Exploring efficient: numerical methods for differential equations

Numerical analysis is a way to do higher mathematical problems on a computer, a technique widely used by scientists and engineers to solve their problems. A major advantage of numerical analysis is that a numerical answer can be obtained even when a problem has no “analytical” solution. Results from numerical analysis are an approximation, which can be made as accurate as desired. The analysis of errors in numerical methods is a critically important part of the study of numerical analysis. Hence, we will see in this research that computation of the error is a must as it is a way to measure the efficiency of the numerical methods developed. Numerical methods require highly tedious and repetitive computations that can only be done using the computer. Hence in this research, it is shown that computer programs must be written for the implementation of numerical methods. In the early part of related research the computer language used was Fortran. Subsequently more and more computer programs used the C programming language. Additionally, now computations can also be carried out using softwares like MATLAB, MATHEMATICA and MAPLE. Many physical problems that arise from ordinary differential equations (ODEs) have magnitudes of eigenvalues which vary greatly, and such systems are commonly known as stiff systems. Stiff systems usually consist of a transient solution, that is, a solution which varies rapidly at the beginning of the integration. This phase is referred to as the transient phase and during this phase, accuracy rather than stability restricts the stepsize of the numerical methods used. Thus the generally the structure of the solutions suggests application of specific methods for non-stiff equations in the transient phase and specific methods for stiff equations during the steady-state phase in a manner whereby computational costs can be reduced. Consequently, in this research we developed embedded Runge-Kutta methods for solving stiff differential equations so that variable stepsize codes can be used in its implementation. We have also included intervalwise partitioning, whereby the system is considered as non-stiff first, and solved using the method with simple iterations, and once stiffness is detected, the system is solved using the same method, but with Newton iterations. By using variable stepsize code and intervalwise partitioning, we have been able to reduce the computational costs. With the aim of increasing the computational efficiency of the Runge-Kutta methods, we have also developed methods of higher order with less number of stages or function evaluations. The method used is an extension of the classical Runge-Kutta method and the approximation at the current point is based on the information at the current internal stage as well as the previous internal stage. This is the idea underlying the construction of Improved Runge-Kutta methods, so that the resulting method will give better accuracy. Usually higher order ordinary differential equations are solved by converting them into a system of first order ODEs and using numerical methods suitable for first order ODEs. However it is more efficient, in terms of accuracy, number of function evaluations as well as computational time, if the higher order ODEs can be solved directly (without being converted to a system of first order ODEs), using numerical methods. In this research we developed numerical methods, particularly Runge-Kutta type methods, which can directly solve special third order and fourth order ODEs. Special second order ODE is an ODE which does not depend on the first derivative. The solution from this type of ODE often exhibits a pronounced oscillatory character. It is well known that it is difficult to obtain accurate numerical results if the ODEs are oscillatory in nature. In order to address this problem a lot of research has been focused on developing methods which have high algebraic order, reduced phase-lag or dispersion and reduced dissipation. Phaselag is the angle between the true and approximate solution, while dissipation is the difference between the approximate solution and the standard cyclic solution. If a method has high algebraic order, high order of dispersion and dissipation, then the numerical solutions obtained will be very accurate. Hence in this research we have developed numerical methods, specifically hybrid methods which have all the above mentioned properties. If the solutions are oscillatory in nature, it means that the solutions will have components which are trigonometric functions, that is, sine and cosine functions. In order to get accurate numerical solutions we thus phase-fitted the methods using trigonometric functions. In this research, it is proven that trigonometrically-fitting the hybrid methods and applying them to solve oscillatory delay differential equations result in better numerical results. These are the highlights of my research journey, though a lot of work has also been done in developing numerical methods which are multistep in nature, for solving higher order ODEs, as well as implementation of methods developed for solving fuzzy differential equations and partial differential equations, which are not covered here

The numerical solution of neural field models posed on realistic cortical domains

