7 research outputs found
Time domain analog circuit simulation
AbstractRecent developments of new methods for simulating electric circuits are described. Emphasis is put on methods that fit existing datastructures for backward differentiation formulae methods. These methods can be modified to apply to hierarchically organized datastructures, which allows for efficient simulation of large designs of circuits in the electronics industry
A Comparison between High-order Temporal Integration Methods Applied to the Discontinuous Galerkin Discretized Euler Equations
Abstract In this work we present a high-order Discontinuous Galerkin (DG) space approximation coupled with two high-order temporal integration methods for the numerical solution of time-dependent compressible flows. The time integration methods analyzed are the explicit Strong-Stability-Preserving Runge-Kutta (SSPRK) and the Two Implicit Advanced Step-point (TIAS) schemes. Their accuracy and efficiency are evaluated by means of an inviscid test case for which an exact solution is available. The study is carried out for several time-steps using different polynomial order approximations and several levels of grid refinement. The effect of mesh irregularities on the accuracy is also investigated by considering randomly perturbed meshes. The analysis of the results has the twofold objective of (i) assessing the performances of the temporal schemes in the context of the high-order DG discretization and(ii) determining if high-order implicit schemes can displace widely used high-order explicit schemes
Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems
In this survey, a recent computational methodology paying a special attention to the separation
of mathematical objects from numeral systems involved in their representation is described.
It has been introduced with the intention to allow one to work with infinities and infinitesimals
numerically in a unique computational framework in all the situations requiring these notions. The
methodology does not contradict Cantor’s and non-standard analysis views and is based on the
Euclid’s Common Notion no. 5 “The whole is greater than the part” applied to all quantities (finite,
infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a
computational device called the Infinity Computer (patented in USA and EU) working numerically
(recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite
and infinitesimal numbers that can be written in a positional numeral system with an infinite radix.
It is argued that numeral systems involved in computations limit our capabilities to compute and lead
to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility
to use the same numeral system for measuring infinite sets, working with divergent series, probability,
fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally
different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical
objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher
accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not
related to their nature but is a consequence of the weakness of traditional numeral systems used to
study them. It is shown that the introduced methodology and numeral system change our perception
of the mathematical objects studied in the two problems
Multi-Value Numerical Modeling for Special Di erential Problems
2013 - 2014The subject of this thesis is the analysis and development of new numerical methods
for Ordinary Di erential Equations (ODEs). This studies are motivated by the
fundamental role that ODEs play in applied mathematics and applied sciences in
general. In particular, as is well known, ODEs are successfully used to describe
phenomena evolving in time, but it is often very di cult or even impossible to nd
a solution in closed form, since a general formula for the exact solution has never
been found, apart from special cases. The most important cases in the applications
are systems of ODEs, whose exact solution is even harder to nd; then the role played
by numerical integrators for ODEs is fundamental to many applied scientists. It is
probably impossible to count all the scienti c papers that made use of numerical
integrators during the last century and this is enough to recognize the importance
of them in the progress of modern science. Moreover, in modern research, models
keep getting more complicated, in order to catch more and more peculiarities of
the physical systems they describe, thus it is crucial to keep improving numerical
integrator's e ciency and accuracy.
The rst, simpler and most famous numerical integrator was introduced by Euler
in 1768 and it is nowadays still used very often in many situations, especially in educational
settings because of its immediacy, but also in the practical integration of
simple and well-behaved systems of ODEs. Since that time, many mathematicians
and applied scientists devoted their time to the research of new and more e cient
methods (in terms of accuracy and computational cost). The development of numerical
integrators followed both the scienti c interests and the technological progress
of the ages during whom they were developed. In XIX century, when most of the calculations
were executed by hand or at most with mechanical calculators, Adams and
Bashfort introduced the rst linear multistep methods (1855) and the rst Runge-
Kutta methods appeared (1895-1905) due to the early works of Carl Runge and
Martin Kutta. Both multistep and Runge-Kutta methods generated an incredible
amount of research and of great results, providing a great understanding of them
and making them very reliable in the numerical integration of a large number of
practical problems.
It was only with the advent of the rst electronic computers that the computational
cost started to be a less crucial problem and the research e orts started to
move towards the development of problem-oriented methods. It is probably possible
to say that the rst class of problems that needed an ad-hoc numerical treatment was
that of sti problems. These problems require highly stable numerical integrators
(see Section ??) or, in the worst cases, a reformulation of the problem itself.
Crucial contributions to the theory of numerical integrators for ODEs were given
in the XX century by J.C. Butcher, who developed a theory of order for Runge-Kutta
methods based on rooted trees and introduced the family of General Linear Methods
together with K. Burrage, that uni ed all the known families of methods for rst
order ODEs under a single formulation. General Linear Methods are multistagemultivalue
methods that combine the characteristics of Runge-Kutta and Linear
Multistep integrators... [edited by Author]XIII n.s
Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review
A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances
Development and evaluation of a wind tunnel manoeuvre rig
EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Perturbed MEBDF methods
We investigate MEBDF methods of Cash from general linear methods point of view. Some perturbations of these methods are constructed which preserve the order of these formulas and improve their stability properties