Multiple Shooting for Unstructured Nonlinear
Differential-Algebraic Equations of Arbitrary Index
Peter Kunkel, Volker Mehrmann & Ronald Stöver
SIAM Journal of Numerical Analysis, Vol. 42(6), pp. 2277-2297, 2005.
Copyright by SIAM. The original publication is available on
SIAM Journals online.
See also
Technical Report 751-2002, Institute of Mathematics,
Technische Universität Berlin, 2002.
Abstract
We study multiple shooting methods for the numerical
solution of nonlinear boundary value problems for unstructured
nonlinear systems of differential-algebraic equations with
arbitrary index. We give a concergence analysis and
demonstrate the results with
some numerical examples.
Key words Differential-algebraic equations, boundary value problems,
multiple shooting
AMS(MOS) subject classifications 65L10, 65L80
Symmetric Collocation for Unstructured Nonlinear
Differential-Algebraic Equations of Arbitrary Index
Peter Kunkel, Volker Mehrmann & Ronald Stöver
Numerische Mathematik, Vol. 98(2), pp. 277-304, 2004.
Copyright by Springer-Verlag.
The original publication is available on
SpringerLink,
DOI 10.1007/s00211-004-0534-9.
See also
Berichte aus der Technomathematik, Report 02-12, November 2002.
Abstract
We examine a class of symmetric collocation schemes for
the solution of nonlinear boundary value problems for unstructured
nonlinear systems of differential-algebraic equations with
arbitrary index. We show that these schemes converge with the same orders
as one would expect for ordinary differential equations.
In particular, we show superconvergence for a special choice
of the collocation points.
We demonstrate the efficiency of the new approach with
some numerical examples.
Key words Differential-algebraic equations, boundary value problems,
collocation methods, symmetric schemes
AMS(MOS) subject classifications 65L10, 65L80
Symmetric collocation methods for solving linear differential-algebraic
boundary value problems
Peter Kunkel & Ronald Stöver
Numerische Mathematik, Vol. 91(3), pp. 475-501, 2002.
Copyright by Springer-Verlag.
The original publication is available on
SpringerLink,
DOI 10.1007/s002110100315.
See also
Berichte aus der Technomathematik, Report 00-15, September 2000.
Abstract
We present symmetric collocation methods for linear differential-algebraic
boundary value problems without restrictions on the index or the structure
of the differential-algebraic equation. In particular, we do not require
a separation into differential and algebraic solution components.
Instead, we use the splitting into differential and algebraic equations
(which arises naturally by index reduction techniques) and apply
Gauß-type (for the differential part) and Lobatto-type (for the
algebraic part) collocation schemes to obtain a symmetric method
which guarantees consistent approximations at the mesh points.
Under standard assumptions, we show solvability and stability
of the discrete problem and determine its order of convergence.
Moreover, we show superconvergence when using the combination of
Gauß and Lobatto schemes and discuss the application of interpolation
to reduce the number of function evaluations.
Finally, we present some numerical comparisons to show the reliability
and efficiency of the new methods.
Key words Differential-algebraic equations, boundary value problems,
collocation methods, symmetric schemes
AMS(MOS) subject classifications 65L10
Collocation methods for solving linear differential-algebraic
boundary value problems
Numerische Mathematik, Vol. 88(4), pp. 771-795, 2001.
Copyright by Springer-Verlag.
The original publication is available on
SpringerLink,
DOI 10.1007/s002110000246.
See also
Berichte aus der Technomathematik, Report 99-08, September 1999.
Abstract
We consider boundary value problems for linear differential-algebraic
equations with variable coefficients
without any restriction for the index.
A well known regularization procedure
yields an equivalent index one problem with d differential
and a=n-d algebraic
equations. Collocation methods based on the regularized BVP approximate
the solution x by a continuous piecewise polynomial of degree k
and deliver, in particular, consistent approximations at mesh points by using
the Radau schemes.
Under weak assumptions the collocation problems are uniquely and
stably solvable and, if the unique solution x is sufficiently smooth,
convergence of order min(k+1,2k-1) and superconvergence at mesh points
of order 2k-1 is shown.
Finally, some numerical experiments illustrating these results are presented.
Key words Differential-algebraic equations, boundary value problems,
collocation methods, Radau schemes
AMS(MOS) subject classifications 65L10
Numerische Lösung von linearen differential-algebraischen
Randwertproblemen
Ph.D. thesis, Universität Bremen, January 1999.
Published by Logos-Verlag 1999
On a Conjugate Gradient-Type Method for Solving Complex Symmetric Linear Systems
Angelika Bunse-Gerstner & Ronald Stöver
Linear Algebra and its Applications , Vol. 287, pp. 105-123, 1999.
Abstract
We consider large sparse linear systems Ax=b with complex symmetric
coefficient matrices A=A^T
which arise, e.g., from the discretization of
partial differential equations with complex coefficients.
For the solution of such systems we
present a new conjugate gradient-type iterative method, CSYM, which is based on
unitary equivalence transformations of A to symmetric tridiagonal form.
An analysis of CSYM shows that its convergence
depends on the singular values of A and that it has both, the
minimal residual property and constant
costs per iteration step. We compare the algorithm with other
methods for solving large sparse complex symmetric systems.
Key words Complex symmetric , CG-type methods, unitary tridiagonalization
AMS(MOS) subject classifications 65F15
Last modified: 13.04.2005