27 research outputs found
Existence of dissipative solutions to the compressible Navier-Stokes system with potential temperature transport
We introduce dissipative solutions to the compressible Navier-Stokes system
with potential temperature transport motivated by the concept of Young
measures. We prove their global-in-time existence by means of convergence
analysis of a mixed finite element-finite volume method. If a classical
solution to the compressible Navier-Stokes system with potential temperature
transport exists, we prove the strong convergence of numerical solutions. Our
results hold for the full range of adiabatic indices including the physically
relevant cases in which the existence of global-in-time weak solutions is open
Energy-stable linear schemes for polymer-solvent phase field models
We present new linear energy-stable numerical schemes for numerical
simulation of complex polymer-solvent mixtures. The mathematical model proposed
by Zhou, Zhang and E (Physical Review E 73, 2006) consists of the Cahn-Hilliard
equation which describes dynamics of the interface that separates polymer and
solvent and the Oldroyd-B equations for the hydrodynamics of polymeric
mixtures. The model is thermodynamically consistent and dissipates free energy.
Our main goal in this paper is to derive numerical schemes for the
polymer-solvent mixture model that are energy dissipative and efficient in
time. To this end we will propose several problem-suited time discretizations
yielding linear schemes and discuss their properties
Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part I: A nonlinear scheme
We present a nonlinear stabilized Lagrange-Galerkin scheme for the Oseen-type
Peterlin viscoelastic model. Our scheme is a combination of the method of
characteristics and Brezzi-Pitk\"aranta's stabilization method for the
conforming linear elements, which yields an efficient computation with a small
number of degrees of freedom. We prove error estimates with the optimal
convergence order without any relation between the time increment and the mesh
size. The result is valid for both the diffusive and non-diffusive models for
the conformation tensor in two space dimensions. We introduce an additional
term that yields a suitable structural property and allows us to obtain
required energy estimate. The theoretical convergence orders are confirmed by
numerical experiments. In a forthcoming paper, Part II, a linear scheme is
proposed and the corresponding error estimates are proved in two and three
space dimensions for the diffusive model.Comment: See arXiv:1603.01074 for Part II: a linear schem
An implicit-explicit solver for a two-fluid single-temperature model
We present an implicit-explicit finite volume scheme for two-fluid
single-temperature flow in all Mach number regimes which is based on a
symmetric hyperbolic thermodynamically compatible description of the fluid
flow. The scheme is stable for large time steps controlled by the interface
transport and is computational efficient due to a linear implicit character.
The latter is achieved by linearizing along constant reference states given by
the asymptotic analysis of the single-temperature model. Thus, the use of a
stiffly accurate IMEX Runge Kutta time integration and the centered treatment
of pressure based quantities provably guarantee the asymptotic preserving
property of the scheme for weakly compressible Euler equations with variable
volume fraction. The properties of the first and second order scheme are
validated by several numerical test cases
Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part II: A linear scheme
This is the second part of our error analysis of the stabilized
Lagrange-Galerkin scheme applied to the Oseen-type Peterlin viscoelastic model.
Our scheme is a combination of the method of characteristics and
Brezzi-Pitk\"aranta's stabilization method for the conforming linear elements,
which leads to an efficient computation with a small number of degrees of
freedom especially in three space dimensions. In this paper, Part II, we apply
a semi-implicit time discretization which yields the linear scheme. We
concentrate on the diffusive viscoelastic model, i.e. in the constitutive
equation for time evolution of the conformation tensor a diffusive effect is
included. Under mild stability conditions we obtain error estimates with the
optimal convergence order for the velocity, pressure and conformation tensor in
two and three space dimensions. The theoretical convergence orders are
confirmed by numerical experiments.Comment: See arXiv:1603.01339 for Part I: a nonlinear schem
A multi-scale method for complex flows of non-Newtonian fluids
We introduce a new heterogeneous multi-scale method for the simulation of
flows of non-Newtonian fluids in general geometries and present its application
to paradigmatic two-dimensional flows of polymeric fluids. Our method combines
micro-scale data from non-equilibrium molecular dynamics (NEMD) with
macro-scale continuum equations to achieve a data-driven prediction of complex
flows. At the continuum level, the method is model-free, since the Cauchy
stress tensor is determined locally in space and time from NEMD data. The
modelling effort is thus limited to the identification of suitable interaction
potentials at the micro-scale. Compared to previous proposals, our approach
takes into account the fact that the material response can depend strongly on
the local flow type and we show that this is a necessary feature to correctly
capture the macroscopic dynamics. In particular, we highlight the importance of
extensional rheology in simulating generic flows of polymeric fluids.Comment: 18 pages, 9 figure