3,499 research outputs found
Reducing the index of differential-algebraic equations by exploiting underlying structures
Differential-algebraic equations arise from the equation based modelling of physical systems, such as those found for example in engineering or physics. This thesis is concerned with square, sufficiently smooth, potentially non-linear differential-algebraic equations. Differential-algebraic equations can be classified by their index. This is a measure of how far a differential-algebraic equation is from an equivalent ordinary differential equation. To solve a differential-algebraic equation one usually transforms the problem to an ordinary differential equation, or something close to one, via an index reduction algorithm. This thesis examines how the index reduction (using dummy derivatives) of differential-algebraic equations can be improved via structural analysis, specifically the Signature Matrix method. Improved and alternative algorithms for finding dummy derivatives are presented and then a new algorithm for finding globally valid universal dummy derivatives is presented. It is also shown that the structural index of a differential-algebraic equation is invariant under order reduction
How AD Can Help Solve Differential-Algebraic Equations
A characteristic feature of differential-algebraic equations is that one
needs to find derivatives of some of their equations with respect to time, as
part of so called index reduction or regularisation, to prepare them for
numerical solution. This is often done with the help of a computer algebra
system. We show in two significant cases that it can be done efficiently by
pure algorithmic differentiation. The first is the Dummy Derivatives method,
here we give a mainly theoretical description, with tutorial examples. The
second is the solution of a mechanical system directly from its Lagrangian
formulation. Here we outline the theory and show several non-trivial examples
of using the "Lagrangian facility" of the Nedialkov-Pryce initial-value solver
DAETS, namely: a spring-mass-multipendulum system, a prescribed-trajectory
control problem, and long-time integration of a model of the outer planets of
the solar system, taken from the DETEST testing package for ODE solvers
Index Reduction for Differential-Algebraic Equations with Mixed Matrices
Differential-algebraic equations (DAEs) are widely used for modeling of
dynamical systems. The difficulty in solving numerically a DAE is measured by
its differentiation index. For highly accurate simulation of dynamical systems,
it is important to convert high-index DAEs into low-index DAEs. Most of
existing simulation software packages for dynamical systems are equipped with
an index-reduction algorithm given by Mattsson and S\"{o}derlind.
Unfortunately, this algorithm fails if there are numerical cancellations.
These numerical cancellations are often caused by accurate constants in
structural equations. Distinguishing those accurate constants from generic
parameters that represent physical quantities, Murota and Iri introduced the
notion of a mixed matrix as a mathematical tool for faithful model description
in structural approach to systems analysis. For DAEs described with the use of
mixed matrices, efficient algorithms to compute the index have been developed
by exploiting matroid theory.
This paper presents an index-reduction algorithm for linear DAEs whose
coefficient matrices are mixed matrices, i.e., linear DAEs containing physical
quantities as parameters. Our algorithm detects numerical cancellations between
accurate constants, and transforms a DAE into an equivalent DAE to which
Mattsson--S\"{o}derlind's index-reduction algorithm is applicable. Our
algorithm is based on the combinatorial relaxation approach, which is a
framework to solve a linear algebraic problem by iteratively relaxing it into
an efficiently solvable combinatorial optimization problem. The algorithm does
not rely on symbolic manipulations but on fast combinatorial algorithms on
graphs and matroids. Furthermore, we provide an improved algorithm under an
assumption based on dimensional analysis of dynamical systems.Comment: A preliminary version of this paper is to appear in Proceedings of
the Eighth SIAM Workshop on Combinatorial Scientific Computing, Bergen,
Norway, June 201
A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations
This paper deals with the index reduction problem for the class of
quasi-regular DAE systems. It is shown that any of these systems can be
transformed to a generically equivalent first order DAE system consisting of a
single purely algebraic (polynomial) equation plus an under-determined ODE
(that is, a semi-explicit DAE system of differentiation index 1) in as many
variables as the order of the input system. This can be done by means of a
Kronecker-type algorithm with bounded complexity
Structural analysis based dummy derivative selection for differential algebraic equations
The signature matrix structural analysis method developed by Pryce provides more structural information than the commonly used Pantelides method and applies to differential-algebraic equations (DAEs) of arbitrary order. It is useful to consider how existing methods using the Pantelides algorithm can benefit from such structural analysis. The dummy derivative method is a technique commonly used to solve DAEs that can benefit from such exploitation of underlying DAE structures and information found in the Signature Matrix method. This paper gives a technique to find structurally necessary dummy derivatives and how to use different block triangular forms effectively when performing the dummy derivative method and then provides a brief complexity analysis of the proposed approach. We finish by outlining an approach that can simplify the task of dummy pivoting
Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations
The Navier--Stokes equations are commonly used to model and to simulate flow
phenomena. We introduce the basic equations and discuss the standard methods
for the spatial and temporal discretization. We analyse the semi-discrete
equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index
and quantify the numerical difficulties in the fully discrete schemes, that are
induced by the strangeness of the system. By analyzing the Kronecker index of
the difference-algebraic equations, that represent commonly and successfully
used time stepping schemes for the Navier--Stokes equations, we show that those
time-integration schemes factually remove the strangeness. The theoretical
considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909,
https://doi.org/10.5281/zenodo.99890
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