7,801 research outputs found
A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations
This paper deals with the index reduction problem for the class of
quasi-regular DAE systems. It is shown that any of these systems can be
transformed to a generically equivalent first order DAE system consisting of a
single purely algebraic (polynomial) equation plus an under-determined ODE
(that is, a semi-explicit DAE system of differentiation index 1) in as many
variables as the order of the input system. This can be done by means of a
Kronecker-type algorithm with bounded complexity
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations
We analyse two practical aspects that arise in the numerical solution of
Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone
approximation schemes known as semi-Lagrangian schemes. These schemes make use
of a wide stencil to achieve convergence and result in discretization matrices
that are less sparse and less local than those coming from standard finite
difference schemes. This leads to computational difficulties not encountered
there. In particular, we consider the overstepping of the domain boundary and
analyse the accuracy and stability of stencil truncation. This truncation
imposes a stricter CFL condition for explicit schemes in the vicinity of
boundaries than in the interior, such that implicit schemes become attractive.
We then study the use of geometric, algebraic and aggregation-based multigrid
preconditioners to solve the resulting discretised systems from implicit time
stepping schemes efficiently. Finally, we illustrate the performance of these
techniques numerically for benchmark test cases from the literature
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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