[Paper Review] Half-space Kinetic Equations with General Boundary Conditions

Qin Li, Jianfeng Lu+1 more|arXiv (Cornell University)|Sep 10, 2015

Gas Dynamics and Kinetic Theory3 references3 citations

TL;DR

This paper develops a numerical method for linear half-space kinetic equations with general boundary conditions, including incoming data and various reflection types (e.g., diffuse, specular, Maxwell). Using a damping-adding-removing procedure, it establishes well-posedness and proves quasi-optimality of the scheme, validated on multi-species, multi-frequency, and linearized BGK models with high accuracy.

ABSTRACT

We study half-space linear kinetic equations with general boundary conditions that consist of both given incoming data and various type of reflections, extending our previous work [LLS14] on half-space equations with incoming boundary conditions. As in [LLS14], the main technique is a damping adding-removing procedure. We establish the well-posedness of linear (or linearized) half-space equations with general boundary conditions and quasi-optimality of the numerical scheme. The numerical method is validated by examples including a two-species transport equation, a multi-frequency transport equation, and the linearized BGK equation in 2D velocity space.

Motivation & Objective

• To extend the analysis and numerical solution of half-space kinetic equations to general boundary conditions that combine incoming data and multiple reflection types.
• To establish well-posedness of linearized half-space equations under such general boundary conditions, including Maxwell and Cercignani-Lampis-type reflections.
• To develop a robust, quasi-optimal numerical scheme based on a damping-adding-removing procedure for solving these equations.
• To validate the method on physically relevant models, including two-species transport, multi-frequency transport, and 2D linearized BGK equations.

Proposed method

• Introduces a damping-adding-removing procedure to handle the well-posedness and numerical solution of half-space kinetic equations with complex boundary conditions.
• Applies a damped version of the kinetic equation with added dissipative terms to stabilize the solution process and enable numerical computation.
• Constructs a special solution $ g_0 $ that satisfies the homogeneous equation and boundary conditions, enabling the exact solution to be expressed as $ f_h = f - c_h g_0 + c_h X_0 $, where $ f $ solves the damped system.
• Uses $ L^2 $-projection operators $ ilde{ ho} $ and $ ilde{ ho}^ot $ onto the null space of $ ilde{ ho} $, and defines $ ilde{ ho}_1 $ to decompose the null space into positive, negative, and zero eigenvalue subspaces.
• Applies the trapezoidal rule for discretization in the frequency variable $ ilde{ ho} $, reducing the problem to a multi-species system amenable to basis function construction.
• Handles boundary operators $ ilde{ ho} $ and $ ilde{ ho}_d $ to model diffuse and specular reflections, and derives invertible forms for $ (I + ar{ ilde{ ho}})^{-1} $ and $ (I + ar{ ilde{ ho}})^{-1}(I - ar{ ilde{ ho}}) $.

Experimental results

Research questions

• RQ1 How can well-posedness be established for linear half-space kinetic equations with general boundary conditions that include both incoming data and various reflection mechanisms?
• RQ2 What is the role of the null space decomposition $ H^+ igoplus H^- igoplus H^0 $ in determining the correct boundary conditions at infinity?
• RQ3 Can a damping-adding-removing procedure be systematically applied to construct a quasi-optimal numerical scheme for such equations?
• RQ4 How does the method perform on models with multi-species, multi-frequency, or multi-dimensional velocity structures?
• RQ5 To what extent can the method recover exact solutions when the exact solution is known, such as in the case of $ f_h = X_0 $?

Key findings

• The method achieves quasi-optimality in the numerical scheme, ensuring convergence rates that are optimal up to logarithmic factors.
• For the two-species transport equation, the numerical solution matches the analytical solution when $ h = X_0 $, with errors in the $ ilde{ ho} $-direction below $ 10^{-10} $.
• In the Maxwell boundary condition case with $ ( ilde{ ho}_d, ilde{ ho}_s) = (0.3, 0.4) $, the method recovers the exact solution $ f_h = X_0 $, with the difference $ f_h - X_0 $ being numerically zero within machine precision.
• The method successfully handles the linearized BGK equation in 2D velocity space, demonstrating robustness across different physical regimes.
• The construction of the special solution $ g_0 $ and the use of the coefficient $ c_h = rac{raket{ ilde{ ho} X_0, f}}{raket{ ilde{ ho} X_0, g_0}} $ enable exact recovery of the solution from the damped system.
• The boundary operator $ ilde{ ho} $ is shown to satisfy the required coercivity condition $ ( ilde{ ho} f, f) eq 0 $, ensuring the invertibility of $ (I + ar{ ilde{ ho}}) $.