1442
1994
eng
postprint
0
20071116


Stabilization of DAEs and invariant manifolds
Many methods have been proposed for the stabilization of higher index differentialalgebraic equations (DAEs). Such methods often involve constraint differentiation and problem stabilization, thus obtaining a stabilized index reduction. A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practice. Here we explain why the Baumgarte method may run into trouble. We then show how to improve it. We further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature. Our approach is to (i) consider stabilization of ODEs with invariants, (ii) discretize the stabilizing term in a simple way, generally different from the ODE discretization, and (iii) use orthogonal projections whenever possible. The best methods thus obtained are related to methods of coordinate projection. We discuss them and make concrete algorithmic suggestions.
urn:nbn:de:kobv:517opus15625
1562
<hr>
ﬁrst published in:<br><a href="http://www.springerlink.com/content/fuyru3vrf0608gl2/">Numerische Mathematik</a>  67 (1994), 2, p. 131149<br>
ISSN: 09453245<br>doi: <a href="http://dx.doi.org/10.1007/s002110050020">10.1007/s002110050020</a><br>
The original publication is available at <a href="http://www.springerlink.com/home/main.mpx">www.springerlink.com</a>.

Uri M. Ascher
Hongsheng Chin
Sebastian Reich
Postprints der Universität Potsdam : MathematischNaturwissenschaftliche Reihe
paper 030
Mathematik
open_access
Institut für Mathematik
Extern
Universität Potsdam
https://publishup.unipotsdam.de/opus4ubp/files/1442/StabilizationDAEs1994.pdf
1443
1995
eng
postprint
0
20071116


Smoothed dynamics of highly oscillatory Hamiltonian systems
We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bondangle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion of the smoothed dynamics of a highly oscillatory Hamiltonian system. Based on our analysis, we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the highfrequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the wellknown SHAKE method.
urn:nbn:de:kobv:517opus15639
1563
<hr>
ﬁrst published in:<br><a href="http://www.sciencedirect.com/science/journal/01672789">Physica D: Nonlinear Phenomena </a>  89 (1995), 12, p. 2842<br>
ISSN: 01672789<br>doi: <a href="http://dx.doi.org/10.1016/01672789(95)00212X">10.1016/01672789(95)00212X</a>

Sebastian Reich
Postprints der Universität Potsdam : MathematischNaturwissenschaftliche Reihe
paper 031
Mathematik
open_access
Institut für Mathematik
Universität Potsdam
https://publishup.unipotsdam.de/opus4ubp/files/1443/SmoothedDynamics1995.pdf
1444
1994
eng
postprint
0
20071116


Symplectic integration of constrained Hamiltonian systems
A Hamiltonian system in potential form (formula in the original abstract) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in Rn. In this paper methods which reduce "Hamiltonian differential algebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraint invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically.
urn:nbn:de:kobv:517opus15653
1565
<hr>
ﬁrst published in:<br><a href="http://www.ams.org/mcom/">Mathematics of Computation</a>  63 (1994), 208, p. 589  605<br>
ISSN: 00255718 <br>Published by the <a href="http://www.ams.org/journals/">American Mathematical Society</a>.

Benedict Leimkuhler
Sebastian Reich
Postprints der Universität Potsdam : MathematischNaturwissenschaftliche Reihe
paper 032
eng
uncontrolled
differentialalgebraic equations
eng
uncontrolled
constrained Hamiltonian systems
eng
uncontrolled
canonical discretization schemes
eng
uncontrolled
symplectic methods
Mathematik
open_access
Institut für Mathematik
Extern
Universität Potsdam
https://publishup.unipotsdam.de/opus4ubp/files/1444/Symplectic_integration1994.pdf
1447
1994
eng
postprint
0
20071121


Stabilization of constrained mechanical systems with DAEs and invariant manifolds
Many methods have been proposed for the simulation of constrained mechanical systems. The most obvious of these have mild instabilities and drift problems. Consequently, stabilization techniques have been proposed A popular stabilization method is Baumgarte's technique, but the choice of parameters to make it robust has been unclear in practice. Some of the simulation methods that have been proposed and used in computations are reviewed here, from a stability point of view. This involves concepts of differentialalgebraic equation (DAE) and ordinary differential equation (ODE) invariants. An explanation of the difficulties that may be encountered using Baumgarte's method is given, and a discussion of why a further quest for better parameter values for this method will always remain frustrating is presented. It is then shown how Baumgarte's method can be improved. An efficient stabilization technique is proposed, which may employ explicit ODE solvers in case of nonstiff or highly oscillatory problems and which relates to coordinate projection methods. Examples of a twolink planar robotic arm and a squeezing mechanism illustrate the effectiveness of this new stabilization method.
urn:nbn:de:kobv:517opus15698
1569
Mechanics Based Design of Structures and Machines.  ISSN 15397742.  23 (1995), p. 135  157
<hr>
This is an electronic version of an article published in <a href="http://www.informaworld.com/smpp/content?content=10.1080/08905459508905232">Mechanics Based Design of Structures and Machines</a>, Volume 23, Issue 2 1995 , pages 135  157. <br>Mechanics Based Design of Structures and Machines is available online at <a href="http://www.informaworld.com/smpp/title~content=t713639027">informaworldTM </a>.

