85 research outputs found
A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry
We develop a high-order kinetic scheme for entropy-based moment models of a
one-dimensional linear kinetic equation in slab geometry. High-order spatial
reconstructions are achieved using the weighted essentially non-oscillatory
(WENO) method, and for time integration we use multi-step Runge-Kutta methods
which are strong stability preserving and whose stages and steps can be written
as convex combinations of forward Euler steps. We show that the moment vectors
stay in the realizable set using these time integrators along with a maximum
principle-based kinetic-level limiter, which simultaneously dampens spurious
oscillations in the numerical solutions. We present numerical results both on a
manufactured solution, where we perform convergence tests showing our scheme
converges of the expected order up to the numerical noise from the numerical
optimization, as well as on two standard benchmark problems, where we show some
of the advantages of high-order solutions and the role of the key parameter in
the limiter
Spatial second-order positive and asymptotic preserving filtered schemes for nonlinear radiative transfer equations
A spatial second-order scheme for the nonlinear radiative transfer equations
is introduced in this paper. The discretization scheme is based on the filtered
spherical harmonics () method for the angular variable and the unified
gas kinetic scheme (UGKS) framework for the spatial and temporal variables
respectively. In order to keep the scheme positive and second-order accuracy,
firstly, we use the implicit Monte Carlo linearization method [6] in the
construction of the UGKS numerical boundary fluxes. Then, by carefully
analyzing the constructed second-order fluxes involved in the macro-micro
decomposition, which is induced by the angular discretization, we
establish the sufficient conditions that guarantee the positivity of the
radiative energy density and material temperature. Finally, we employ linear
scaling limiters for the angular variable in the reconstruction and for
the spatial variable in the piecewise linear slopes reconstruction
respectively, which are shown to be realizable and reasonable to enforce the
sufficient conditions holding. Thus, the desired scheme, called the
-based UGKS, is obtained. Furthermore, in the regime
and the regime , a simplified spatial second-order scheme,
called the -based SUGKS, is presented, which possesses all the
properties of the non-simplified one. Inheriting the merit of UGKS, the
proposed schemes are asymptotic preserving. By employing the method for
the angular variable, the proposed schemes are almost free of ray effects. To
our best knowledge, this is the first time that spatial second-order, positive,
asymptotic preserving and almost free of ray effects schemes are constructed
for the nonlinear radiative transfer equations without operator splitting.
Various numerical experiments are included to validate the properties of the
proposed schemes
A Finite Element Method for Angular Discretization of the Radiation Transport Equation on Spherical Geodesic Grids
Discrete ordinate () and filtered spherical harmonics () based
schemes have been proven to be robust and accurate in solving the Boltzmann
transport equation but they have their own strengths and weaknesses in
different physical scenarios. We present a new method based on a finite element
approach in angle that combines the strengths of both methods and mitigates
their disadvantages. The angular variables are specified on a spherical
geodesic grid with functions on the sphere being represented using a finite
element basis. A positivity-preserving limiting strategy is employed to prevent
non-physical values from appearing in the solutions. The resulting method is
then compared with both and schemes using four test problems and
is found to perform well when one of the other methods fail.Comment: 24 pages, 13 figure
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