Abstract
Two-dimensional dissipative and isotropic kinetic models, like the ones used in neutron transport theory, are considered. Especially, steady-states are expressed for constant opacity and damping, allowing to derive a scattering S-matrix and corresponding "truly 2D well-balanced" numerical schemes. A first scheme is obtained by directly implementing truncated Fourier–Bessel series, whereas another proceeds by applying an exponential modulation to a former, conservative, one. Consistency with the asymptotic damped parabolic approximation is checked for both algorithms. A striking property of some of these schemes is that they can be proved to be both 2D well-balanced and asymptotic-preserving in the parabolic limit, even when setting up IMEX time-integrators: see Corollaries 3.4 and A.1. These findings are further confirmed by means of practical benchmarks carried out on coarse Cartesian computational grids.
Highlights
Dissipative (2 + 1)-dimensional kinetic models Consider a damped kinetic model, where σ(x) ≥ 0 is the opacity, ∂tf + v · ∇f = (︂ ∫︁ σ(x) c(x)S1 f (t, x, v′) dv′ 2π − )︂ f
Two distinct numerical processes will be introduced: the first one, given in Section 3, strongly relies on the data of two-dimensional steady-states for (1.1). Such steady-states are derived in Section 2, following original ideas given in [5,6]: in both papers, it is shown that solutions of stationary elliptic equations yield, thanks to Laplace transforms, microscopic steady-states of related kinetic models
Two numerical schemes endowed with both 2D well-balanced (WB) and asymptotic-preserving (AP) properties, extending the one previously in [25], were studied in this paper
Summary
Dissipative (2 + 1)-dimensional kinetic models Consider a damped kinetic model, where σ(x) ≥ 0 is the opacity,. Following the roadmap proposed in [25], we intend to study numerical approximations of (1.2) endowed with both 2D well-balanced (WB) and asymptotic-preserving (AP) properties. To this end, two distinct numerical processes will be introduced: the first one, given, strongly relies on the data of two-dimensional steady-states for (1.1). Two distinct numerical processes will be introduced: the first one, given, strongly relies on the data of two-dimensional steady-states for (1.1) Such steady-states are derived, following original ideas given in [5,6]: in both papers, it is shown that solutions of stationary elliptic equations yield, thanks to Laplace transforms, microscopic steady-states of related kinetic models.
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