thesis, Durham University, 1993. http://etheses.dur.ac.uk/5664/.
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We derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent results and proposed numerical methods is given in chapter 2. Chapter 3 is devoted to the analysis of a family of implicit Chebyshev methods proposed by Panovsky k Richardson. We show that for each non-negative integer r, there are two methods of order 2r from this family which possess non-vanishing intervals of periodicity. The equivalence of these methods with one-step collocation methods is also established, and these methods are shown to be neither P-stable nor symplectic. In chapters 4 and 5, two families of multistep collocation methods are derived, and their order and stability properties are investigated. A detailed analysis of the two-step symmetric methods from each class is also given. The multistep Runge-Kutta-Nystrom methods of chapter 4 are found to be difficult to analyse, and the specific examples considered are found to perform poorly in the areas of both accuracy and stability. By contrast, the two-step symmetric hybrid methods of chapter 5 are shown to have excellent stability properties, in particular we show that all two-step 27V-point methods of this type possess non-vanishing intervals of periodicity, and we give conditions under which these methods are almost P-stable. P-stable and efficient methods from this family are obtained and demonstrated in numerical experiments. A simple, cheap and effective error estimator for these methods is also given.
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Zamri, Mohd Y. "An improved treatment of two-dimensional two-phase flows of steam by a Runge-Kutta method." Thesis, University of Birmingham, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.251270.
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Mayo, Colin F. "Implementation of the Runge-Kutta-Fehlberg method for solution of ordinary differential equations on a parallel processor." Thesis, Monterey, California. Naval Postgraduate School, 1987. http://hdl.handle.net/10945/22285.
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Malroy, Eric Thomas. "Solution of the ideal adiabatic stirling model with coupled first order differential equations by the Pasic method." Ohio : Ohio University, 1998. http://www.ohiolink.edu/etd/view.cgi?ohiou1176410606.
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Boat, Matthew. "The time-domain numerical solution of Maxwell's electromagnetic equations, via the fourth order Runge-Kutta discontinuous Galerkin method." Thesis, Swansea University, 2008. https://cronfa.swan.ac.uk/Record/cronfa42532.
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This thesis presents a high-order numerical method for the Time-Domain solution of Maxwell's Electromagnetic equations in both one- and two-dimensional space. The thesis discuses the validity of high-order representation and improved boundary representation. The majority of the theory is concerned with the formulation of a high-order scheme which is capable of providing a numerical solution for specific two-dimensional scattering problems. Specifics of the theory involve the selection of a suitable numerical flux, the choice of appropriate boundary conditions, mapping between coordinate systems and basis functions. The effectiveness of the method is then demonstrated through a series of examples.
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Jewell, Jeffrey Steven. "Higher-order Runge--Kutta type schemes based on the Method of Characteristics for hyperbolic equations with crossing characteristics." ScholarWorks @ UVM, 2019. https://scholarworks.uvm.edu/graddis/1028.
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The Method of Characteristics (MoC) is a well-known procedure used to find the numerical solution of systems of hyperbolic partial differential equations (PDEs). The main idea of the MoC is to integrate a system of ordinary differential equations (ODEs) along the characteristic curves admitted by the PDEs. In principle, this can be done by any appropriate numerical method for ODEs. In this thesis, we will examine the MoC applied to systems of hyperbolic PDEs with straight-line and crossing characteristics. So far, only first- and second-order accurate explicit MoC schemes for these types of systems have been reported. As such, the purpose of this thesis is to develop MoC schemes which are of an order greater than two.
The order of the global truncation error of an MoC scheme goes hand-in-hand with the order of the ODE solver used. The MoC schemes which have already been developed use the first-order Simple Euler (SE) and second-order Modified Euler (ME) methods as the ODE solvers. The SE and ME methods belong to a larger family of numerical methods for ODEs known as the Runge--Kutta (RK) methods. First, we will attempt to develop third- and fourth-order MoC schemes by using the classical third- and fourth-order RK methods as the ODE solver. We will show that the resulting MoC schemes can be strongly unstable, meaning that the error in the numerical solution becomes unbounded rather quickly.
We then turn our attention to the so-called pseudo-RK (pRK) methods for ODEs. The pRK methods are at the intersection of RK and multistep methods, and a variety of third- and fourth-order schemes can be constructed. We show that when certain pRK schemes are used in the MoC, at most a weak instability, or no instability at all, is present, and thus the resulting methods are suitable for long-time computations. Finally, we present some numerical results confirming that the MoC using third- and fourth-order pRK schemes have the desired accuracy.
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Voonna, Kiran. "Development of discontinuous galerkin method for 1-D inviscid burgers equation." ScholarWorks@UNO, 2003. http://louisdl.louislibraries.org/u?/NOD,75.
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Thesis (M.S.)--University of New Orleans, 2003.Title from electronic submission form. "A thesis ... in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mechanical Engineering"--Thesis t.p. Vita. Includes bibliographical references.
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Roberts, Steven Byram. "Multimethods for the Efficient Solution of Multiscale Differential Equations." Diss., Virginia Tech, 2021. http://hdl.handle.net/10919/104872.
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Mathematical models involving ordinary differential equations (ODEs) play a critical role in scientific and engineering applications. Advances in computing hardware and numerical methods have allowed these models to become larger and more sophisticated. Increasingly, problems can be described as multiphysics and multiscale as they combine several different physical processes with different characteristics. If just one part of an ODE is stiff, nonlinear, chaotic, or rapidly-evolving, this can force an expensive method or a small timestep to be used. A method which applies a discretization and timestep uniformly across a multiphysics problem poorly utilizes computational resources and can be prohibitively expensive.