Uri M. Ascher
Hongsheng Chin
Linda R. Petzold
Sebastian Reich
Postprints der Universität Potsdam : MathematischNaturwissenschaftliche Reihe
paper 033
Mathematik
open_access
Institut für Mathematik
Extern
Universität Potsdam
https://publishup.unipotsdam.de/opus4ubp/files/1447/Stabilization_with_DAEs_1994.pdf
1549
1994
eng
postprint
0
20080319


Momentum conserving symplectic integrators
In this paper, we show that symplectic partitioned RungeKutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of LiePoisson systems on certain submanifolds of the general matrix group GL(n).
urn:nbn:de:kobv:517opus16824
1682
Physica D: Nonlinear Phenomena.  76 (1994), 4, p. 375  383.  ISSN 01672789
<hr>
ﬁrst published in:<br><a href="http://www.sciencedirect.com/science/journal/01672789"> Physica D: Nonlinear Phenomena </a>  76 (1994), 4, p. 375  383
<br>
ISSN: 01672789 <a href="http://dx.doi.org/doi:10.1016/01672789(94)900469 ">doi:10.1016/01672789(94)900469 </a>
Keine Nutzungslizenz vergeben  es gilt das deutsche Urheberrecht
Sebastian Reich
Postprints der Universität Potsdam : MathematischNaturwissenschaftliche Reihe
paper 044
Mathematik
open_access
Institut für Mathematik
Universität Potsdam
https://publishup.unipotsdam.de/opus4ubp/files/1549/reich_1994.pdf
12018
2006
2006
eng
article
1



Linear PDEs and numerical methods that preserve a multisymplectic conservation law
Multisymplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multisymplectic methods is still largely missing. In this paper, we study linear multisymplectic PDEs and their discretization by means of numerical dispersion relations. It is found that multisymplectic methods in the sense of Bridges and Reich [Phys. Lett. A, 284 ( 2001), pp. 184193] and Reich [J. Comput. Phys., 157 (2000), pp. 473499], such as GaussLegendre RungeKutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFLtype restriction on Delta t/Delta x might be required for methods higher than second order in time. It is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of Delta t/Delta x despite being multisymplectic in the sense of discrete variational mechanics [J. E. Marsden, G. P. Patrick, and S. Shkoller, Commun. Math. Phys., 199 (1999), pp. 351395]
http://epubs.siam.org/sisc/
10.1137/050628271
10648275
allegro:19912014
10102143
Siam journal on scientific computing.  ISSN 10648275.  28 (2006), 1, S. 260  277
Jason Frank
Brian E. Moore
Sebastian Reich
Institut für Mathematik
Referiert
12173
2010
2010
eng
article
1



Evaluation of three spatial discretization schemes with the Galewsky et al. test
We evaluate the Hamiltonian particle methods (HPM) and the Nambu discretization applied to shallowwater equations on the sphere using the test suggested by Galewsky et al. (2004). Both simulations show excellent conservation of energy and are stable in longterm simulation. We repeat the test also using the ICOSWP scheme to compare with the two conservative spatial discretization schemes. The HPM simulation captures the main features of the reference solution, but wave 5 pattern is dominant in the simulations applied on the ICON grid with relatively low spatial resolutions. Nevertheless, agreement in statistics between the three schemes indicates their qualitatively similar behaviors in the longterm integration.
http://www3.interscience.wiley.com/cgibin/jhome/106562719
10.1002/Asl.279
1530261X
allegro:19912014
10107777
Atmospheric science letters.  ISSN 1530261X.  11 (2010), 3, S. 223  228
Seoleun Shin
Matthias Sommer
Sebastian Reich
Peter Névir
Institut für Mathematik
Referiert
35783
2012
2012
eng
1388
1399
12
666
138
article
WileyBlackwell
Hoboken
1



