394 research outputs found
Continuous-time link-based kinematic wave model: formulation, solution existence, and well-posedness
We present a continuous-time link-based kinematic wave model (LKWM) for
dynamic traffic networks based on the scalar conservation law model. Derivation
of the LKWM involves the variational principle for the Hamilton-Jacobi equation
and junction models defined via the notions of demand and supply. We show that
the proposed LKWM can be formulated as a system of differential algebraic
equations (DAEs), which captures shock formation and propagation, as well as
queue spillback. The DAE system, as we show in this paper, is the
continuous-time counterpart of the link transmission model. In addition, we
present a solution existence theory for the continuous-time network model and
investigate continuous dependence of the solution on the initial data, a
property known as well-posedness. We test the DAE system extensively on several
small and large networks and demonstrate its numerical efficiency.Comment: 39 pages, 14 figures, 2 tables, Transportmetrica B: Transport
Dynamics 201
Geometric Generalisations of SHAKE and RATTLE
A geometric analysis of the Shake and Rattle methods for constrained
Hamiltonian problems is carried out. The study reveals the underlying
differential geometric foundation of the two methods, and the exact relation
between them. In addition, the geometric insight naturally generalises Shake
and Rattle to allow for a strictly larger class of constrained Hamiltonian
systems than in the classical setting.
In order for Shake and Rattle to be well defined, two basic assumptions are
needed. First, a nondegeneracy assumption, which is a condition on the
Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy
assumption, which is a condition on the geometry of the constrained phase
space. Non-trivial examples of systems fulfilling, and failing to fulfill,
these assumptions are given
Index-2 hybrid DAE: a case study with well-posedness and numerical analysis
In this work, we study differential algebraic equations with constraints defined in a piecewise manner using a conditional statement. Such models classically appear in systems where constraints can evolve in a very small time frame compared to the observed time scale. The use of conditional statements or hybrid automata are a powerful way to describe such systems and are, in general, well suited to simulation with event driven numerical schemes. However, such methods are often subject to chattering at mode switch in presence of sliding modes, or can result in Zeno behaviours. In contrast, the representation of such systems using differential inclusions and method from non-smooth dynamics are often closer to the physical theory but may be harder to interpret. Associated time-stepping numerical methods have been extensively used in mechanical modelling with success and then extended to other fields such as electronics and system biology. In a similar manner to the previous application of non-smooth methods to the simulation of piecewise linear ODEs, non-smooth event-capturing numerical scheme are applied to piecewise linear DAEs. In particular, the study of a 2-D dynamical system of index-2 with a switching constraint using set-valued operators, is presented
Constraint-consistent Runge-Kutta methods for one-dimensional incompressible multiphase flow
New time integration methods are proposed for simulating incompressible
multiphase flow in pipelines described by the one-dimensional two-fluid model.
The methodology is based on 'half-explicit' Runge-Kutta methods, being explicit
for the mass and momentum equations and implicit for the volume constraint.
These half-explicit methods are constraint-consistent, i.e., they satisfy the
hidden constraints of the two-fluid model, namely the volumetric flow
(incompressibility) constraint and the Poisson equation for the pressure. A
novel analysis shows that these hidden constraints are present in the
continuous, semi-discrete, and fully discrete equations.
Next to constraint-consistency, the new methods are conservative: the
original mass and momentum equations are solved, and the proper shock
conditions are satisfied; efficient: the implicit constraint is rewritten into
a pressure Poisson equation, and the time step for the explicit part is
restricted by a CFL condition based on the convective wave speeds; and
accurate: achieving high order temporal accuracy for all solution components
(masses, velocities, and pressure). High-order accuracy is obtained by
constructing a new third order Runge-Kutta method that satisfies the additional
order conditions arising from the presence of the constraint in combination
with time-dependent boundary conditions.
Two test cases (Kelvin-Helmholtz instabilities in a pipeline and liquid
sloshing in a cylindrical tank) show that for time-independent boundary
conditions the half-explicit formulation with a classic fourth-order
Runge-Kutta method accurately integrates the two-fluid model equations in time
while preserving all constraints. A third test case (ramp-up of gas production
in a multiphase pipeline) shows that our new third order method is preferred
for cases featuring time-dependent boundary conditions
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