The focus of this dissertation is on "multimethods" which apply different methods to different partitions of an ODE. Well-designed multimethods can drastically reduce the computation costs by matching methods to the individual characteristics of each partition while making minimal concessions to stability and accuracy. However, they are not without their limitations. High order methods are difficult to derive and may suffer from order reduction. Also, the stability of multimethods is difficult to characterize and analyze.
The goals of this work are to develop new, practical multimethods and to address these issues. First, new implicit multirate Runge–Kutta methods are analyzed with a special focus on stability. This is extended into implicit multirate infinitesimal methods. We introduce approaches for constructing implicit-explicit methods based on Runge–Kutta and general linear methods. Finally, some unique applications of multimethods are considered including using surrogate models to accelerate Runge–Kutta methods and eliminating order reduction on linear ODEs with time-dependent forcing.Doctor of Philosophy
Almost all time-dependent physical phenomena can be effectively described via ordinary differential equations. This includes chemical reactions, the motion of a pendulum, the propagation of an electric signal through a circuit, and fluid dynamics. In general, it is not possible to find closed-form solutions to differential equations. Instead, time integration methods can be employed to numerically approximate the solution through an iterative procedure. Time integration methods are of great practical interest to scientific and engineering applications because computational modeling is often much cheaper and more flexible than constructing physical models for testing. Large-scale, complex systems frequently combine several coupled processes with vastly different characteristics. Consider a car where the tires spin at several hundred revolutions per minute, while the suspension has oscillatory dynamics that is orders of magnitude slower. The brake pads undergo periods of slow cooling, then sudden, rapid heating. When using a time integration scheme for such a simulation, the fastest dynamics require an expensive and small timestep that is applied globally across all aspects of the simulation. In turn, an unnecessarily large amount of work is done to resolve the slow dynamics. The goal of this dissertation is to explore new "multimethods" for solving differential equations where a single time integration method using a single, global timestep is inadequate. Multimethods combine together existing time integration schemes in a way that is better tailored to the properties of the problem while maintaining desirable accuracy and stability properties. This work seeks to overcome limitations on current multimethods, further the understanding of their stability, present new applications, and most importantly, develop methods with improved efficiency.
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Park, Jinwon. "A Runge Kutta Discontinuous Galerkin-Direct Ghost Fluid (RKDG-DGF) Method to Near-field Early-time Underwater Explosion (UNDEX) Simulations." Diss., Virginia Tech, 2008. http://hdl.handle.net/10919/28905.
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A coupled solution approach is presented for numerically simulating a near-field underwater explosion (UNDEX). An UNDEX consists of a complicated sequence of events over a wide range of time scales. Due to the complex physics, separate simulations for near/far-field and early/late-time are common in practice. This work focuses on near-field early-time UNDEX simulations. Using the assumption of compressible, inviscid and adiabatic flow, the fluid flow is governed by a set of Euler fluid equations. In practical simulations, we often encounter computational difficulties that include large displacements, shocks, multi-fluid flows with cavitation, spurious waves reflecting from boundaries and fluid-structure coupling. Existing methods and codes are not able to simultaneously consider all of these characteristics.
A robust numerical method that is capable of treating large displacements, capturing shocks, handling two-fluid flows with cavitation, imposing non-reflecting boundary conditions (NRBC) and allowing the movement of fluid grids is required. This method is developed by combining numerical techniques that include a high-order accurate numerical method with a shock capturing scheme, a multi-fluid method to handle explosive gas-water flows and cavitating flows, and an Arbitrary Lagrangian Eulerian (ALE) deformable fluid mesh. These combined approaches are unique for numerically simulating various near-field UNDEX phenomena within a robust single framework. A review of the literature indicates that a fully coupled methodology with all of these characteristics for near-field UNDEX phenomena has not yet been developed.
A set of governing equations in the ALE description is discretized by a Runge Kutta Discontinuous Galerkin (RKDG) method. For multi-fluid flows, a Direct Ghost Fluid (DGF) Method coupled with the Level Set (LS) interface method is incorporated in the RKDG framework. The combination of RKDG and DGF methods (RKDG-DGF) is the main contribution of this work which improves the quality and stability of near-field UNDEX flow simulations. Unlike other methods, this method is simpler to apply for various UNDEX applications and easier to extend to multi-dimensions.Ph. D.
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Alhojilan, Yazid Yousef M. "Higher-order numerical scheme for solving stochastic differential equations." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/15973.
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We present a new pathwise approximation method for stochastic differential equations driven by Brownian motion which does not require simulation of the stochastic integrals. The method is developed to give Wasserstein bounds O(h3/2) and O(h2) which are better than the Euler and Milstein strong error rates O(√h) and O(h) respectively, where h is the step-size. It assumes nondegeneracy of the diffusion matrix. We have used the Taylor expansion but generate an approximation to the expansion as a whole rather than generating individual terms. We replace the iterated stochastic integrals in the method by random variables with the same moments conditional on the linear term. We use a version of perturbation method and a technique from optimal transport theory to find a coupling which gives a good approximation in Lp sense. This new method is a Runge-Kutta method or so-called derivative-free method. We have implemented this new method in MATLAB. The performance of the method has been studied for degenerate matrices. We have given the details of proof for order h3/2 and the outline of the proof for order h2.
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Akman, Makbule. "Differential Quadrature Method For Time-dependent Diffusion Equation." Master's thesis, METU, 2003. http://etd.lib.metu.edu.tr/upload/1224559/index.pdf.
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This thesis presents the Differential Quadrature Method (DQM) for solving time-dependent or heat conduction problem. DQM discretizes the space derivatives giving a system of ordinary differential equations with respect to time and the fourth order Runge Kutta Method (RKM) is employed for solving this system. Stabilities of the ordinary differential equations system and RKM are considered and step sizes are arranged accordingly.
