115 research outputs found
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Integrating-factor-based 2-additive Runge-Kutta methods for advection-reaction-diffusion equations
There are three distinct processes that are predominant in models of flowing
media with interacting components: advection, reaction, and diffusion.
Collectively, these processes are typically modelled with partial differential
equations (PDEs) known as advection-reaction-diffusion (ARD) equations.
To solve most PDEs in practice, approximation methods known as numerical methods
are used. The method of lines is used to approximate PDEs with systems of
ordinary differential equations (ODEs) by a process known as
semi-discretization. ODEs are more readily analysed and benefit from
well-developed numerical methods and software. Each term of an ODE that
corresponds to one of the processes of an ARD equation benefits from particular
mathematical properties in a numerical method. These properties are often
mutually exclusive for many basic numerical methods.
A limitation to the widespread use of more complex numerical methods is that the
development of the appropriate software to provide comparisons to existing
numerical methods is not straightforward. Scientific and numerical software is
often inflexible, motivating the development of a class of software known as
problem-solving environments (PSEs). Many existing PSEs such as Matlab have
solvers for ODEs and PDEs but lack specific features, beyond a scripting
language, to readily experiment with novel or existing solution methods. The PSE
developed during the course of this thesis solves ODEs known as initial-value
problems, where only the initial state is fully known. The PSE is used to assess
the performance of new numerical methods for ODEs that integrate each term of a
semi-discretized ARD equation. This PSE is part of the PSE pythODE that uses
object-oriented and software-engineering techniques to allow implementations of
many existing and novel solution methods for ODEs with minimal effort spent on
code modification and integration.
The new numerical methods use a commutator-free exponential Runge-Kutta (CFERK)
method to solve the advection term of an ARD equation. A matrix exponential is
used as the exponential function, but CFERK methods can use other numerical
methods that model the flowing medium. The reaction term is solved separately
using an explicit Runge-Kutta method because solving it along with the
diffusion term can result in stepsize restrictions and hence inefficiency. The
diffusion term is solved using a Runge-Kutta-Chebyshev method that takes
advantage of the spatially symmetric nature of the diffusion process to avoid
stepsize restrictions from a property known as stiffness. The resulting methods,
known as Integrating-factor-based 2-additive Runge-Kutta methods, are shown to be able to find higher-accuracy
solutions in less computational time than competing methods for certain
challenging semi-discretized ARD equations. This demonstrates the practical
viability both of using CFERK methods for advection and a 3-splitting in
general
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SciCADE 95: International conference on scientific computation and differential equations
This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included
Backward error analysis for variational discretisations of partial differential equations
In backward error analysis, an approximate solution to an equation is
compared to the exact solution to a nearby "modified" equation. In numerical
ordinary differential equations, the two agree up to any power of the step
size. If the differential equation has a geometric property then the modified
equation may share it. In this way, known properties of differential equations
can be applied to the approximation. But for partial differential equations,
the known modified equations are of higher order, limiting applicability of the
theory. Therefore, we study symmetric solutions of discretized partial
differential equations that arise from a discrete variational principle. These
symmetric solutions obey infinite-dimensional functional equations. We show
that these equations admit second-order modified equations which are
Hamiltonian and also possess first-order Lagrangians in modified coordinates.
The modified equation and its associated structures are computed explicitly for
the case of rotating travelling waves in the nonlinear wave equation
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods
Computational and Theoretical Developements for (Time Dependent) Density Functional Theory
En esta tesis se presentan avances computacionales y teoricos en la teoria de funcionales de la densidad (DFT) y en la teoria de funcionales de la densidad dependientes del tiempo (TDDFT). Hemos explorado una posible nueva ruta para la mejora de los funcionales de intercambio y correlacion (XCF) en DFT, comprobado y desarrollado propagadores numericos para TDDFT, y aplicado una combinacion de la teoria de control optimo con TDDFT.En los ultimos anos, DFT se ha convertido en el metodo mas utilizado en el area de estructura electronica gracias a su inigualable relacion entre coste y precision. Podemos usar DFT para calcular multitud de propiedades fisicas y quimicas de atomos, moleculas, nanoestructuras, y materia macroscopica. El factor principal que determina la precision que podemos alcanzar usando DFT es el XCF, un objeto desconocido para el cual se han propuesto cientos de aproximaciones distintas. Algunas de estas aproximaciones funcionan correctamente en ciertas situaciones, pero a dia de hoy no existe un XCF que pueda aplicarse con certeza sobre su validez a un sistema arbitrario. Mas aun, no hay una forma sistematica de refinar estos funcionales. Proponemos y exploramos, para sistemas unidimensionales, una nueva manera de estudiarlos y optimizarlos basada en establecer una relacion con la interaccion entre electrones.TDDFT es la extension de DFT a problemas dependientes del tiempo y problemas conestados excitados, y es tambien uno de los metodos mas