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

Implicit-explicit (IMEX) linear multistep time-discretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable time-step restrictions when applied to convection-diffusion problems, unless diffusion strongly dominates and an appropriate BDF-based scheme is selected [2]. In this paper, we develop Runge-Kutta-based IMEX schemes that have better stability regions than the best known IMEX multistep schemes over a wide parameter range. 1 Introduction When a time-dependent partial differential equation (PDE) involves terms of different types, it is a natural idea to employ different discretizations for them. Implicit-explicit (IMEX) time-discretization 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...

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 high-tech 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 life-cycle from model to realization is covered. The resulting model-driven design approach fits the current engineering practice in industry and is both flexible and effective.

In many applications of atmospheric transport-chemistry 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 gas-phase problems. However, like all other dedicated explicit solvers, it cannot efficiently...

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 Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part.

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 all-atom resolution at small timesteps are sought. Mathematical techniques that have worked well in other multiple-timescale 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, reduced-variable, implicit, symplectic, multipletimestep, and normal-mode-based 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

Many problems of practical interest can be modeled by differential systems where the solution lies on an invariant manifold defined explicitly by algebraic equations. In computer simulations, it is often important to take into account the invariant's information, either in order to improve upon the stability of the discretization (which is especially important in cases of long time integration) or because a more precise conservation of the invariant is needed for the given application. In this paper we review and discuss methods for stabilizing such an invariant. Particular attention is paid to post-stabilization techniques, where the stabilization steps are applied to the discretized differential system. We summarize theoretical convergence results for these methods and describe the application of this technique to multibody systems with holonomic constraints. We then briefly consider collocation methods which automatically satisfy certain, relatively simple invariants. Finally, we co...

Implicit integrators are very useful in efficiently solving stiff systems of ODEs arising from atmospheric chemistry kinetics, provided that they are modified to take full advantage of the structure of the problem. A systematic way of treating sparsity for reducing the linear algebra cost is presented. 3 Efficient implementation of fully implicit methods 4 1 Introduction It is well known that the equations arising from chemical kinetics comprise a system of stiff ordinary differential equations (ODE). For solving these equations numerically, implicit integrators with infinite stability regions are likely to work with relatively large step-sizes when the accuracy requirements are not too stringent. However, at each integration step, a nonlinear system of equations has to be solved. This involves the repeated evaluation of Jacobians and the solution of linear algebraic systems of dimension n, the number of species considered in the model. General stiff ODE solvers do not take advanta...

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A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the Euler-Lagrange equations is presented. A new class of integrators is defined: the super partitioned additive Runge-Kutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structure-preservation, the class of s-stage Lobatto IIIA-B-C-C* SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.