The procedure is applied to several time dependent diffusion problems and the solutions are presented in terms of graphics comparing with the exact solutions. This method exhibits high accuracy and efficiency comparing to the other numerical methods.
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Doricio, José Laércio. "Estudo da aplicabilidade do método de fronteira imersa no cálculo de derivadas de Flutter com as equações de Euler para fluxo compressível." Universidade de São Paulo, 2009. http://www.teses.usp.br/teses/disponiveis/18/18148/tde-19012011-105736/.
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Neste trabalho, desenvolve-se um método de fronteira imersa para o estudo de escoamento compressível modelado pelas equações de Euler bidimensionais. O método de discretização de diferenças finitas é empregado, usando o método de Steger-Warming de ordem dois para discretizar as variáveis espaciais e o esquema de Runge-Kutta de ordem quatro para discretizar as variáveis temporais. O método da fronteira imersa foi empregado para o estudo de aeroelasticidade computacional em uma seção típica de aerofólio bidimensional com dois movimentos prescritos: torsional e vertical, com o objetivo de se verifcar a eficiência do método e sua aplicabilidade para problemas em aeroelasticidade computacional. Neste estudo desenvolveu-se também um programa de computador para simular escoamentos compressíveis de fluido invíscido utilizando a metodologia proposta. A verificação do código gerado foi feita utilizando o método das soluções manufaturadas e o problema de reflexão de choque oblíquo. A validação foi realizada comparando-se os resultados obtidos para o escoamento ao redor de uma seção circular e de uma seção de aerofólio NACA 0012 com os resultados experimentais, para cada caso.In this work, an immersed boundary method is developed to study compressible flow modeled by the two-dimensional Euler equations. The finite difference method is employed, using the second order Steger-Warming method to discretizate the space variables and the fourth order Runge-Kutta method to discretizate the time variables. The immersed boundary method was employed to study computational aeroelasticity on a typical two-dimensional airfoil section with two prescribed motion: pitching and plunging, in order to verify the efficiency of the numerical method and its applicability in computational aeroelasticity problems. In this work, a computer program was developed to simulate compressible flows for inviscid fluids using the methodology proposed. The verification of the computational code was performed using the method of manufactured solutions and the oblique shock wave reflection problem. The validation was performed comparing the obtained results for flows around a circular section and a NACA 0012 airfoil section with the experimental results, for each case.
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Stumpf, Felipe Tempel. "Implementação numérica de problemas de viscoelasticidade finita utilizando métodos de Runge-Kutta de altas ordens e interpolação consistente entre as discretizações temporal e espacial." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2013. http://hdl.handle.net/10183/75757.
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Em problemas de viscoelasticidade computacional, a discretização espacial para a solução global das equações de equilíbrio é acoplada à discretização temporal para a solução de um problema de valor inicial local do fluxo viscoelástico. É demonstrado que este acoplamento espacial-temporal (ou global-local) éconsistente se o tensor de deformação total, agindo como elemento acoplador, tem uma aproximação de ordem p ao longo do tempo igual à ordem de convergência do método de integração de Runge-Kutta (RK). Para a interpolação da deformação foram utilizados polinômios baseados em soluções obtidas nos tempos tn+1, tn, . . ., tn+2−p, p ≥ 2, fornecendo dados consistentes de deformação nos estágios do RK. Em uma situação onde tal regra para a interpolação da deformação não é satisfeita, a integração no tempo apresentará, consequentemente, redução de ordem, baixa precisão e, por conseguinte, eficiência inferior. Em termos gerais, o propósito é generalizar esta condição de consistência proposta pela literatura, formalizando-a matematicamente e o demonstrando através da utilização de métodos de Runge-Kutta diagonalmente implícitos (DIRK) até ordem p = 4, aplicados a modelos viscoelásticos não-lineares sujeitos a deformações finitas. Através de exemplos numéricos, os algoritmos de integração temporal adaptados apresentaram ordem de convergência nominal e, portanto, comprovam a validade da formalização do conceito de interpolação consistente da deformação. Comparado com o método de integração de Euler implícito, é demonstrado que os métodos DIRK aqui aplicados apresentam um ganho considerável em eficiência, comprovado através dos fatores de aceleração atingidos.In computational viscoelasticity, spatial discretization for the solution of the weak form of the balance of linear momentum is coupled to the temporal discretization for solving a local initial value problem (IVP) of the viscoelastic flow. It is shown that this spatial- temporal (or global-local) coupling is consistent if the total strain tensor, acting as the coupling agent, exhibits the same approximation of order p in time as the convergence order of the Runge-Kutta (RK) integration algorithm. To this end we construct interpolation polynomials based on data at tn+1, tn, . . ., tn+2−p, p ≥ 2, which provide consistent strain data at the RK stages. If this novel rule for strain interpolation is not satisfied, time integration shows order reduction, poor accuracy and therefore less efficiency. Generally, the objective is to propose a generalization of this consistency idea proposed in the literature, formalizing it mathematically and testing it using diagonally implicit Runge-Kutta methods (DIRK) up to order p = 4 applied to a nonlinear viscoelasticity model subjected to finite strain. In a set of numerical examples, the adapted time integrators obtain full convergence order and thus approve the novel concept of consistency. Substantially high speed-up factors confirm the improvement in the efficiency compared with Backward Euler algorithm.
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Nguyen, Hoan Kim Huynh. "Volterra Systems with Realizable Kernels." Diss., Virginia Tech, 2004. http://hdl.handle.net/10919/11153.
