61 research outputs found
Entropy conserving/stable schemes for a vector-kinetic model of hyperbolic systems
The moment of entropy equation for vector-BGK model results in the entropy
equation for macroscopic model. However, this is usually not the case in
numerical methods because the current literature consists only of entropy
conserving/stable schemes for macroscopic model (to the best of our knowledge).
In this paper, we attempt to fill this gap by developing an entropy conserving
scheme for vector-kinetic model, and we show that the moment of this results in
an entropy conserving scheme for macroscopic model. With the numerical
viscosity of entropy conserving scheme as reference, the entropy stable scheme
for vector-kinetic model is developed in the spirit of [33]. We show that the
moment of this scheme results in an entropy stable scheme for macroscopic
model. The schemes are validated on several benchmark test problems for scalar
and shallow water equations, and conservation/stability of both kinetic and
macroscopic entropies are presented
A stability property for a mono-dimensional three velocities scheme with relative velocity
In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative velocity schemes, we derive necessary and sufficient conditions for the non-negativity preserving property. These conditions are then expressed in a simple way when the relative velocity is reduced to zero. For the general case, we propose some simple necessary conditions on the relaxation parameters and we put in evidence numerically the non-negativity preserving regions. Numerical experiments show finally that no oscillations occur for the propagation of a non-smooth profile if the non-negativity preserving property is satisfied
A kinetic scheme with variable velocities and relative entropy
A new kinetic model is proposed where the equilibrium distribution with
bounded support has a range of velocities about two average velocities in 1D.
In 2D, the equilibrium distribution function has a range of velocities about
four average velocities, one in each quadrant. In the associated finite volume
scheme, the average velocities are used to enforce the Rankine-Hugoniot jump
conditions for the numerical diffusion at cell-interfaces, thereby capturing
steady discontinuities exactly. The variable range of velocities is used to
provide additional diffusion in smooth regions. Further, a novel kinetic theory
based expression for relative entropy is presented which, along with an
additional criterion, is used to identify expansions and smooth flow regions.
Appropriate flow tangency and far-field boundary conditions are formulated for
the proposed kinetic model. Several benchmark 1D and 2D compressible flow test
cases are solved to demonstrate the efficacy of the proposed solver.Comment: 53 page
A new discrete velocity method for Navier-Stokes equations
The relation between Latttice Boltzmann Method, which has recently become
popular, and the Kinetic Schemes, which are routinely used in Computational
Fluid Dynamics, is explored. A new discrete velocity model for the numerical
solution of the Navier-Stokes equations for incompressible fluid flow is
presented by combining both the approaches. The new scheme can be interpreted
as a pseudo-compressibility method and, for a particular choice of parameters,
this interpretation carries over to the Lattice Boltzmann Method.Comment: 28 pages, 8 figure
- …