TY - GEN
A1 - Ascher, Uri M.
A1 - Chin, Hongsheng
A1 - Reich, Sebastian
T1 - Stabilization of DAEs and invariant manifolds
N2 - Many methods have been proposed for the stabilization of higher index differential-algebraic 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.
T3 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 030
Y1 - 1994
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15625
ER -
TY - GEN
A1 - Reich, Sebastian
T1 - Smoothed dynamics of highly oscillatory Hamiltonian systems
N2 - 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 bond-angle 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 high-frequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the well-known SHAKE method.
T3 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 031
Y1 - 1995
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15639
ER -
TY - GEN
A1 - Leimkuhler, Benedict
A1 - Reich, Sebastian
T1 - Symplectic integration of constrained Hamiltonian systems
N2 - 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.
T3 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 032
KW - differential-algebraic equations
KW - constrained Hamiltonian systems
KW - canonical discretization schemes
KW - symplectic methods
Y1 - 1994
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15653
ER -
TY - GEN
A1 - Ascher, Uri M.
A1 - Chin, Hongsheng
A1 - Petzold, Linda R.
A1 - Reich, Sebastian
T1 - Stabilization of constrained mechanical systems with DAEs and invariant manifolds
N2 - 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 differential-algebraic 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 two-link planar robotic arm and a squeezing mechanism illustrate the effectiveness of this new stabilization method.
T3 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 033
Y1 - 1994
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15698
ER -
TY - GEN
A1 - Reich, Sebastian
T1 - Momentum conserving symplectic integrators
N2 - In this paper, we show that symplectic partitioned Runge-Kutta 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 Lie-Poisson systems on certain submanifolds of the general matrix group GL(n).
T3 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 044
Y1 - 1994
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-16824
ER -
TY - JOUR
A1 - Frank, Jason
A1 - Moore, Brian E.
A1 - Reich, Sebastian
T1 - Linear PDEs and numerical methods that preserve a multisymplectic conservation law
N2 - 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. 184-193] and Reich [J. Comput. Phys., 157 (2000), pp. 473-499], such as Gauss-Legendre Runge-Kutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFL-type 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. 351-395]
Y1 - 2006
UR - http://epubs.siam.org/sisc/
U6 - http://dx.doi.org/10.1137/050628271
SN - 1064-8275
ER -
TY - JOUR
A1 - Shin, Seoleun
A1 - Sommer, Matthias
A1 - Reich, Sebastian
A1 - Névir, Peter
T1 - Evaluation of three spatial discretization schemes with the Galewsky et al. test
N2 - We evaluate the Hamiltonian particle methods (HPM) and the Nambu discretization applied to shallow-water equations on the sphere using the test suggested by Galewsky et al. (2004). Both simulations show excellent conservation of energy and are stable in long-term 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 long-term integration.
Y1 - 2010
UR - http://www3.interscience.wiley.com/cgi-bin/jhome/106562719
U6 - http://dx.doi.org/10.1002/Asl.279
SN - 1530-261X
ER -
TY - JOUR
A1 - Shin, Seoleun
A1 - Reich, Sebastian
A1 - Frank, Jason
T1 - Hydrostatic Hamiltonian particle-mesh (HPM) methods for atmospheric modelling
JF - Quarterly journal of the Royal Meteorological Society
N2 - We develop a hydrostatic Hamiltonian particle-mesh (HPM) method for efficient long-term 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 vertical-slice 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.
KW - conservative discretization
KW - Lagrangian modeling
KW - holonomic constraints
KW - fluid mechanics
Y1 - 2012
U6 - http://dx.doi.org/10.1002/qj.982
SN - 0035-9009 (print)
VL - 138
IS - 666
SP - 1388
EP - 1399
PB - Wiley-Blackwell
CY - Hoboken
ER -
TY - JOUR
A1 - Amezcua, Javier
A1 - Ide, Kayo
A1 - Kalnay, Eugenia
A1 - Reich, Sebastian
T1 - Ensemble transform Kalman-Bucy filters
JF - Quarterly journal of the Royal Meteorological Society
N2 - Two recent works have adapted the Kalman-Bucy 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 pseudo-time, 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 pseudo-time. 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 transform-based alternatives for these Bucy-type 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 Kalman-Bucy implementations is evaluated using three models: the 3-variable Lorenz 1963 model, the 40-variable 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.
KW - Kalman-Bucy Filter
KW - Ensemble Kalman Filter
KW - stiff ODE
KW - weight-based formulations
Y1 - 2014
U6 - http://dx.doi.org/10.1002/qj.2186
SN - 0035-9009 (print)
SN - 1477-870X (online)
VL - 140
IS - 680
SP - 995
EP - 1004
PB - Wiley-Blackwell
CY - Hoboken
ER -
TY - JOUR
A1 - Bergemann, Kay
A1 - Reich, Sebastian
T1 - An ensemble Kalman-Bucy filter for continuous data assimilation
JF - Meteorologische Zeitschrift
N2 - 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 well-known Kalman-Bucy filter for linear differential equations subject to continous observations to the ensemble setting and nonlinear differential equations. The proposed filter is called the ensemble Kalman-Bucy filter and its performance is demonstrated for a simple mechanical model (Langevin dynamics) subject to incremental observations of its velocity.
Y1 - 2012
U6 - http://dx.doi.org/10.1127/0941-2948/2012/0307
SN - 0941-2948 (print)
VL - 21
IS - 3
SP - 213
EP - 219
PB - Schweizerbart
CY - Stuttgart
ER -