populares (a veces el unico metodo que se puede poner en practica) en la comunidad de estructura electronica para tratar conellos. De nuevo, la razon detras de su popularidad reside en su relacion precision/coste computacional, que nos permite tratar sistemas mayores y mas complejos. Puede usarse en combinacion con la dinamica de Ehrenfest, un tipo de dinamica molecular no adiabatica.Hemos ido mas alla y hemos combinado TDDFT y la dinamica de Ehrenfest con la teoria de control optimo, creando un instrumento que nos permite, por ejemplo, predecir la forma de los pulsos laser que inducen una explosion de Coulomb en clusters de sodio. A pesar del buen rendimiento computacional de TDDFT en comparacion con otros metodos, hallamos que el coste de estos calculos era bastante elevado.Motivados por este hecho, tambien dedicamos una parte del trabajo de la tesis a la investigacion computacional. En particular, hemos estudiado e implementado familias de propagadores numericos que no se habian examinado en el contexto de TDDFT. Mas concretamente, metodos con varios pasos previos, formulas Runge-Kutta exponenciales, y las expansiones de Magnus sin conmutadores. Finalmente, hemos implementado modificaciones de estas expansiones de Magnus sin conmutadores para la propagacion de las ecuaciones clasico-cuanticas que resultan de la combinacion de la dinamica de Ehrenfest con TDDFT.In this thesis we present computational and theoretical developments for density functional theory (DFT) and time dependent density functional theory (TDDFT). We have explored a new possible route to improve exchange and correlation functionals (XCF) in DFT, tested and developed numerical propagators for TDDFT, and applied a combination of optimal control theory with TDDFT. In recent years, DFT has become the most used method in the electronic structure field thanks to its unparalleled precision/computational cost relationship. We can use DFT to accurately calculate many physical and chemical properties of atoms, molecules, nanostructures, and bulk materials. The main factor that determines the precision that we can obtain using DFT is the XCF, an unknown object for which hundreds of different approximations have been proposed. Some of these approximations work well enough for certain situations, but to this day there is no XCF that can be reliably applied to any arbitrary system. Moreover, there is no clear way for a systematic refinement of these functionals. We propose and explore, for one-dimensional systems, a new way to optimize them, based on establishing a relationship with the electron-electron interaction. TDDFT is the extension of DFT to time-dependent and excited-states problems, and it is also one of the most popular methods (sometimes the only practical one) in the electronic structure community to deal with them. Once again, the reason behind its popularity is its accuracy/computational cost ratio, which allows us to tackle bigger, more complex systems. It can be used in combination with Ehrenfest dynamics, a non-adiabatic type of molecular dynamics. We have furthermore combined both TDDFT and Ehrenfest dynamics with optimal control theory, a scheme that has allowed us, for example, to predict the shapes of the laser pulses that induce a Coulomb explosion in different sodium clusters. Despite the good numerical performance of TDDFT compared to other methods, we found that these computations were still quite expensive. Motivated by this fact, we have also dedicated a part of the thesis work to computational research. In particular, we have studied and implemented families of numerical propagators that had not been tested in the context of TDDFT. More concretely, linear multistep schemes, exponential Runge-Kutta formulas, and commutator-free Magnus expansions. Moreover, we have implemented modifications of these commutator-free Magnus methods for the propagation of the classical-quantum equations that result of combining Ehrenfest dynamics with TDDFT.<br /
A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions
A new parallel, computationally efficient immersed boundary method for
solving three-dimensional, viscous, incompressible flows on unbounded domains
is presented. Immersed surfaces with prescribed motions are generated using the
interpolation and regularization operators obtained from the discrete delta
function approach of the original (Peskin's) immersed boundary method. Unlike
Peskin's method, boundary forces are regarded as Lagrange multipliers that are
used to satisfy the no-slip condition. The incompressible Navier-Stokes
equations are discretized on an unbounded staggered Cartesian grid and are
solved in a finite number of operations using lattice Green's function
techniques. These techniques are used to automatically enforce the natural
free-space boundary conditions and to implement a novel block-wise adaptive
grid that significantly reduces the run-time cost of solutions by limiting
operations to grid cells in the immediate vicinity and near-wake region of the
immersed surface. These techniques also enable the construction of practical
discrete viscous integrating factors that are used in combination with
specialized half-explicit Runge-Kutta schemes to accurately and efficiently
solve the differential algebraic equations describing the discrete momentum
equation, incompressibility constraint, and no-slip constraint. Linear systems
of equations resulting from the time integration scheme are efficiently solved
using an approximation-free nested projection technique. The algebraic
properties of the discrete operators are used to reduce projection steps to
simple discrete elliptic problems, e.g. discrete Poisson problems, that are
compatible with recent parallel fast multipole methods for difference
equations. Numerical experiments on low-aspect-ratio flat plates and spheres at
Reynolds numbers up to 3,700 are used to verify the accuracy and physical
fidelity of the formulation.Comment: 32 pages, 9 figures; preprint submitted to Journal of Computational
Physic
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