6,167 research outputs found
A general framework for solving Riemann-Hilbert problems\ud numerically
A new, numerical framework for the approximation of solutions to matrix-valued Riemann-Hilbert problems is developed, based on a recent method for the homogeneous Painlev\'e II Riemann- Hilbert problem. We demonstrate its effectiveness by computing solutions to other Painlev\'e transcendents.\ud
\ud
An implementation in MATHEMATICA is made available online
Entropy-Preserving Coupling Conditions for One-dimensional Euler Systems at Junctions
This paper is concerned with a set of novel coupling conditions for the
one-dimensional Euler system with source terms at a junction of
pipes with possibly different cross-sectional areas. Beside conservation of
mass, we require the equality of the total enthalpy at the junction and that
the specific entropy for pipes with outgoing flow equals the convex combination
of all entropies that belong to pipes with incoming flow. Previously used
coupling conditions include equality of pressure or dynamic pressure. They are
restricted to the special case of a junction having only one pipe with outgoing
flow direction. Recently, Reigstad [SIAM J. Appl. Math., 75:679--702, 2015]
showed that such pressure-based coupling conditions can produce non-physical
solutions for isothermal flows through the production of mechanical energy. Our
new coupling conditions ensure energy as well as entropy conservation and also
apply to junctions connecting an arbitrary number of pipes with flexible flow
directions. We prove the existence and uniqueness of solutions to the
generalised Riemann problem at a junction in the neighbourhood of constant
stationary states which belong to the subsonic region. This provides the basis
for the well-posedness of the homogeneous and inhomogeneous Cauchy problems for
initial data with sufficiently small total variation.Comment: 17 pages, 2 figure
On the optimization of conservation law models at a junction with inflow and flow distribution controls
The paper proposes a general framework to analyze control problems for
conservation law models on a network. Namely we consider a general class of
junction distribution controls and inflow controls and we establish the
compactness in of a class of flux-traces of solutions. We then derive the
existence of solutions for two optimization problems: (I) the maximization of
an integral functional depending on the flux-traces of solutions evaluated at
points of the incoming and outgoing edges; (II) the minimization of the total
variation of the optimal solutions of problem (I). Finally we provide an
equivalent variational formulation of the min-max problem (II) and we discuss
some numerical simulations for a junction with two incoming and two outgoing
edges.Comment: 29 pages, 14 figure
Coupling conditions for the 3x3 Euler system
This paper is devoted to the extension to the full Euler system of
the basic analytical properties of the equations governing a fluid flowing in a
duct with varying section. First, we consider the Cauchy problem for a pipeline
consisting of 2 ducts joined at a junction. Then, this result is extended to
more complex pipes. A key assumption in these theorems is the boundedness of
the total variation of the pipe's section. We provide explicit examples to show
that this bound is necessary.Comment: 21 pages, 6 figure
A Smooth Lattice construction of the Oppenheimer-Snyder spacetime
We present test results for the smooth lattice method using an
Oppenheimer-Snyder spacetime. The results are in excellent agreement with
theory and numerical results from other authors.Comment: 60 pages, 28 figure
Balance laws with integrable unbounded sources
We consider the Cauchy problem for a strictly hyperbolic system
of balance laws each characteristic field being genuinely nonlinear or linearly
degenerate. Assuming that the norm of
and \|u_o\|_{BV(\reali)} are small enough, we
prove the existence and uniqueness of global entropy solutions of bounded total
variation extending the result in [1] to unbounded (in ) sources.
Furthermore, we apply this result to the fluid flow in a pipe with
discontinuous cross sectional area, showing existence and uniqueness of the
underlying semigroup.Comment: 26 pages, 4 figure
Smooth and discontinuous junctions in the p-system
Consider the p-system describing the subsonic flow of a fluid in a pipe with
section a = a(x). We prove that the resulting Cauchy problem generates a
Lipschitz semigroup, provided the total variation of the initial datum and the
oscillation of a are small. An explicit estimate on the bound of the total
variation of a is provided, showing that at lower fluid speeds, higher total
variations of a are acceptable. An example shows that the bound on TV(a) is
mandatory, for otherwise the total variation of the solution may grow
arbitrarily.Comment: 27 pages, 4 figure
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