7,429 research outputs found
On discretely entropy conservative and entropy stable discontinuous Galerkin methods
High order methods based on diagonal-norm summation by parts operators can be
shown to satisfy a discrete conservation or dissipation of entropy for
nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as
nodal discontinuous Galerkin methods with diagonal mass matrices. In this work,
we describe how use flux differencing, quadrature-based projections, and
SBP-like operators to construct discretely entropy conservative schemes for DG
methods under more arbitrary choices of volume and surface quadrature rules.
The resulting methods are semi-discretely entropy conservative or entropy
stable with respect to the volume quadrature rule used. Numerical experiments
confirm the stability and high order accuracy of the proposed methods for the
compressible Euler equations in one and two dimensions
Data Assimilation for hyperbolic conservation laws. A Luenberger observer approach based on a kinetic description
Developing robust data assimilation methods for hyperbolic conservation laws
is a challenging subject. Those PDEs indeed show no dissipation effects and the
input of additional information in the model equations may introduce errors
that propagate and create shocks. We propose a new approach based on the
kinetic description of the conservation law. A kinetic equation is a first
order partial differential equation in which the advection velocity is a free
variable. In certain cases, it is possible to prove that the nonlinear
conservation law is equivalent to a linear kinetic equation. Hence, data
assimilation is carried out at the kinetic level, using a Luenberger observer
also known as the nudging strategy in data assimilation. Assimilation then
resumes to the handling of a BGK type equation. The advantage of this framework
is that we deal with a single "linear" equation instead of a nonlinear system
and it is easy to recover the macroscopic variables. The study is divided into
several steps and essentially based on functional analysis techniques. First we
prove the convergence of the model towards the data in case of complete
observations in space and time. Second, we analyze the case of partial and
noisy observations. To conclude, we validate our method with numerical results
on Burgers equation and emphasize the advantages of this method with the more
complex Saint-Venant system
Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds
We investigate a class of scalar conservation laws on manifolds driven by
multiplicative Gaussian (Ito) noise. The Cauchy problem defined on a Riemannian
manifold is shown to be well-posed. We prove existence of generalized kinetic
solutions using the vanishing viscosity method. A rigidity result is derived,
which implies that generalized solutions are kinetic solutions and that kinetic
solutions are uniquely determined by their initial data ( contraction
principle). Deprived of noise, the equations we consider coincide with those
analyzed by Ben-Artzi and LeFloch (2007), who worked with Kruzkov-DiPerna
solutions. In the Euclidian case, the stochastic equations agree with those
examined by Debussche and Vovelle (2010).Comment: Submitted for publication on 23.09.1
Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms
We consider nonlinear hyperbolic conservation laws, posed on a differential
(n+1)-manifold with boundary referred to as a spacetime, and in which the
"flux" is defined as a flux field of n-forms depending on a parameter (the
unknown variable). We introduce a formulation of the initial and boundary value
problem which is geometric in nature and is more natural than the vector field
approach recently developed for Riemannian manifolds. Our main assumption on
the manifold and the flux field is a global hyperbolicity condition, which
provides a global time-orientation as is standard in Lorentzian geometry and
general relativity. Assuming that the manifold admits a foliation by compact
slices, we establish the existence of a semi-group of entropy solutions.
Moreover, given any two hypersurfaces with one lying in the future of the
other, we establish a "contraction" property which compares two entropy
solutions, in a (geometrically natural) distance equivalent to the L1 distance.
To carry out the proofs, we rely on a new version of the finite volume method,
which only requires the knowledge of the given n-volume form structure on the
(n+1)-manifold and involves the {\sl total flux} across faces of the elements
of the triangulations, only, rather than the product of a numerical flux times
the measure of that face.Comment: 26 page
Affordable, Entropy Conserving and Entropy Stable Flux Functions for the Ideal MHD Equations
In this work, we design an entropy stable, finite volume approximation for
the ideal magnetohydrodynamics (MHD) equations. The method is novel as we
design an affordable analytical expression of the numerical interface flux
function that discretely preserves the entropy of the system. To guarantee the
discrete conservation of entropy requires the addition of a particular source
term to the ideal MHD system. Exact entropy conserving schemes cannot dissipate
energy at shocks, thus to compute accurate solutions to problems that may
develop shocks, we determine a dissipation term to guarantee entropy stability
for the numerical scheme. Numerical tests are performed to demonstrate the
theoretical findings of entropy conservation and robustness.Comment: arXiv admin note: substantial text overlap with arXiv:1509.06902;
text overlap with arXiv:1007.2606 by other author
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