# A convergent method for linear half-space kinetic equations

@article{Li2014ACM, title={A convergent method for linear half-space kinetic equations}, author={Qin Li and Jianfeng Lu and Weiran Sun}, journal={arXiv: Analysis of PDEs}, year={2014} }

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… Expand

#### 19 Citations

Half-space kinetic equations with general boundary conditions

- Mathematics, Physics
- Math. Comput.
- 2017

The main technique is a damping adding-removing procedure, which establishes the well-posedness of linear (or linearized) half-space equations with general boundary conditions and quasi-optimality of the numerical scheme. Expand

Error analysis of an asymptotic preserving dynamical low-rank integrator for the multi-scale radiative transfer equation

- Mathematics, Computer Science
- ArXiv
- 2019

This work performs an error analysis for a dynamical low-rank algorithm applied to a classical model in kinetic theory, namely the radiative transfer equation, and proves that the scheme dynamically and automatically captures the low rank structure of the solution, and preserves the fluid limit on the numerical level. Expand

A numerical method for coupling the BGK model and Euler equations through the linearized Knudsen layer

- Mathematics, Physics
- J. Comput. Phys.
- 2019

A full domain numerical solver is developed with a domain-decomposition approach, where the Euler solver and kinetic solver are applied on the appropriate subdomains and connected via the half-space solver. Expand

Diffusion approximations and domain decomposition method of linear transport equations: Asymptotics and numerics

- Computer Science, Mathematics
- J. Comput. Phys.
- 2015

This paper construct numerical schemes to approximate linear transport equations with slab geometry by diffusion equations by applying the half-space solver to resolve the boundary layer equation and obtain the boundary data for the diffusion equation. Expand

Random Sampling and Efficient Algorithms for Multiscale PDEs

- Computer Science, Mathematics
- SIAM J. Sci. Comput.
- 2020

A numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition and achieves the asymptotic preserving property for the kinetic equations and numerical homogenization for the elliptic equations is described. Expand

Extending the range of validity of Fourier's law into the kinetic transport regime via asymptotic solution of the phonon Boltzmann transport equation

- Physics, Mathematics
- 2016

We derive the continuum equations and boundary conditions governing phonon-mediated heat transfer in the limit of small but finite mean free path from asymptotic solution of the linearized Boltzmann… Expand

Kinetic Layers and Coupling Conditions for Macroscopic Equations on Networks I: The Wave Equation

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 2018

A new approximate method for the solution of kinetic half-space problems is derived and used for the determination of the coupling conditions, and numerical comparisons between the solutions of the macroscopic equation with different coupling conditions and the kinetic solution are presented. Expand

Stability of Stationary Inverse Transport Equation in Diffusion Scaling

- Mathematics
- 2017

We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales… Expand

Asymptotic Analysis of Unsteady Neutron Transport Equation

- Physics, Mathematics
- 2018

Consider the unsteady neutron transport equation with diffusive boundary condition in 2D convex domains. We establish the diffusive limit with both initial layer and boundary layer corrections. The… Expand

Boundary Layer of Transport Equation with In-Flow Boundary

- Physics, Mathematics
- Archive for Rational Mechanics and Analysis
- 2019

Consider the steady neutron transport equation in two dimensional convex domains with an in-flow boundary condition. We establish the diffusive limit while the boundary layers are present. Our… Expand

#### References

SHOWING 1-10 OF 32 REFERENCES

A classification of well‐posed kinetic layer problems

- Mathematics
- 1988

In the first part of this paper, we study the half space boundary value problem for the Boltzmann equation with an incoming distribution, obtained when considering the boundary layer arising in the… Expand

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations

- Mathematics, Computer Science
- Math. Comput.
- 2012

This work extends the maximum-principle-satisfying scheme for scalar conservation laws in a recent work by X. Zhang and C. Shu to include the linear Boltzmann collision term and develops a high-order positivity-preserving discontinuous Galerkin scheme for linear Vlasov-Boltzmann transport equations. Expand

A numerical method for computing asymptotic states and outgoing distributions for kinetic linear half-space problems

- Mathematics
- 1995

Linear half-space problems can be used to solve domain decomposition problems between Boltzmann and aerodynamic equations. A new fast numerical method computing the asymptotic states and outgoing… Expand

Nonlinear Boundary Layers of the Boltzmann Equation: I. Existence

- Mathematics
- 2003

Abstract: We study the half-space problem of the nonlinear Boltzmann equation, assigning the Dirichlet data for outgoing particles at the boundary and a Maxwellian as the far field. We will show that… Expand

Existence of boundary layer solutions to the Boltzmann equation

- Mathematics
- 2004

In this paper, we consider the existence of boundary layer solutions to the Boltzmann equation for a hard potential with angular cut-off. The boundary condition is imposed for incoming particles of… Expand

Errata: Computation of the asymptotic states for linear half space kinetic problems

- Mathematics
- 1990

Abstract A spectral numerical scheme computing the asymptotic states for linear half space problems is described in the case of a simple transport equation and the linearized Bhatnagar-Gross-Krook… Expand

A MIXED VARIATIONAL FRAMEWORK FOR THE RADIATIVE TRANSFER EQUATION

- Mathematics
- 2012

We present a rigorous variational framework for the analysis and discretization of the radiative transfer equation. Existence and uniqueness of weak solutions are established under rather general… Expand

A smooth transition model between kinetic and hydrodynamic equations

- Mathematics
- 2005

This paper presents a model which provides a smooth transition between a kinetic and a hydrodynamic domain. The idea is to use a buffer zone, in which both hydrodynamics and kinetic equations will be… Expand

Existence of solutions and diffusion approximation for a model Fokker-Planck equation

- Mathematics
- 1987

Abstract We study a simplified model of the Fokker-Planck equation of plasma physics. This model only involves a linear angular diffusion for a monoenergetic beam. We discuss the problem of existence… Expand

Nonlinear stability of boundary layers of the Boltzmann equation for cutoff hard potentials

- Physics
- 2006

Many physical models have boundaries. For the Boltzmann equation, the study on the boundary layer in the region of the width in the order of the Kundsen number along the boundary is important both in… Expand