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372
Discrete mechanics and variational integrators
 Acta Numer
, 2001
"... This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the disc ..."
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Cited by 172 (30 self)
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This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic
ImplicitExplicit RungeKutta Methods for TimeDependent Partial Differential Equations
 Appl. Numer. Math
, 1997
"... Implicitexplicit (IMEX) linear multistep timediscretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable timestep restrictions when applied to convectiondiffusion problems, unless diffusion strongly dominates and an ap ..."
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Cited by 94 (4 self)
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Implicitexplicit (IMEX) linear multistep timediscretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable timestep restrictions when applied to convectiondiffusion problems, unless diffusion strongly dominates and an appropriate BDFbased scheme is selected [2]. In this paper, we develop RungeKuttabased IMEX schemes that have better stability regions than the best known IMEX multistep schemes over a wide parameter range. 1 Introduction When a timedependent partial differential equation (PDE) involves terms of different types, it is a natural idea to employ different discretizations for them. Implicitexplicit (IMEX) timediscretization schemes are an example of such a strategy. Linear multistep IMEX schemes have been used by many researchers, especially in conjunction with spectral methods [10, 3]. Some schemes of this type were proposed and analyzed as far back as the late 1970's [15, 5]. Instances of...
Benchmarking Stiff ODE Solvers for Atmospheric Chemistry Problems I: Implicit versus Explicit
, 1996
"... In many applications of atmospheric transportchemistry problems, a major task is the numerical integration of the stiff systems of ordinary differential equations describing the chemical transformations. This paper presents a comprehensive numerical comparison between five dedicated explicit and fo ..."
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Cited by 44 (17 self)
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In many applications of atmospheric transportchemistry problems, a major task is the numerical integration of the stiff systems of ordinary differential equations describing the chemical transformations. This paper presents a comprehensive numerical comparison between five dedicated explicit and four implicit solvers for a set of seven benchmark problems from actual applications. The implicit solvers use sparse matrix techniques to economize on the numerical linear algebra overhead. As a result they are often more efficient than the dedicated explicit ones, particularly when approximately two or more figures of accuracy are required. In most test cases, sparse rodas, a Rosenbrock solver, came out as most competitive in the 1% error region. Of the dedicated explicit solvers, twostep came out as best. When less than 1% accuracy is aimed at, this solver performs very efficiently for tropospheric gasphase problems. However, like all other dedicated explicit solvers, it cannot efficiently...
The dynamics of legged locomotion: Models, analyses, and challenges
 SIAM Review
, 2006
"... Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we ..."
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Cited by 42 (6 self)
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Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we explore a corner of this fascinating world. We describe mathematical models for legged animal locomotion, focusing on rapidly running insects, and highlighting achievements and challenges that remain. Newtonian bodylimb dynamics are most naturally formulated as piecewiseholonomic rigid body mechanical systems, whose constraints change as legs touch down or lift off. Central pattern generators and proprioceptive sensing require models of spiking neurons, and simplified phase oscillator descriptions of ensembles of them. A full neuromechanical model of a running animal requires integration of these elements, along with proprioceptive feedback and models of goaloriented sensing, planning and learning. We outline relevant background material from neurobiology and biomechanics, explain key properties of the hybrid dynamical systems that 1 underlie legged locomotion models, and provide numerous examples of such models, from the simplest, completely soluble ‘pegleg walker ’ to complex neuromuscular subsystems that are yet to be assembled into models of behaving animals. 1
Modeling and Validating Distributed Embedded RealTime Control Systems
, 2008
"... The development of complex embedded control systems can be improved significantly by applying formal techniques from control engineering and software engineering. It is shown how these approaches can be combined to improve the design and analysis of hightech systems, both in theory and practice. Th ..."