Hydrostatic Hamiltonian particlemesh (HPM) methods for atmospheric modelling
We develop a hydrostatic Hamiltonian particlemesh (HPM) method for efficient longterm numerical integration of the atmosphere. In the HPM method, the hydrostatic approximation is interpreted as a holonomic constraint for the vertical position of particles. This can be viewed as defining a set of vertically buoyant horizontal meshes, with the altitude of each mesh point determined so as to satisfy the hydrostatic balance condition and with particles modelling horizontal advection between the moving meshes. We implement the method in a verticalslice model and evaluate its performance for the simulation of idealized linear and nonlinear orographic flow in both dry and moist environments. The HPM method is able to capture the basic features of the gravity wave to a degree of accuracy comparable with that reported in the literature. The numerical solution in the moist experiment indicates that the influence of moisture on wave characteristics is represented reasonably well and the reduction of momentum flux is in good agreement with theoretical analysis.
Quarterly journal of the Royal Meteorological Society
10.1002/qj.982
00359009 (print)
wos:20112013
WOS:000306859800020
Shin, S (reprint author), Univ Potsdam, Inst Math, Neuen Palais 10, D14469 Potsdam, Germany., seshin@unipotsdam.de
Seoleun Shin
Sebastian Reich
Jason Frank
eng
uncontrolled
conservative discretization
eng
uncontrolled
Lagrangian modeling
eng
uncontrolled
holonomic constraints
eng
uncontrolled
fluid mechanics
Institut für Mathematik
Referiert
37953
2014
2014
eng
995
1004
10
680
140
article
WileyBlackwell
Hoboken
1



Ensemble transform KalmanBucy filters
Two recent works have adapted the KalmanBucy filter into an ensemble setting. In the first formulation, the ensemble of perturbations is updated by the solution of an ordinary differential equation (ODE) in pseudotime, while the mean is updated as in the standard Kalman filter. In the second formulation, the full ensemble is updated in the analysis step as the solution of single set of ODEs in pseudotime. Neither requires matrix inversions except for the frequently diagonal observation error covariance.
We analyse the behaviour of the ODEs involved in these formulations. We demonstrate that they stiffen for large magnitudes of the ratio of background error to observational error variance, and that using the integration scheme proposed in both formulations can lead to failure. A numerical integration scheme that is both stable and is not computationally expensive is proposed. We develop transformbased alternatives for these Bucytype approaches so that the integrations are computed in ensemble space where the variables are weights (of dimension equal to the ensemble size) rather than model variables.
Finally, the performance of our ensemble transform KalmanBucy implementations is evaluated using three models: the 3variable Lorenz 1963 model, the 40variable Lorenz 1996 model, and a medium complexity atmospheric general circulation model known as SPEEDY. The results from all three models are encouraging and warrant further exploration of these assimilation techniques.
Quarterly journal of the Royal Meteorological Society
10.1002/qj.2186
00359009 (print)
1477870X (online)
wos:2014
WOS:000334926800023
Amezcua, J (reprint author), Univ Reading, Dept Meteorol, POB 243, Reading RG66BB, Berks, England., j.amezcuaespinosa@reading.ac.uk
NASA [NNX07AM97G, NNX08AD40G]; DOE [DEFG0207ER64437]; ONR
[N000140910418, N000141010557]; NOAA [NA09OAR4310178]
Javier Amezcua
Kayo Ide
Eugenia Kalnay
Sebastian Reich
eng
uncontrolled
KalmanBucy Filter
eng
uncontrolled
Ensemble Kalman Filter
eng
uncontrolled
stiff ODE
eng
uncontrolled
weightbased formulations
Institut für Mathematik
Referiert
35837
2012
2012
eng
213
219
7
3
21
article
Schweizerbart
Stuttgart
1



An ensemble KalmanBucy filter for continuous data assimilation
The ensemble Kalman filter has emerged as a promising filter algorithm for nonlinear differential equations subject to intermittent observations. In this paper, we extend the wellknown KalmanBucy filter for linear differential equations subject to continous observations to the ensemble setting and nonlinear differential equations. The proposed filter is called the ensemble KalmanBucy filter and its performance is demonstrated for a simple mechanical model (Langevin dynamics) subject to incremental observations of its velocity.
Meteorologische Zeitschrift
10.1127/09412948/2012/0307
09412948 (print)
wos:20112013
WOS:000306436800002
Reich, S (reprint author), Univ Potsdam, Inst Math, Neuen Palais 10, D14469 Potsdam, Germany., sereich@unipotsdam.de
Kay Bergemann
Sebastian Reich
Institut für Mathematik
Referiert