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We compare an internal state method and a direct Runge-Kutta method for solving Volterra integro-differential equations and Volterra delay differential equations. The internal state method requires the kernel of the Volterra integral to be realizable as an impulse response function. We discover that when applicable, the internal state method is orders of magnitude more efficient than the direct numerical method. However, constructing state representation for realizable kernels can be challenging at times; therefore, we propose a rational approximation approach to avoid the problem. That is, we approximate the transfer function by a rational function, construct the corresponding linear system, and then approximate the Volterra integro-differential equation. We show that our method is convergent for the case where the kernel is nuclear. We focus our attention on time-invariant realizations but the case where the state representation of the kernel is a time-variant linear system is briefly discussed.Ph. D.
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Palmerini, Claudia. "On the smoothed finite element method in dynamics: the role of critical time step for linear triangular elements." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017.
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Il metodo agli elementi finiti (FEM) è molto utilizzato per risolvere problemi strutturali in diversi ambiti dell’ingegneria. Negli anni, è stata sviluppata una famiglia di nuovi metodi ottenuta combinando il FEM standard con la tecnica “strain smoothing”, giungendo ai cosiddetti “smoothed finite element method” (SFEM). In questa tesi, l’attenzione è stata concentrata sul node-based SFEM (NS-FEM) e sull'edge-based SFEM (ES-FEM), che appartengono a questa nuova famiglia di metodi. Dopo una literature review, volta a metterne in luce le proprietà e gli aspetti fondamenti, i due metodi sono stati confrontati con il FEM standard. L'implementazione dei due metodi è stata eseguita con il software MATLAB. Lo studio è stato fatto in ambito dinamico, utilizzando due metodi di integrazione numerica nel tempo: il metodo delle differenze centrali e il metodo di Runge-Kutta. Come problema test è stato studiato il problema delle vibrazioni libere di un elemento strutturale in stato piano di tensione. Il confronto è stato portato avanti su due fronti: il costo computazionale dei metodi ed il calcolo del “critical time step”. I risultati hanno mostrato che il NS-FEM e l'ES-FEM hanno un costo maggiore rispetto al FEM standard, mentre, lato critical time step, sono paragonabil al FEM standard.
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Moretto, Irene. "Aspetti numerici nell'applicazione del modello SIR." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/22165/.
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Questo lavoro si concentra sull'applicazione del classico modello a compartimenti di Kermack–McKendrick (SIR), per controllare l'andamento dell'epidemia COVID-19 in Italia, utilizzando i dati messi a disposizione dal Dipartimento della Protezione Civile italiano, relativi alla regione Veneto. L'implementazione di tale modello pone interessanti problemi legati sia alla soluzione numerica del problema inverso, attraverso cui avviene la calibrazione dei parametri del modello, che alla soluzione numerica dei problemi differenziali richiesti all'interno della calibrazione e successivamente in fase di previsione.
Un primo obiettivo di questa tesi è quello di fare un'analisi dei metodi numerici per risolvere il sistema di equazioni differenziali che descrive il modello SIR e determinare, attraverso opportuni confronti, l'algoritmo più adatto a risolvere il problema di calibrazione. Inoltre, si è studiata una modifica del SIR che prevede l'utilizzo di parametri come funzioni dipendenti dal tempo, nota come modello forzato, con l'obiettivo di ottenere un migliore adattamento della curva epidemiologica in base alle misure di contenimento introdotte.
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Kachanovska, Maryna. "Fast, Parallel Techniques for Time-Domain Boundary Integral Equations." Doctoral thesis, Universitätsbibliothek Leipzig, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-132183.
Full textAbstract:
This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion.
As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems:
the need to construct many dense matrices and to evaluate many singular and near-singular integrals.
The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested.
The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal.
In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.
23
Kachanovska, Maryna. "Fast, Parallel Techniques for Time-Domain Boundary Integral Equations." Doctoral thesis, Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2013. https://ul.qucosa.de/id/qucosa%3A12278.
Full textAbstract:
This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion.
As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems:
the need to construct many dense matrices and to evaluate many singular and near-singular integrals.
The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested.
The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal.
In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.
24
Elmikkawy, M. E. A. "Embedded Runge-Kutta-Nystrom methods." Thesis, Teesside University, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.371400.
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Alahmadi, Eyman Salem M. "1D model for flow in the pulmonary airway system." Thesis, University of Manchester, 2012. https://www.research.manchester.ac.uk/portal/en/theses/1d-model-for-flow-in-the-pulmonary-airway-system(a44e1ea2-a93e-4ed2-b2b0-bc8aa6edd039).html.
Full textAbstract:
Voluntary coughs are used as a diagnostic tool to detect lung diseases. Understanding the mechanics of a cough is therefore crucial to accurately interpreting the test results. A cough is characterised by a dynamic compression of the airways, resulting in large flow velocities and producing transient peak expiratory flows. Existing models for pulmonary flow have one or more of the following limitations: 1) they assume quasi-steady flows, 2) they assume low speed flows, 3) they assume a symmetrical branching airway system. The main objective of this thesis is to develop a model for a cough in the branching pulmonary airway system. First, the time-dependent one-dimensional equations for flow in a compliant tube is used to simulate a cough in a single airway. Using anatomical and physiological data, the tube law coupling the fluid and airway mechanics is constructed to accurately mimic the airway behaviour in its inflated and collapsed states. Next, a novel model for air flow in an airway bifurcation is constructed. The model is the first to capture successfully subcritical and supercritical flows across the bifurcation and allows for free time evolution from one case to another. The model is investigated by simulating a cough in both symmetric and asymmetric airway bifurcations. Finally, a cough model for the complete branching airway system is developed. The model takes into account the key factors involved in a cough; namely, the compliance of the lungs and the airways, the coughing effort and the sudden opening of the glottis. The reliability of the model is assessed by comparing the model predictions with previous experimental results. The model captures the main characteristics of forced expiatory flows; namely, the flow limitation phenomenon (the flow out of the lungs becomes independent of the applied expiratory effort) and the negative effort dependence phenomenon (the flow out of the lungs decreases with increasing expiratory effort). The model also gives a good qualitative agreement with the measured values of airway resistance. The location of the collapsed airway segment during forced expiration is, however, inconsistent with previous experimental results. The effect of changing the model parameters on the model predictions is therefore discussed.
