1,069 research outputs found
A three-dimensional multidimensional gas-kinetic scheme for the Navier-Stokes equations under gravitational fields
This paper extends the gas-kinetic scheme for one-dimensional inviscid
shallow water equations (J. Comput. Phys. 178 (2002), pp. 533-562) to
multidimensional gas dynamic equations under gravitational fields. Four
important issues in the construction of a well-balanced scheme for gas dynamic
equations are addressed. First, the inclusion of the gravitational source term
into the flux function is necessary. Second, to achieve second-order accuracy
of a well-balanced scheme, the Chapman-Enskog expansion of the Boltzmann
equation with the inclusion of the external force term is used. Third, to avoid
artificial heating in an isolated system under a gravitational field, the
source term treatment inside each cell has to be evaluated consistently with
the flux evaluation at the cell interface. Fourth, the multidimensional
approach with the inclusion of tangential gradients in two-dimensional and
three-dimensional cases becomes important in order to maintain the accuracy of
the scheme. Many numerical examples are used to validate the above issues,
which include the comparison between the solutions from the current scheme and
the Strang splitting method. The methodology developed in this paper can also
be applied to other systems, such as semi-conductor device simulations under
electric fields.Comment: The name of first author was misspelled as C.T.Tian in the published
paper. 35 pages,9 figure
A Comparative Study of an Asymptotic Preserving Scheme and Unified Gas-kinetic Scheme in Continuum Flow Limit
Asymptotic preserving (AP) schemes are targeting to simulate both continuum
and rarefied flows. Many AP schemes have been developed and are capable of
capturing the Euler limit in the continuum regime. However, to get accurate
Navier-Stokes solutions is still challenging for many AP schemes. In order to
distinguish the numerical effects of different AP schemes on the simulation
results in the continuum flow limit, an implicit-explicit (IMEX) AP scheme and
the unified gas kinetic scheme (UGKS) based on Bhatnagar-Gross-Krook (BGk)
kinetic equation will be applied in the flow simulation in both transition and
continuum flow regimes. As a benchmark test case, the lid-driven cavity flow is
used for the comparison of these two AP schemes. The numerical results show
that the UGKS captures the viscous solution accurately. The velocity profiles
are very close to the classical benchmark solutions. However, the IMEX AP
scheme seems have difficulty to get these solutions. Based on the analysis and
the numerical experiments, it is realized that the dissipation of AP schemes in
continuum limit is closely related to the numerical treatment of collision and
transport of the kinetic equation. Numerically it becomes necessary to couple
the convection and collision terms in both flux evaluation at a cell interface
and the collision source term treatment inside each control volume
A Compact Third-order Gas-kinetic Scheme for Compressible Euler and Navier-Stokes Equations
In this paper, a compact third-order gas-kinetic scheme is proposed for the
compressible Euler and Navier-Stokes equations. The main reason for the
feasibility to develop such a high-order scheme with compact stencil, which
involves only neighboring cells, is due to the use of a high-order gas
evolution model. Besides the evaluation of the time-dependent flux function
across a cell interface, the high-order gas evolution model also provides an
accurate time-dependent solution of the flow variables at a cell interface.
Therefore, the current scheme not only updates the cell averaged conservative
flow variables inside each control volume, but also tracks the flow variables
at the cell interface at the next time level. As a result, with both cell
averaged and cell interface values the high-order reconstruction in the current
scheme can be done compactly. Different from using a weak formulation for
high-order accuracy in the Discontinuous Galerkin (DG) method, the current
scheme is based on the strong solution, where the flow evolution starting from
a piecewise discontinuous high-order initial data is precisely followed. The
cell interface time-dependent flow variables can be used for the initial data
reconstruction at the beginning of next time step. Even with compact stencil,
the current scheme has third-order accuracy in the smooth flow regions, and has
favorable shock capturing property in the discontinuous regions. Many test
cases are used to validate the current scheme. In comparison with many other
high-order schemes, the current method avoids the use of Gaussian points for
the flux evaluation along the cell interface and the multi-stage Runge-Kutta
time stepping technique.Comment: 27 pages, 38 figure
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