The mathematical modelling of neural activity is a hugely complex and prominent area of exploration that has been the focus of many researchers since the mid 1900s. Although many advancements and scientific breakthroughs have been made, there is still a great deal that is not yet understood about the brain. There have been a considerable amount of studies in mathematical neuroscience that consider the brain as a simple one-dimensional or two-dimensional domain; however, this is not biologically realistic and is primarily selected as the domain of choice to aid analytical progress. The primary aim of this thesis is to develop and provide a novel suite of codes to facilitate the computationally efficient numerical solution of large-scale delay differential equations, and utilise this to explore both neural mass and neural field models with space-dependent delays. Through this, we seek to widen the scope of models of neural activity by posing them on realistic cortical domains and incorporating real brain data to describe non-local cortical connections. The suite is validated using a selection of examples that compare numerical and analytical results, along with recreating existing results from the literature. The relationship between structural connectivity and functional connectivity is then analysed as we use an eigenmode fitting approach to inform the desired stability regimes of a selection of neural mass models with delays. Here, we explore a next-generation neural mass model developed by Coombes and Byrne [36], and compare results to the more traditional Wilson-Cowan formulation [180, 181]. Finally, we examine a variety of solutions to three different neural field models that incorporate real structural connectivity, path length, and geometric surface data, using our NFESOLVE library to efficiently compute the numerical solutions. We demonstrate how the field version of the next-generation model can yield intricate and detailed solutions which push us closer to recreating observed brain dynamics

The numerical solution of neural field models posed on realistic cortical domains

The mathematical modelling of neural activity is a hugely complex and prominent area of exploration that has been the focus of many researchers since the mid 1900s. Although many advancements and scientific breakthroughs have been made, there is still a great deal that is not yet understood about the brain. There have been a considerable amount of studies in mathematical neuroscience that consider the brain as a simple one-dimensional or two-dimensional domain; however, this is not biologically realistic and is primarily selected as the domain of choice to aid analytical progress. The primary aim of this thesis is to develop and provide a novel suite of codes to facilitate the computationally efficient numerical solution of large-scale delay differential equations, and utilise this to explore both neural mass and neural field models with space-dependent delays. Through this, we seek to widen the scope of models of neural activity by posing them on realistic cortical domains and incorporating real brain data to describe non-local cortical connections. The suite is validated using a selection of examples that compare numerical and analytical results, along with recreating existing results from the literature. The relationship between structural connectivity and functional connectivity is then analysed as we use an eigenmode fitting approach to inform the desired stability regimes of a selection of neural mass models with delays. Here, we explore a next-generation neural mass model developed by Coombes and Byrne [36], and compare results to the more traditional Wilson-Cowan formulation [180, 181]. Finally, we examine a variety of solutions to three different neural field models that incorporate real structural connectivity, path length, and geometric surface data, using our NFESOLVE library to efficiently compute the numerical solutions. We demonstrate how the field version of the next-generation model can yield intricate and detailed solutions which push us closer to recreating observed brain dynamics

Hybrid Frequency-Time Analysis and Numerical Methods for Time-Dependent Wave Propagation