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Macdougall, Thomas Anthony. "Global error estimators for explicit Runge-Kutta methods." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0001/MQ28227.pdf.
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Tanner, Gregory Mark. "Generalized additive Runge-Kutta methods for stiff odes." Diss., University of Iowa, 2018. https://ir.uiowa.edu/etd/6507.
Full textAbstract:
In many applications, ordinary differential equations can be additively partitioned
\[y'=f(y)=\sum_{m=1}^{N}\f{}{m}(y).]
It can be advantageous to discriminate between the different parts of the right-hand side according to stiffness, nonlinearity, evaluation cost, etc. In 2015, Sandu and G\"{u}nther \cite{sandu2015gark} introduced Generalized Additive Runge-Kutta (GARK) methods which are given by
\begin{eqnarray*}
Y_{i}^{\{q\}} & = & y_{n}+h\sum_{m=1}^{N}\sum_{j=1}^{s^{\{m\}}}a_{i,j}^{\{q,m\}}f^{\{m\}}\left(Y_{j}^{\{m\}}\right)\\
& & \text{for } i=1,\dots,s^{\{q\}},\,q=1,\dots,N\\
y_{n+1} & = & y_{n}+h\sum_{m=1}^{N}\sum_{j=1}^{s^{\{m\}}}b_{j}^{\{m\}}f^{\{m\}}\left(Y_{j}^{\{m\}}\right)\end{eqnarray*}
with the corresponding generalized Butcher tableau
\[\begin{array}{c|ccc} \c{}{1} & \A{1,1} & \cdots & \A{1,N}\\\vdots & \vdots & \ddots & \vdots\\
\c{}{N} & \A{N,1} & \cdots & \A{N,N}\\\hline & \b{}{1} & \cdots & \b{}{N}\end{array}\]
The diagonal blocks $\left(\A{q,q},\b{}{q},\c{}{q}\right)$ can be chosen for example from standard Runge-Kutta methods, and the off-diagonal blocks $\A{q,m},\:q\neq m,$ act as coupling coefficients between the underlying methods. The case when $N=2$ and both diagonal blocks are implicit methods (IMIM) is examined.
This thesis presents order conditions and simplifying assumptions that can be used to choose the off-diagonal coupling blocks for IMIM methods. Error analysis is performed for stiff problems of the form
\begin{eqnarray*}\dot{y} & = & f(y,z)\\
\epsilon\dot{z} & = & g(y,z)\end{eqnarray*}
with small stiffness parameter $\epsilon.$ As $\epsilon\to 0,$ the problem reduces to an index 1 differential algebraic equation provided
$g_{z}(y,z)$ is invertible in a neighborhood of the solution. A tree theory is developed for IMIM methods applied to the reduced problem. Numerical results will be presented for several IMIM methods applied to the Van der Pol equation.
28
Sehnalová, Pavla. "Stabilita a konvergence numerických výpočtů." Doctoral thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2011. http://www.nusl.cz/ntk/nusl-261248.
Full textAbstract:
Tato disertační práce se zabývá analýzou stability a konvergence klasických numerických metod pro řešení obyčejných diferenciálních rovnic. Jsou představeny klasické jednokrokové metody, jako je Eulerova metoda, Runge-Kuttovy metody a nepříliš známá, ale rychlá a přesná metoda Taylorovy řady. V práci uvažujeme zobecnění jednokrokových metod do vícekrokových metod, jako jsou Adamsovy metody, a jejich implementaci ve dvojicích prediktor-korektor. Dále uvádíme generalizaci do vícekrokových metod vyšších derivací, jako jsou např. Obreshkovovy metody. Dvojice prediktor-korektor jsou často implementovány v kombinacích modů, v práci uvažujeme tzv. módy PEC a PECE. Hlavním cílem a přínosem této práce je nová metoda čtvrtého řádu, která se skládá z dvoukrokového prediktoru a jednokrokového korektoru, jejichž formule využívají druhých derivací. V práci je diskutována Nordsieckova reprezentace, algoritmus pro výběr proměnlivého integračního kroku nebo odhad lokálních a globálních chyb. Navržený přístup je vhodně upraven pro použití proměnlivého integračního kroku s přístupe vyšších derivací. Uvádíme srovnání s klasickými metodami a provedené experimenty pro lineární a nelineární problémy.
29
Ijaz, Muhammad. "Implicit runge-kutta methods to simulate unsteady incompressible flows." Texas A&M University, 2007. http://hdl.handle.net/1969.1/85850.
Full textAbstract:
A numerical method (SIMPLE DIRK Method) for unsteady incompressible
viscous flow simulation is presented. The proposed method can be used to achieve
arbitrarily high order of accuracy in time-discretization which is otherwise limited to
second order in majority of the currently used simulation techniques. A special class of
implicit Runge-Kutta methods is used for time discretization in conjunction with finite
volume based SIMPLE algorithm. The algorithm was tested by solving for velocity field
in a lid-driven square cavity. In the test case calculations, power law scheme was used in
spatial discretization and time discretization was performed using a second-order implicit
Runge-Kutta method. Time evolution of velocity profile along the cavity centerline was
obtained from the proposed method and compared with that obtained from a commercial
computational fluid dynamics software program, FLUENT 6.2.16. Also, steady state
solution from the present method was compared with the numerical solution of Ghia, Ghia,
and Shin and that of Erturk, Corke, and Goökçöl. Good agreement of the solution of the
proposed method with the solutions of FLUENT; Ghia, Ghia, and Shin; and Erturk, Corke,
and Goökçöl establishes the feasibility of the proposed method.
