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KRM

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. 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 has the expected
order up to the numerical noise from the optimization routine,
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.

keywords:
moment models
,
realizability-preserving
,
kinetic scheme
,
Radiation transport
,
high order
,
WENO.
,
realizability

KRM

We consider the simplest member of the hierarchy of the extended quadrature
method of moments (EQMOM), which gives equations for the zeroth-, first-, and
second-order moments of the energy density of photons in the radiative
transfer equations in slab geometry.
First we show that the equations are well-defined for all moment vectors
consistent with a nonnegative underlying distribution, and that the
reconstruction is explicit and therefore computationally inexpensive.
Second, we show that the resulting moment equations are hyperbolic.
These two properties make this moment method quite similar to the attractive
but far more expensive $M_2$ method.
We confirm through numerical solutions to several benchmark problems that the
methods give qualitatively similar results.

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