This thesis focuses on the solution of causal, time-dependent wave propagation and scattering problems, in two- and three-dimensional spatial domains. This important and long-lasting problem has attracted a great deal of interest reflecting not only its use as a model problem but also the prevalence of wave phenomena in diverse areas of modern science, technology and engineering. Essentially all prior methods rely on "time-stepping" in one form or another, which involves local-in-time approximation of the evolution of the solution of the partial differential equation (PDE) based on the immediate time history and temporal finite-difference approximation. In addition to the need to manage the accumulation of (dispersion) error and the burdensome increase in computational cost over time, there are additionally difficult issues of stability, time-domain boundary conditions, and absorbing boundary conditions which often need to be addressed. To sidestep many of these problems, this thesis develops a novel highly-efficient approach for time-dependent wave scattering problems employing the global-in-time techniques of Fourier transformation and leading to a frequency/time hybrid method for the time-dependent wave equation. Thus, relying on Fourier Transformation in time and utilizing a fixed (time-independent) number of frequency-domain solutions, the method evaluates the desired time-domain evolution with errors that both, decay faster than any negative power of the temporal sampling rate, and that, for a given sampling rate, are additionally uniform in time for all time. The fast error decay guarantees that high accuracies can be attained on the basis of relatively coarse temporal and frequency discretizations. The uniformity of the error for all time with fixed sampling rate, a property known as dispersionlessness, plays a crucial role, together with other properties of the Fourier transform, in enabling the evaluation of solutions for long times at O(1) cost. In particular, this thesis demonstrates the significant advantages enjoyed by the proposed methods over alternative approaches based on volumetric discretizations, time-domain integral equations, and convolution-quadrature. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider in detail the Laplace-based method, and only briefly point out its essential features and associated challenges. The proposed frequency/time Fourier-transform methods for obstacle scattering problems are easily generalizable to any linear partial differential equation in the time domain for which frequency-domain solutions can readily be obtained, including e.g. the time-domain Maxwell equations, the linear elasticity equations, inhomogeneous and/or frequency-dependent dispersive media, etc. Further, the proposed approach can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping—that is, solution sampling at any given time T at O(1)-bounded sampling cost, for arbitrarily large values of T, and without requirement of evaluation of the solution at intermediate times. In particular, effective algorithms are introduced that, relying on use of time-asymptotics, compute two-dimensional solutions at O(1) cost despite the very slow time-decay that takes place in the two-dimensional case. A significant portion of this thesis is devoted to a theoretical study of the validity of a certain stopping criterion used by the algorithm, which guarantees that certain field contributions can safely be neglected after certain stopping times. Roughly speaking, the theoretical results guarantee that, after the incident field is turned off, the magnitude of the future scattering density (and thus the magnitudes of the fields) can be estimated by the magnitude of the integral density over a time period comparable to the time required by a wave to travel a distance equal to the diameter of the scatterer. The criterion, which is crucial in ensuring the O(1) computational cost of the algorithm, is closely related to the well-known scattering theory developed in the 1960s and '70s by Lax, Morawetz, Phillips, Strauss and others. Our approach to the decay problem is based on use of frequency-domain estimates (developed previously in the context of numerical analysis of frequency-domain problems) on integral operators in the high-frequency regime for obstacles of various trapping classes. In particular, our theory yields, for the first time, decay estimates for a class of connected trapping obstacles: all previous estimates of scattered-field decay for connected obstacles are restricted to nontrapping structures. In all, the proposed approach leverages the power of the Fourier transformation together with a range of newly developed spectrally convergent numerical methods in both the frequency and time domain and a variety of novel theoretical results in the general area of scattering theory to produce a radically-new framework for the solution of time-dependent wave propagation and scattering problems.</p

Universal dynamics and thermalization in isolated quantum systems

The goal of this work is to explore the relaxation dynamics of isolated quantum systems driven out of equilibrium, focusing on nonequilibrium phenomena emerging on the way to thermal equilibrium. We study close-to-equilibrium states relaxing to thermal equilibrium directly as well as far-from-equilibrium states approaching a transient regime characterized by a self-similar time evolution. One of the most persistent challenges concerns the thermalization process of the quark-gluon plasma in heavy-ion collisions. We address this issue by investigating the equilibration process of the quark-meson model, an effective low-energy theory of quantum chromodynamics (QCD) that captures important features of QCD including the chiral phase transition. Our simulations probe the approach of quantum thermal equilibrium, characterized by the emergence of Bose-Einstein and Fermi-Dirac distribution functions, in different regions of the phase diagram. We find additional light fermionic degrees of freedom in the crossover region of the quasiparticle excitation spectrum. A remarkable feature of isolated quantum systems is that far-from-equilibrium states can approach nonthermal fixed points, where the dynamics becomes self-similar and universal across disparate physical systems. We study the infrared nonthermal fixed point by investigating the scaling properties of distribution functions in a relativistic scalar field theory as well as a spin-1 Bose gas. For the scalar field theory we also compute the effective four-vertex and unequal-time two-point correlation functions entailing the nonthermal properties of the system in terms of a strongly violated fluctuation-dissipation theorem. In the spin-1 Bose gas quickly emerging long-range correlations indicate the formation of a condensate out of equilibrium

