24 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 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
An application of 3-D kinematical conservation laws: propagation of a 3-D wavefront
Three-dimensional (3-D) kinematical conservation laws (KCL) are equations of evolution of a propagating surface Omega(t) in three space dimensions. We start with a brief review of the 3-D KCL system and mention some of its properties relevant to this paper. The 3-D KCL, a system of six conservation laws, is an underdetermined system to which we add an energy transport equation for a small amplitude 3-D nonlinear wavefront propagating in a polytropic gas in a uniform state and at rest. We call the enlarged system of 3-D KCL with the energy transport equation equations of weakly nonlinear ray theory (WNLRT). We highlight some interesting properties of the eigenstructure of the equations of WNLRT, but the main aim of this paper is to test the numerical efficacy of this system of seven conservation laws. We take several initial shapes for a nonlinear wavefront with a suitable amplitude distribution on it and let it evolve according to the 3-D WNLRT. The 3-D WNLRT is a weakly hyperbolic 7 × 7 system that is highly nonlinear. Here we use the staggered Lax–Friedrichs and Nessyahu–Tadmor central schemes and have obtained some very interesting shapes of the wavefronts. We find the 3-D KCL to be suitable for solving many complex problems for which there presently seems to be no other method capable of giving such physically realistic features
A genuinely multi-dimensional relaxation scheme for hyperbolic conservation laws
A novel genuinely multi-dimensional relaxation scheme is proposed. Based on a new discrete velocity Boltzmann equation, which is an improvement over previously introduced relaxation systems in terms of isotropic coverage of the multi-dimensional domain by the foot of the characteristic, a finite volume method is developed in which the fluxes at the cell interfaces are evaluated in a genuinely multi-dimensional way, in contrast to the traditional dimension-by-dimension treatment. This algorithm is tested on some bench-mark test problems for hyperbolic conservation laws