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Cited by 34 (14 self)
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The development of complex embedded control systems can be improved significantly by applying formal techniques from control engineering and software engineering. It is shown how these approaches can be combined to improve the design and analysis of hightech systems, both in theory and practice. The semantics of the integration of two established rigorous techniques has been defined formally in this work. The strength of this integrated semantics is demonstrated by means of a significant industrial case study: the embedded control of a printer paper path, whereby the full development lifecycle from model to realization is covered. The resulting modeldriven design approach fits the current engineering practice in industry and is both flexible and effective.
Residual Based A Posteriori Error Estimators For Eddy Current Computation
, 1999
"... We consider H(curl;\Omega\Gamma3932/608 problems that have been discretized by means of N'ed'elec's edge elements on tetrahedral meshes. Such problems occur in the numerical compuation of eddy currents. From the defect equation we derive localized expressions that can be used as a ..."
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Cited by 32 (7 self)
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We consider H(curl;\Omega\Gamma3932/608 problems that have been discretized by means of N'ed'elec's edge elements on tetrahedral meshes. Such problems occur in the numerical compuation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori error estimators to control adaptive refinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtztype decomposition of the error into an irrotational part and a weakly solenoidal part.
A New Discrete Transparent Boundary Condition for Standard and Wide Angle "Parabolic" Equations in Underwater Acoustics
"... This paper is concerned with transparent boundary conditions (TBCs) for standard and wide angle “parabolic” equations (SPE, WAPE) in the application to underwater acoustics (assuming cylindrical symmetry). Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nic ..."
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Cited by 31 (14 self)
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This paper is concerned with transparent boundary conditions (TBCs) for standard and wide angle “parabolic” equations (SPE, WAPE) in the application to underwater acoustics (assuming cylindrical symmetry). Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nicolson finite difference method only conditionally stable. Here, a novel discrete TBC is derived from the discrete whole–space problem that yields an unconditionally stable scheme. The superiority of the new discrete TBC over existing discretizations is illustrated on several benchmark problems.
Inversion of 3D electromagnetic data in frequency and time domain using an inexact allatonce approach
 Geophys
"... approach ..."
Biomolecular dynamics at long timesteps: Bridging the timescale gap between simulation and experimentation
 ANNU. REV. BIOPHYS. BIOMOL. STRUCT
, 1997
"... Innovative algorithms have been developed during the past decade for simulating Newtonian physics for macromolecules. A major goal is alleviation of the severe requirement that the integration timestep be small enough to resolve the fastest components of the motion and thus guarantee numerical stab ..."
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Cited by 25 (10 self)
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Innovative algorithms have been developed during the past decade for simulating Newtonian physics for macromolecules. A major goal is alleviation of the severe requirement that the integration timestep be small enough to resolve the fastest components of the motion and thus guarantee numerical stability. This timestep problem is challenging if strictly faster methods with the same allatom resolution at small timesteps are sought. Mathematical techniques that have worked well in other multipletimescale contexts—where the fast motions are rapidly decaying or largely decoupled from others—have not been as successful for biomolecules, where vibrational coupling is strong. This review examines general issues that limit the timestep and describes available methods (constrained, reducedvariable, implicit, symplectic, multipletimestep, and normalmodebased schemes). A section compares results of selected integrators for a model dipeptide, assessing physical and numerical performance. Included is our dual timestep method LN, which relies on an approximate linearization of the equations of motion every �t interval (5 fs or less), the solution of which is obtained by explicit integration at the inner timestep �τ (e.g., 0.5 fs). LN is computationally competitive, providing 4–5 speedup factors, and results are in good agreement, in comparison to 0.5 fs trajectories. These collective algorithmic efforts help fill the gap between the time range that can be simulated and the timespans of major biological interest (milliseconds and longer). Still, only a hierarchy of models and methods, along with
Dynamical Modeling and MultiExperiment Fitting with PottersWheel – Supplement
, 2008
"... This supplement provides detailed information about the functionalities of the PottersWheel toolbox as described in the main text. For further information please use the ..."
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Cited by 19 (5 self)
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This supplement provides detailed information about the functionalities of the PottersWheel toolbox as described in the main text. For further information please use the