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

State Of the Art Report in the fields of numerical analysis and scientific computing. Final version as of 16/02/2020 deliverable D4.1 of the HORIZON 2020 project EURAD.: European Joint Programme on Radioactive Waste Management

Document information Project Acronym EURAD Project Title European Joint Programme on Radioactive Waste Management Project Type European Joint Programme (EJP) EC grant agreement No. 847593 Project starting / end date 1 st June 2019-30 May 2024 Work Package No. 4 Work Package Title Development and Improvement Of NUmerical methods and Tools for modelling coupled processes Work Package Acronym DONUT Deliverable No. 4.

Book of abstracts of the 10th International Chemical and Biological Engineering Conference: CHEMPOR 2008

This book contains the extended abstracts presented at the 10th International Chemical and Biological Engineering Conference - CHEMPOR 2008, held in Braga, Portugal, over 3 days, from the 4th to the 6th of September, 2008. Previous editions took place in Lisboa (1975, 1889, 1998), Braga (1978), Póvoa de Varzim (1981), Coimbra (1985, 2005), Porto (1993), and Aveiro (2001). The conference was jointly organized by the University of Minho, “Ordem dos Engenheiros”, and the IBB - Institute for Biotechnology and Bioengineering with the usual support of the “Sociedade Portuguesa de Química” and, by the first time, of the “Sociedade Portuguesa de Biotecnologia”. Thirty years elapsed since CHEMPOR was held at the University of Minho, organized by T.R. Bott, D. Allen, A. Bridgwater, J.J.B. Romero, L.J.S. Soares and J.D.R.S. Pinheiro. We are fortunate to have Profs. Bott, Soares and Pinheiro in the Honor Committee of this 10th edition, under the high Patronage of his Excellency the President of the Portuguese Republic, Prof. Aníbal Cavaco Silva. The opening ceremony will confer Prof. Bott with a “Long Term Achievement” award acknowledging the important contribution Prof. Bott brought along more than 30 years to the development of the Chemical Engineering science, to the launch of CHEMPOR series and specially to the University of Minho. Prof. Bott’s inaugural lecture will address the importance of effective energy management in processing operations, particularly in the effectiveness of heat recovery and the associated reduction in greenhouse gas emission from combustion processes. The CHEMPOR series traditionally brings together both young and established researchers and end users to discuss recent developments in different areas of Chemical Engineering. The scope of this edition is broadening out by including the Biological Engineering research. One of the major core areas of the conference program is life quality, due to the importance that Chemical and Biological Engineering plays in this area. “Integration of Life Sciences & Engineering” and “Sustainable Process-Product Development through Green Chemistry” are two of the leading themes with papers addressing such important issues. This is complemented with additional leading themes including “Advancing the Chemical and Biological Engineering Fundamentals”, “Multi-Scale and/or Multi-Disciplinary Approach to Process-Product Innovation”, “Systematic Methods and Tools for Managing the Complexity”, and “Educating Chemical and Biological Engineers for Coming Challenges” which define the extended abstracts arrangements along this book. A total of 516 extended abstracts are included in the book, consisting of 7 invited lecturers, 15 keynote, 105 short oral presentations given in 5 parallel sessions, along with 6 slots for viewing 389 poster presentations. Full papers are jointly included in the companion Proceedings in CD-ROM. All papers have been reviewed and we are grateful to the members of scientific and organizing committees for their evaluations. It was an intensive task since 610 submitted abstracts from 45 countries were received. It has been an honor for us to contribute to setting up CHEMPOR 2008 during almost two years. We wish to thank the authors who have contributed to yield a high scientific standard to the program. We are thankful to the sponsors who have contributed decisively to this event. We also extend our gratefulness to all those who, through their dedicated efforts, have assisted us in this task. On behalf of the Scientific and Organizing Committees we wish you that together with an interesting reading, the scientific program and the social moments organized will be memorable for all.Fundação para a Ciência e a Tecnologia (FCT