27,877 research outputs found
Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems
On computers, discrete problems are solved instead of continuous ones. One
must be sure that the solutions of the former problems, obtained in real time
(i.e., when the stepsize h is not infinitesimal) are good approximations of the
solutions of the latter ones. However, since the discrete world is much richer
than the continuous one (the latter being a limit case of the former), the
classical definitions and techniques, devised to analyze the behaviors of
continuous problems, are often insufficient to handle the discrete case, and
new specific tools are needed. Often, the insistence in following a path
already traced in the continuous setting, has caused waste of time and efforts,
whereas new specific tools have solved the problems both more easily and
elegantly. In this paper we survey three of the main difficulties encountered
in the numerical solutions of ODEs, along with the novel solutions proposed.Comment: 25 pages, 4 figures (typos fixed
Introduction to hyperbolic equations and fluid-structure interaction
In this semester project we deal with hyperbolic partial differential equations and Fluid-Structure Interactio
Difference Methods and Deferred Corrections for Ordinary Boundary Value Problems
Compact as possible difference schemes for systems of nth order equations are developed. Generalizations of the Mehrstellenverfahren and simple theoretically sound implementations of deferred corrections are given. It is shown that higher order systems are more efficiently solved as given rather than as reduced to larger lower order systems. Tables of coefficients to implement these methods are included and have been derived using symbolic computations
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
XMDS2: Fast, scalable simulation of coupled stochastic partial differential equations
XMDS2 is a cross-platform, GPL-licensed, open source package for numerically
integrating initial value problems that range from a single ordinary
differential equation up to systems of coupled stochastic partial differential
equations. The equations are described in a high-level XML-based script, and
the package generates low-level optionally parallelised C++ code for the
efficient solution of those equations. It combines the advantages of high-level
simulations, namely fast and low-error development, with the speed, portability
and scalability of hand-written code. XMDS2 is a complete redesign of the XMDS
package, and features support for a much wider problem space while also
producing faster code.Comment: 9 pages, 5 figure
Towards a unified linear kinetic transport model with the trace ion module for EIRENE
Linear kinetic Monte Carlo particle transport models are frequently employed
in fusion plasma simulations to quantify atomic and surface effects on the main
plasma flow dynamics. Separate codes are used for transport of neutral
particles (incl. radiation) and charged particles (trace impurity ions).
Integration of both modules into main plasma fluid solvers provides then self
consistent solutions, in principle. The required interfaces are far from
trivial, because rapid atomic processes in particular in the edge region of
fusion plasmas require either smoothing and resampling, or frequent transfer of
particles from one into the other Monte Carlo code. We propose a different
scheme here, in which despite the inherently different mathematical form of
kinetic equations for ions and neutrals (e.g. Fokker-Planck vs. Boltzmann
collision integrals) both types of particle orbits can be integrated into one
single code. We show that the approximations and shortcomings of this "single
sourcing" concept (e.g., restriction to explicit ion drift orbit integration)
can be fully tolerable in a wide range of typical fusion edge plasma
conditions, and be overcompensated by the code-system simplicity, as well as by
inherently ensured consistency in geometry (one single numerical grid only) and
(the common) atomic and surface process modulesComment: 15 pages, 7 figure
Characteristic Evolution and Matching
I review the development of numerical evolution codes for general relativity
based upon the characteristic initial value problem. Progress in characteristic
evolution is traced from the early stage of 1D feasibility studies to 2D
axisymmetric codes that accurately simulate the oscillations and gravitational
collapse of relativistic stars and to current 3D codes that provide pieces of a
binary black hole spacetime. Cauchy codes have now been successful at
simulating all aspects of the binary black hole problem inside an artificially
constructed outer boundary. A prime application of characteristic evolution is
to extend such simulations to null infinity where the waveform from the binary
inspiral and merger can be unambiguously computed. This has now been
accomplished by Cauchy-characteristic extraction, where data for the
characteristic evolution is supplied by Cauchy data on an extraction worldtube
inside the artificial outer boundary. The ultimate application of
characteristic evolution is to eliminate the role of this outer boundary by
constructing a global solution via Cauchy-characteristic matching. Progress in
this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note:
updated version of arXiv:gr-qc/050809
Numerical relativity with the conformal field equations
I discuss the conformal approach to the numerical simulation of radiating
isolated systems in general relativity. The method is based on conformal
compactification and a reformulation of the Einstein equations in terms of
rescaled variables, the so-called ``conformal field equations'' developed by
Friedrich. These equations allow to include ``infinity'' on a finite grid,
solving regular equations, whose solutions give rise to solutions of the
Einstein equations of (vacuum) general relativity. The conformal approach
promises certain advantages, in particular with respect to the treatment of
radiation extraction and boundary conditions. I will discuss the essential
features of the analytical approach to the problem, previous work on the
problem - in particular a code for simulations in 3+1 dimensions, some new
results, open problems and strategies for future work.Comment: 34 pages, submitted to the Proceedings of the 2001 Spanish Relativity
meeting, eds. L. Fernandez and L. Gonzalez, to be published by Springer,
Lecture Notes in Physics serie
- …