BAIL 2006 Book of Abstracts - Institut für Numerische und ...

N. PARUMASUR, J. BANASIAK, J.M. KOZAKIEWICZ: Numerical and Asymptotic
Analysis of Singularly Perturbed PDEs of Kinetic Theory
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where the superscript N indicates the order of the approximation and W are time-independent
bounded linear operators from V to W . Substituting this expansion into the first equation in
(3) yields
N�
∂tρ = PAPρ + ε n PSQ(Wnρ). (6)
n=0
Expressing the time derivative ∂tρ in (6) in powers of ε and comparing terms of the same power
in ε yields at first order
W0 = 0, W1 = −(QCQ) −1 QSP. (7)
The operator W1 can be evaluated since QCQ is invertible on the subspace W. Using (7) in (6)
gives the equation
∂tρ = PAPρ − εPSQ(QCQ) −1 QSPρ. (8)
A similar procedure yields the initial layer terms
and the initial condition for (8)
˜v0(τ) ≡ 0, ˜v1(τ) = PSQ(QCQ) −1 e τQCQ o w,
¯v(0) = o v −εPSQ(QCQ) −1 o w . (9)
We apply the procedure to a wide range of problems of kinetic theory.
References
[1] J. Banasiak, J. Kozakiewicz, N. Parumasur, Diffusion Approximation of Linear Kinetic
Equations with Non-equilibrium Data – Computational Experiments, Transport Theory
Statist. Phys. (accepted).
[2] J. R. Mika, New asymptotic expansion algorithm for singularly perturbed evolution equations,
Math. Methods Appl. Sci. 3 (1981) 172-188.
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Speaker: PARUMASUR, N. 131 BAIL 2006
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