30
Fletcher, Matthew T. "Discovery and optimization of low-storage Runge-Kutta methods." Thesis, Monterey, California: Naval Postgraduate School, 2015. http://hdl.handle.net/10945/45852.
Full textAbstract:
Approved for public release; distribution is unlimitedRunge-Kutta (RK) methods are an important family of iterative methods for approximating the solutions of ordinary differential equations (ODEs) and differential algebraic equations (DAEs). It is common to use an RK method to discretize in time when solving time dependent partial differential equations (PDEs) with a method-of-lines approach as in Maxwell’s Equations. Different types of PDEs are discretized in such a way that could result in a high dimensional ODE or DAE.We use a low-storage RK (LSRK) method that stores two registers per ODE dimension, which limits the impact of increased storage requirements. Classical RK methods, however, have one storage variable per stage. In this thesis we compare the efficiency and accuracy of LSRK methods to RK methods. We then focus on optimizing the truncation error coefficients for LSRK to discover new methods. Reusing the tools from the optimization method, we discover new methods for low-storage half-explicit RK (LSHERK) methods for solving DAEs.
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Fenton, P. "The dynamics of variable time-stepping Runge-Kutta methods." Thesis, University of Sussex, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.394994.
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Mugg, Patrick R. "Construction and Analysis of Multi-Rate Partitioned Runge-Kutta Methods." Thesis, Monterey, California. Naval Postgraduate School, 2012. http://hdl.handle.net/10945/7390.
Full textAbstract:
Adaptive mesh refinement (AMR) of hyperbolic systems allows us to refine the spatial grid of an initial value problem (IVP), in order to obtain better accuracy and improved efficiency of the numerical method being used. However, the solutions obtained are still limited to the local Courant-Friedrichs-Lewy (CFL) time-step restrictions of the smallest element within the spatial domain. Therefore, we look to construct a multi-rate time-integration scheme capable of solving an IVP within each spatial sub-domain that is congruent with that sub-domains respective time-step size. The primary objective for this research is to construct a multi-rate method for use with AMR. In this thesis we will focus on constructing a 2nd order, multi-rate partitioned Runge-Kutta (MPRK2) scheme, such that the non-uniform mesh is constructed with the coarse and fine elements at a two-to-one ratio. We will use general 2nd and 4th order finite differences (FD) methods for non-uniform grids to discretize the spatial derivative, and then use this model to compare the MPRK2 time-integrator against three explicit, 2nd order, single-rate time-integrators Adams-Bashforth 2 (AB2), Backward Differentiation Formula 2 (BDF2), and Runge-Kutta 2 (RK2).
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Small, Scott Joseph. "Runge-Kutta type methods for differential-algebraic equations in mechanics." Diss., University of Iowa, 2011. https://ir.uiowa.edu/etd/1082.
Full textAbstract:
Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and electrical circuits. This thesis concentrates on systems of index 2, originally index 3, and mixed index 2 and 3.
Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions. We focus primarily on the class of Gauss-Lobatto SPARK methods. However, we also introduce an extension to methods proposed by Murua for solving index 2 systems to systems of mixed index 2 and 3. An analysis of these methods is also presented in this thesis. We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, and the local error and global convergence of the methods.
When applied to index 2 DAEs, SPARK methods are shown to be equivalent to a class of collocation type methods. When applied to originally index 3 and mixed index 2 and 3 DAEs, they are equivalent to a class of discontinuous collocation methods. Using these equivalences, (s,s)--Gauss-Lobatto SPARK methods can be shown to be superconvergent of order 2s.
Symplectic SPARK methods applied to Hamiltonian systems with holonomic constraints preserve well the total energy of the system. This follows from a backward error analysis approach. SPARK methods and our proposed EMPRK methods are shown to be Lagrange-d'Alembert integrators.
This thesis also presents some numerical results for Gauss-Lobatto SPARK and EMPRK methods. A few problems from mechanics are considered.
34
Pornsawad, Pornsarp, and Christine Böckmann. "Modified iterative Runge-Kutta-type methods for nonlinear ill-posed problems." Universität Potsdam, 2014. http://opus.kobv.de/ubp/volltexte/2014/7083/.
Full textAbstract:
This work is devoted to the convergence analysis of a modified Runge-Kutta-type
iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under Hölder-type source-wise condition if the Fréchet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt and Radau methods.
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JAGOSICH, FABIO H. "Estudos dos processos de transferência de energia dos íons de Ersup(3+) e Hosup(3+) para os íons de Ndsup(3+), Tbsup(3+) e Eusup(3+) no cristal de LiYFsub(4) e no vidro ZBLAN para a otimização de meios laser ativos que operam na região de 3 microns." reponame:Repositório Institucional do IPEN, 2006. http://repositorio.ipen.br:8080/xmlui/handle/123456789/11432.
Full textAbstract:
Made available in DSpace on 2014-10-09T12:51:51Z (GMT). No. of bitstreams: 0Made available in DSpace on 2014-10-09T14:06:32Z (GMT). No. of bitstreams: 0
Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
Tese (Doutoramento)
IPEN/T
Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP
FAPESP:00/11446-0
36
Biehn, Neil David. "Implicit Runge-Kutta Methods for Stiff and Constrained Optimal Control Problems." NCSU, 2001. http://www.lib.ncsu.edu/theses/available/etd-20010322-165913.
