A convergent method for linear half-space kinetic equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1583-1615.

We give a unified proof for the well-posedness of a class of linear half-space equations with general incoming data and construct a Galerkin method to numerically resolve this type of equations in a systematic way. Our main strategy in both analysis and numerics includes three steps: adding damping terms to the original half-space equation, using an inf-sup argument and even-odd decomposition to establish the well-posedness of the damped equation, and then recovering solutions to the original half-space equation. The proposed numerical methods for the damped equation is shown to be quasi-optimal and the numerical error of approximations to the original equation is controlled by that of the damped equation. This efficient solution to the half-space problem is useful for kinetic-fluid coupling simulations.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016076
Classification : 35F15, 35Q79
Mots clés : Half-space equations, boundary layer, kinetic-fluid coupling, Galerkin method
Li, Qin 1 ; Lu, Jianfeng 2 ; Sun, Weiran 3

1 Computing and Mathematical Sciences, California Institute of Technology, 1200 E California Blvd. MC 305-16, Pasadena, CA 91125 USA.
2 Department of Mathematics, Department of Physics, and Department of Chemistry, Duke University, Box 90320, Durham, NC 27708 USA.
3 Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada.
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     title = {A convergent method for linear half-space kinetic equations},
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     pages = {1583--1615},
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Li, Qin; Lu, Jianfeng; Sun, Weiran. A convergent method for linear half-space kinetic equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1583-1615. doi : 10.1051/m2an/2016076. http://www.numdam.org/articles/10.1051/m2an/2016076/

I. Babuška and A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method. In The Mathematical foundation of the Finite Element method with Applications to Partial Differential Equations, edited by A.K. Aziz. Academic Press, New York (1972) 1–359. | MR | Zbl

C. Besse, S. Borghol, T. Goudon, I. Lacroix-Violet and J.-P. Dudon, Hydrodynamic regimes, Knudsen layer, numerical schemes: definition of boundary fluxes. Adv. Appl. Math. Mech. 3 (2011) 519–561. | DOI | MR | Zbl

A. Bensoussan, J.L. Lions and G.C. Papanicolaou, Boundary-layers and homogenization of transport processes. J. Publ. RIMS Kyoto Univ. 15 (1979) 53–157. | DOI | MR | Zbl

C. Bardos, R. Santos and R Sentis, Diffusion approximation and computation of the critical size. Trans. Amer. Math. Soc. 284 (1984) 617–649. | DOI | MR | Zbl

C. Bardos and X. Yang, The classification of well-posed kinetic boundary layer for hard sphere gas mixtures. Commun. Partial Differ. Equ. 37 (2012) 1286–1314. | DOI | MR | Zbl

Y. Cheng, I. Gamba and J. Proft, Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Bboltzmann transport equations. Math. Comput. 81 (2012) 153–190. | DOI | MR | Zbl

F. Coron, F. Golse and C. Sulem, A classification of well-posed kinetic layer problems. Commun. Pure Appl. Math. 41 (1988) 409–435. | DOI | MR | Zbl

C.-C. Chen, T.-P. Liu and T. Yang, Existence of boundary layer solutions to the Boltzmann equation. Anal. Appl. 2 (2004) 337–363. | DOI | MR | Zbl

F. Coron, Computation of the asymptotic states for linear half space kinetic problems. Transport Theory Statist. Phys. 19 (1990) 89–114. | DOI | MR | Zbl

Z. Chen and C.-W. Shu, Recovering exponential accuracy from collocation point values of smooth functions with end-point singularities. J. Comput. Appl. Math. 265 (2014) 83–95. | DOI | MR | Zbl

Z. Chen and C.-W. Shu, Recovering exponential accuracy in Fourier spectral methods involving piecewise smooth functions with unbounded derivative singularities. J. Sci. Comput. (2015) 121. | MR

S. Dellacherie, Coupling of the Wang Chang-Uhlenbeck equations with the multispecies Euler system. J. Comput. Phys. (2003) 189. | MR | Zbl

P. Degond and S. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker-Planck equation. Transport Theory Statist. Phys. 16 (1987) 589–636. | DOI | MR | Zbl

H. Egger and M. Schlottbom, A mixed variational framework for the radiative transfer equation. Math. Models Methods Appl. Sci. 22 (2012) 1150014. | DOI | MR | Zbl

F. Golse and A. Klar, A numerical method for computing asymptotic states and outgoing distributions for kinetic linear half-space problems. J. Stat. Phys. 80 (1995) 1033–1061. | DOI | MR | Zbl

F. Golse. Analysis of the boundary layer equation in the kinetic theory of gases. Bull. Inst. Math. Acad. Sin. (N.S.) 3 (2008) 211–242. | MR | Zbl

D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon and its resolution. SIAM Rev. 30 (1997) 644–668. | DOI | MR | Zbl

S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations. SIAM J. Numer. Anal. 38 (2001) 913–936. | DOI | MR | Zbl

Q. Li, J. Lu and W. Sun, Half-space kinetic equations with general boundary conditions. Math. Comp. 86 (2017) 1269–1301 | DOI | MR | Zbl

R.E. Marshak, Note on the spherical harmonic method as applied to the Milne problem for a sphere. Phys. Rev. 71 (1947) 443–446. | DOI | MR | Zbl

B. Shizgal. A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems. J. Comput. Phys. 41 (1981) 309–328. | DOI | MR | Zbl

S. Ukai, T. Yang and S.-H. Yu, Nonlinear boundary layers of the Boltzmann equation. I. Existence. Commun. Math. Phys. 236 (2003) 373–393. | DOI | MR | Zbl

W. Wang, T. Yang and X. Yang, Nonlinear stability of boundary layers of the Boltzmann equation for cutoff hard potentials. J. Math. Phys. 47 (2006) 083301. | DOI | MR | Zbl

W. Wang, T. Yang and X. Yang, Existence of boundary layers to the Boltzmann equation with cutoff soft potentials. J. Math. Phys. 48 (2007) 073304. | DOI | MR | Zbl

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