65L05 Initial value problems
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The dynamics of pressurized water distribution networks are naturally modeled by differential algebraic equations (DAE). This paper investigates fundamental structural properties of such a DAE model under weak regularity assumptions. The usual partial derivative-based index-1 condition is shown to be necessary and sufficient for several index concepts, as well as sufficient for solvability in a strong sense. Using the physical properties of nonlinear network elements and the inherent saddle point structure of network hydraulics, we then derive purely topological index criteria based on the network graph and the choice of control variables. Several examples illustrate the theoretical results and explore different non-index-1 situations. A brief discussion of the implications for operative planning by discrete time DAE boundary value problems concludes the paper.
Pseudo--transient continuation methods are quite popular for the numerical solution of steady state problems, typically in PDEs. They are based on an embedding into a time dependent initial value problem. In the presence of dynamical invariants the Jacobian matrix of the nonlinear equation system is bound to be singular. The paper presents a convergence analysis which takes this property into account -- in contrast to known approaches. On the basis of the new analysis adaptive algorithms are suggested in detail. These include a variant with Jacobian approximations as well as inexact pseudo--transient continuation, both of which play an important role in discretized PDEs. Numerical experiments are left to future work.
This series of lectures has been given to a class of mathematics postdocs at a European summer school on Computational Mathematics Driven by Industrial Applications in Martina Franca, Italy (organized by CIME). It deals with a variety of challenging real life problems selected from clinical cancer therapy, communication technology, polymer production, and pharmaceutical drug design. All of these problems from rather diverse application areas share two common features: (a) they have been modelled by various differential equations -- elliptic, parabolic, or Schrödinger--type partial differential equations, countable ordinary diffential equations, or Hamiltonian systems, (b) their numerical solution has turned out to be real challenge to computational mathematics.
Two polynomial expansions of the time-evolution superoperator to directly integrate Markovian Liouville-von Neumann (LvN) equations for quantum open systems, namely the Newton interpolation and the Faber approximation, are presented and critically compared. Details on the numerical implementation including error control, and on the performance of either method are given. In a first physical application, a damped harmonic oscillator is considered. Then, the Faber approximation is applied to compute a condensed phase absorption spectrum, for which a semi--analytical expression is derived. Finally, even more general applications are discussed. In all applications considered here it is found that both the Newton and Faber integrators are fast, general, stable, and accurate.
Dynamical systems with two well-separated time-scales are investigated using normal form theory. Exponential estimates for the normal form truncation error are derived and applied to the numerical integration of differential equations (backward error analysis) and the reduction of highly oscillatory Hamiltonian systems (constrained dynamics and correcting potentials). The theoretical results are used to formulate new algorithms for the time integration of conservative Hamiltonian systems (projected multiple time stepping, soft constraints, rigid bodies, symplectic variable step-size methods).
A variable step-size, semi-explicit variant of the explicit Störmer-Verlet method has been proposed for the time-reversible integration of Newton's equations of motion by Huang & Leimkuhler. Here we propose a fully explicit version of this approach applicable to explicit and symmetric integration methods for general time-reversible differential equations. As applications, we discuss the variable step-size, time-reversible, and fully explicit integration of rigid body motion and reversible Nos\'e-Hoover dynamics.
A Hamiltonian system subject to smooth constraints can typically be viewed as a Hamiltonian system on a manifold. Numerical computations, however, must be performed in $ R^n$. In this paper, canonical transformations from ``Hamiltonian differential--algebraic equations'' to ODEs in Euclidean space are considered. In \S2, canonical parameterizations or local charts are developed and it is shown how these can be computed in a practical framework. In \S3 we consider the construction of unconstrained Hamiltonian ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an integral invariant and whose flow reduces to the flow of the constrained system along the manifold. It is shown that certain of these unconstrained Hamiltonian systems force Lyapunov stability of the constraint--invariants, while others lead to an unstable invariant. In \S4, we compare various projection techniques which might be incorporated to better insure preservation of the constraint--invariants in the context of numerical discretization. Numerical experiments illustrate the degree to which the constraint and symplectic invariants are maintained under discretization of various formulations. {\bf Keywords:} differential--algebraic equations, Hamiltonian systems, canonical discretization schemes. {\bf AMS(MOS):} subject classification 65L05.