Full textAbstract:
The purpose of the research presented in this thesis is to better understand and improve direct transcription methods for stiff and state constrained optimal control problems. When some implicit Runge-Kutta methods are implemented as approximations to the dynamics of an optimal control problem, a loss of accuracy occurs when the dynamics are stiff or constrained. A new grid refinement strategy which exploits the variation of accuracy is discussed. In addition, the use of a residual function in place of classical error estimation techniques is proven to work well for stiff systems. Computational experience reveals the improvement in efficiency and reliability when the new strategies are incorporated as part of a direct transcription algorithm. For index three differential-algebraic equations, the solutions of some implicit Runge-Kutta methods may not converge. However, computational experience reveals apparent convergence for the same methods used when index three state inequality constraints become active. It is shown that the solution chatters along the constraint boundary allowing for better approximations. Moreover, the consistency of the nonlinear programming problem formed by a direct transcription algorithm using an implicit Runge-Kutta approximation is proven for state constraints of arbitrary index.
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Mohr, Darin Griffin. "Hybrid Runge-Kutta and quasi-Newton methods for unconstrained nonlinear optimization." Diss., University of Iowa, 2011. https://ir.uiowa.edu/etd/1249.
Full textAbstract:
Finding a local minimizer in unconstrained nonlinear optimization and a fixed point of a gradient system of ordinary differential equations (ODEs) are two closely related problems. Quasi-Newton algorithms are widely used in unconstrained nonlinear optimization while Runge-Kutta methods are widely used for the numerical integration of ODEs. In this thesis, hybrid algorithms combining low-order implicit Runge-Kutta methods for gradient systems and quasi-Newton type updates of the Jacobian matrix such as the BFGS update are considered. These hybrid algorithms numerically approximate the gradient flow, but the exact Jacobian matrix is not used to solve the nonlinear system at each step. Instead, a quasi-Newton matrix is used to approximate the Jacobian matrix and matrix-vector multiplications are performed in a limited memory setting to reduce storage, computations, and the need to calculate Jacobian information. For hybrid algorithms based on Runge-Kutta methods of order at least two, a curve search is implemented instead of the standard line search used in quasi-Newton algorithms. Stepsize control techniques are also performed to control the stepsize associated with the underlying Runge-Kutta method. These hybrid algorithms are tested on a variety of test problems and their performance is compared with that of the limited memory BFGS algorithm.
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Keeping, Benjamin Rolf. "Efficient solution methods for large systems of differential-algebraic equations." Thesis, Imperial College London, 1996. http://hdl.handle.net/10044/1/8851.
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Schwanenberg, Dirk [Verfasser]. "Die Runge-Kutta-Discontinuous-Galerkin-Methode zur Lösung konvektionsdominierter tiefengemittelter Flachwasserprobleme / Dirk Schwanenberg." Aachen : Shaker, 2005. http://d-nb.info/1172614334/34.
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Rössler, Andreas [Verfasser]. "Runge-Kutta Methods for the Numerical Solution of Stochastic Differential Equations / Andreas Rössler." Aachen : Shaker, 2003. http://d-nb.info/1179021118/34.
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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.
Full textAbstract:
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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Higham, D. J. "Error control in nonstiff initial value solvers." Thesis, University of Manchester, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.234210.
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Brachet, Jean-Baptiste. "A dynamic multiscale viscosity algorithm for shock capturing in Runge Kutta Discontinuous Galerkin methods." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/32441.
Full textAbstract:
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2005.Includes bibliographical references (p. 69-70).
In order to improve the performance of higher-order Discontinuous Galerkin finite element solvers, a shock capturing procedure has been developed for hyperbolic equations. The Dynamic Multiscale Viscosity method, originally presented by Oberai and Wanderer [8, 9] in a Fourier Galerkin context, is adapted to the Discontinuous Galerkin discretization. The notions of diffusive model term, artificial viscosities, and the Germano identity are introduced. A general technique for the evaluation of the multiscale model term's parameters is then presented. This technique is used to perform efficient shock capturing on an one-dimensional stationary Burgers' equation with 1-parameter and 2-parameter model terms. Corresponding numerical results are shown.
by Jean-Baptiste Brachet.
S.M.
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Debrabant, Kristian. "Numerische Behandlung linearer und semilinearer partieller differentiell-algebraischer Systeme mit Runge-Kutta-Methoden." [S.l. : s.n.], 2004. http://deposit.ddb.de/cgi-bin/dokserv?idn=97414178X.
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Cheng, Ping. "Evaluation of a family of Runge-Kutta oriented parallel methods for the solution of ODE's." Thesis, University of Ottawa (Canada), 1995. http://hdl.handle.net/10393/9622.
Full textAbstract:
The numerical solution of ordinary differential equations (ODE's) can be a computationally intensive task. It is becoming widely believed that the only feasible means for solving such computationally intensive problems in science and engineering is to use parallel computers efficiently. As a result, there is an increasing interest in the development of parallel methods for the numerical solution of ODE's. This research is, for the most part, still in its preliminary stages. Our goal in this thesis is to contribute to the evolving knowledge about parallel methods for the solution of ODE's. In this context, we examine in detail one particular class of methods. This class is Runge-Kutta oriented in the sense that the underlying computational process is based on Runge-Kutta formulas. However, from a broader perspective, the methods in this family also have an essential predictor-corrector feature. Our study examines stability and performance aspects of this class of methods as originally proposed. In addition, a modification to the approach is suggested and similarly evaluated. Performance is examined in the context of a suite of test problems and results are compared to previously obtained results with two families of parallel Predictor-Corrector oriented methods.
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Guibert, David. "Analyse de méthodes de résolution parallèles d’EDO/EDA raides." Thesis, Lyon 1, 2009. http://www.theses.fr/2009LYO10138/document.
