The Special Case of 2-Velocity Kinetic Models
This chapter deals mainly with the numerical analysis of the following one-dimensional system of semilinear equations,$$ \begin{array}{ccc} {\partial}_t{f}^{\pm}\pm {\partial}_x{f}^{\pm }=G\left({f}^{+},{f}^{-}\right), & x\in \mathbb{R}, & t>0,
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The Special Case of 2-Velocity Kinetic Models
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This chapter deals mainly with the numerical analysis of the following one-dimensional system of semilinear equations,
∂t f ± ± ∂x f ± = ∓G(f + , f − ),
x ∈ R,
t > 0,
(8.1)
in both rarefied and diffusive regimes, the latter being obtained through the transformation t → t/ε 2 , x → x/ε , see (8.27). The kinetic densities 0 ≤ f ± are supposed to be at least bounded variation (BV) functions, [40], in the space variable. In order to ensure some stability properties, namely L1 (R)-contraction, [26, 34, 39], we ask for the so-called quasi-monotonicity of the right-hand side which reads: G ∈ C1 (R2 ); G(0, 0) = 0,
def
∂+ G =
∂G ≥ 0, ∂f+
def
∂− G =
∂G < 0. ∂f−
(8.2)
This matches essentially the standard hypotheses encountered in [12, 30, 31], with the notable exception of [32] in which compactness results are established by means of a different methodology. The objective is then to develop and study robust numerical processes for (8.1), (8.27), stable and reliable on the whole range 0 ≤ ε ≤ 1. Before stating anything, let’s remark that a 2-velocity kinetic model possesses the same amount of both microscopic and macroscopic variables; hence its relevance when it comes to modeling real-life processes remains doubtful. However, its peculiar structure allows for interesting mathematical calculations which led to a major rethinking about well-balanced methods, namely about their ability to handle correctly asymptotic limits of kinetic equations within a diffusive rescaling.
8.1 A Localization Process for the Collisional Term We mimic hereafter the localization procedure which has been set up in Chapter 2 for building the well-balanced Godunov scheme on solid mathematical ground. L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws, SIMAI Springer Series 2, DOI 10.1007/978-88-470-2892-0_8, © Springer-Verlag Italia 2013
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8 The Special Case of 2-Velocity Kinetic Models
8.1.1 Uniform BV Estimates and Strong Compactness Given a parameter h > 0, one considers the Cauchy problem for 1 ≥ ε > 0:
∂t f ± ± ∂x f ± = ∓G(f + , f − )∂x aε ,
0 ≤ f ± (0, x) = f0± (x) ∈ L1 ∩ BV(R).
(8.3)
We assume that aε is Lipschitz continuous for ε > 0, more precisely: ⎧ 1 ε ⎪ ⎪ jh, for x ∈ jh, j + − h , ⎪ ⎪ ⎪ 2 2 ⎨ aε (x) = x + j + 1 h 1 − 1 , for x ∈ j + 1 − ε h, j + 1 + ε h , (8.4) ⎪ ε 2 ε 2 2 2 2 ⎪ ⎪ 1 ε ⎪ ⎪ ⎩ h, (j + 1)h . (j + 1)h, for x ∈ j+ + 2 2 This means that aε =1 (x) = x, aε ∈ BVloc (R) uniformly in ε , ∂x aε ≥ 0 and moreover, (1A stands for the characteristic function of a set A) ε →0
aε −→
∑ jh1](j− 12 )h,(j+ 12 )h] ,
h > 0.
j∈Z
From [26, 34], the Cauchy problem (8.3) is well-posed for any ε > 0 but becomes ambiguous in the limit ε → 0 because a so-called nonconservati
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