128 research outputs found
Relative entropy in multi-phase models of 1d elastodynamics: Convergence of a non-local to a local model
In this paper we study a local and a non-local regularization of the system
of nonlinear elastodynamics with a non-convex energy. We show that solutions of
the non-local model converge to those of the local model in a certain regime.
The arguments are based on the relative entropy framework and provide an
example how local and non-local regularizations may compensate for
non-convexity of the energy and enable the use of the relative entropy
stability theory -- even if the energy is not quasi- or poly-convex
Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model
We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of Allen--Cahn/Cahn--Hilliard/Navier--Stokes--Korteweg type which allows for phase transitions. We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system
A posteriori analysis of fully discrete method of lines DG schemes for systems of conservation laws
We present reliable a posteriori estimators for some fully discrete schemes
applied to nonlinear systems of hyperbolic conservation laws in one space
dimension with strictly convex entropy. The schemes are based on a method of
lines approach combining discontinuous Galerkin spatial discretization with
single- or multi-step methods in time. The construction of the estimators
requires a reconstruction in time for which we present a very general framework
first for odes and then apply the approach to conservation laws. The
reconstruction does not depend on the actual method used for evolving the
solution in time. Most importantly it covers in addition to implicit methods
also the wide range of explicit methods typically used to solve conservation
laws. For the spatial discretization, we allow for standard choices of
numerical fluxes. We use reconstructions of the discrete solution together with
the relative entropy stability framework, which leads to error control in the
case of smooth solutions. We study under which conditions on the numerical flux
the estimate is of optimal order pre-shock. While the estimator we derive is
computable and valid post-shock for fixed meshsize, it will blow up as the
meshsize tends to zero. This is due to a breakdown of the relative entropy
framework when discontinuities develop. We conclude with some numerical
benchmarking to test the robustness of the derived estimator
Geometric Error of Finite Volume Schemes for Conservation Laws on Evolving Surfaces
This paper studies finite volume schemes for scalar hyperbolic conservation
laws on evolving hypersurfaces of . We compare theoretical
schemes assuming knowledge of all geometric quantities to (practical) schemes
defined on moving polyhedra approximating the surface. For the former schemes
error estimates have already been proven, but the implementation of such
schemes is not feasible for complex geometries. The latter schemes, in
contrast, only require (easily) computable geometric quantities and are thus
more useful for actual computations. We prove that the difference between
approximate solutions defined by the respective families of schemes is of the
order of the mesh width. In particular, the practical scheme converges to the
entropy solution with the same rate as the theoretical one. Numerical
experiments show that the proven order of convergence is optimal.Comment: 23 pages, 5 figures, to appear in Numerische Mathemati
Formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary
We study nonlinear hyperbolic conservation laws posed on a differential
(n+1)-manifold with boundary referred to as a spacetime, and defined from a
prescribed flux field of n-forms depending on a parameter (the unknown
variable), a class of equations proposed by LeFloch and Okutmustur in 2008. Our
main result is a proof of the convergence of the finite volume method for weak
solutions satisfying suitable entropy inequalities. A main difference with
previous work is that we allow for slices with a boundary and, in addition,
introduce a new formulation of the finite volume method involving the notion of
total flux functions. Under a natural global hyperbolicity condition on the
flux field and the spacetime and by assuming that the spacetime admits a
foliation by compact slices with boundary, we establish an existence and
uniqueness theory for the initial and boundary value problem, and we prove a
contraction property in a geometrically natural L1-type distance.Comment: 32 page
On cavitation in Elastodynamics
Motivated by the works of Ball (1982) and Pericak-Spector and Spector (1988), we investigate singular solutions of the compressible nonlinear elastodynamics equations.
These singular solutions contain discontinuities in the displacement field and
can be seen as describing fracture or cavitation.
We explore a definition of singular solution via approximating sequences of smooth functions.
We use these approximating sequences to investigate the energy of such solutions, taking into account the energy needed to open a crack or hole.
In particular, we find that the existence of singular solutions and the finiteness of their energy
is strongly related to the behavior of the stress response function for infinite stretching, i.e.
the material has to display a sufficient amount of softening.
In this note we detail our findings in one space dimension
Singular limiting induced from continuum solutions and the problem of dynamic cavitation
In the works of
K.A. Pericak-Spector and S. Spector [Pericak-Spector, Spector 1988, 1997] a class of self-similar
solutions are constructed for the equations of radial isotropic elastodynamics
that describe cavitating solutions. Cavitating solutions decrease the total
mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions
(for polyconvex energies) due to point-singularities at the cavity. To resolve
this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution),
according to which a discontinuous motion is a slic-solution if its averages
form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for
creating the cavity, which is captured by the notion of slic-solution but neglected by the
usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the
cavitating solution is in fact larger than that of the homogeneously deformed state.
We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture,
and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan
Stability properties of the Euler-Korteweg system with nonmonotone pressures
We establish a relative energy framework for the Euler-Korteweg system with
non-convex energy. This allows us to prove weak-strong uniqueness and to show
convergence to a Cahn-Hilliard system in the large friction limit. We also use
relative energy to show that solutions of Euler-Korteweg with convex energy
converge to solutions of the Euler system in the vanishing capillarity limit,
as long as the latter admits sufficiently regular strong solutions
A compressible mixture model with phase transition
We introduce a new thermodynamically consistent diffuse interface model of Allen--Cahn/Navier--Stokes type for multi-component flows with phase transitions and chemical reactions.
For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques.
We consider two scaling regimes, i.e.~a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized
Allen-Cahn/Euler system for mixtures with chemical
reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a Young--Laplace and a Stefan type law
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