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La simulation numérique de systèmes d’équations différentielles raides ordinaires ou algébriques est devenue partie intégrante dans le processus de conception des systèmes mécaniques à dynamiques complexes. L’objet de ce travail est de développer des méthodes numériques pour réduire les temps de calcul par le parallélisme en suivant deux axes : interne à l’intégrateur numérique, et au niveau de la décomposition de l’intervalle de temps. Nous montrons l’efficacité limitée au nombre d’étapes de la parallélisation à travers les méthodes de Runge-Kutta et DIMSIM. Nous développons alors une méthodologie pour appliquer le complément de Schur sur le système linéarisé intervenant dans les intégrateurs par l’introduction d’un masque de dépendance construit automatiquement lors de la mise en équations du modèle. Finalement, nous étendons le complément de Schur aux méthodes de type "Krylov Matrix Free". La décomposition en temps est d’abord vue par la résolution globale des pas de temps dont nous traitons la parallélisation du solveur non-linéaire (point fixe, Newton-Krylov et accélération de Steffensen). Nous introduisons les méthodes de tirs à deux niveaux, comme Parareal et Pita dont nous redéfinissons les finesses de grilles pour résoudre les problèmes raides pour lesquels leur efficacité parallèle est limitée. Les estimateurs de l’erreur globale, nous permettent de construire une extension parallèle de l’extrapolation de Richardson pour remplacer le premier niveau de calcul. Et nous proposons une parallélisation de la méthode de correction du résiduThis PhD Thesis deals with the development of parallel numerical methods for solving Ordinary and Algebraic Differential Equations. ODE and DAE are commonly arising when modeling complex dynamical phenomena. We first show that the parallelization across the method is limited by the number of stages of the RK method or DIMSIM. We introduce the Schur complement into the linearised linear system of time integrators. An automatic framework is given to build a mask defining the relationships between the variables. Then the Schur complement is coupled with Jacobian Free Newton-Krylov methods. As time decomposition, global time steps resolutions can be solved by parallel nonlinear solvers (such as fixed point, Newton and Steffensen acceleration). Two steps time decomposition (Parareal, Pita,...) are developed with a new definition of their grids to solved stiff problems. Global error estimates, especially the Richardson extrapolation, are used to compute a good approximation for the second grid. Finally we propose a parallel deferred correction
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Aubry, Anne. "Methodes de runge-kutta pour les equations differentielles algebriques d'indice deux et les systemes hamiltoniens." Rennes 1, 1997. http://www.theses.fr/1997REN10183.
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Nous nous interessons a la resolution numerique d'equations differentielles ordinaires possedant des invariants. D'un point de vue general, le probleme est de construire des methodes numeriques preservant au mieux ces invariants. La premiere partie est consacree aux systemes hamiltoniens. Ces systemes sont en fait caracterises par la symplecticite de leur flot, qui est une propriete de conservation des aires. Lorsque le flot numerique satisfait cette propriete, on parle de methodes symplectiques. A part pour des problemes specifiques et des methodes particulieres, les methodes de runge-kutta symplectiques sont necessairement implicites. Nous construisons des methodes de runge-kutta explicites qui miment le comportement des methodes symplectiques, essentiellement par la conservation de la croissance lineaire de l'erreur globale. De telles methodes d'ordre p, doivent etre d'ordre de pseudo-symplecticite 2p, c'est a dire qu'elles preservent la forme symplectique jusqu'a l'ordre 2p. La deuxieme partie est consacree aux equations differentielles algebriques sous forme d'hessenberg d'indice deux. Elle s'articule autour du phenomene de reduction d'ordre. Pour ces problemes, l'ordre de convergence des methodes de runge-kutta est generalement plus faible que lorsqu'elles sont appliquees a une equation differentielle ordinaire. De plus, il varie suivant les composantes et est souvent plus faible pour la composante dite algebrique. Ce phenomene est une consequence de la non preservation des invariants (ici la contrainte et/ou la contrainte cachee). Nous developpons un outil pour generer automatiquement les conditions d'ordre de ces methodes appliquees aux edas d'indice deux. Celui-ci nous permet en outre de denombrer ces conditions. Puis, nous definissons une nouvelle technique pour eliminer le phenomene de reduction d'ordre. Elle concerne les methodes de radau iia et s'inspire des techniques dites de symetrisation. Finalement, une nouvelle version du code radau5 sans reduction d'ordre pour les edas d'indice deux est presentee.
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Milet, Tomáš. "Grafické intro 64kB s použitím OpenGL." Master's thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2012. http://www.nusl.cz/ntk/nusl-236545.
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This thesis deals with the creation of the intro with limited size. This work describes methods for reducing the size of the final application. The main part describes methods for generating graphic content and methods for its animation. It deals with creation of textures and geometry. Another part is aimed on the physical simulation of particle and elastic systems.
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Amir, Taher Kolar. "Comparison of numerical methods for solving a system of ordinary differential equations: accuracy, stability and efficiency." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-48211.
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In this thesis, we compute approximate solutions to initial value problems of first-order linear ODEs using five explicit Runge-Kutta methods, namely the forward Euler method, Heun's method, RK4, RK5, and RK8. This thesis aims to compare the accuracy, stability, and efficiency properties of the five explicit Runge-Kutta methods. For accuracy, we carry out a convergence study to verify the convergence rate of the five explicit Runge-Kutta methods for solving a first-order linear ODE. For stability, we analyze the stability of the five explicit Runge-Kutta methods for solving a linear test equation. For efficiency, we carry out an efficiency study to compare the efficiency of the five explicit Runge-Kutta methods for solving a system of first-order linear ODEs, which is the main focus of this thesis. This system of first-order linear ODEs is a semi-discretization of a two-dimensional wave equation.
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Wood, Dylan M. "Solving Unsteady Convection-Diffusion Problems in One and More Dimensions with Local Discontinuous Galerkin Methods and Implicit-Explicit Runge-Kutta Time Stepping." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1461181441.
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