Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical boundary control of multi-dimensional discrete-velocity kinetic models

Haitian Yang, Wen-An Yong

TL;DR

The paper addresses numerical boundary control for multi-D discrete-velocity kinetic systems with stiff collision terms. It develops an operator-splitting scheme that combines upwind discretization for advection with forward-Euler collision steps and introduces a discrete Lyapunov function to prove exponential decay of the numerical solution under appropriate boundary conditions. A semi-implicit variant is proposed to handle stiffness, with stability proven independent of the stiffness parameter at the cost of growing auxiliary parameters as the stiffness increases. Validations on the 2-D coplanar model demonstrate feasible numerical boundary controls, rapid decay, and the practical trade-offs between explicit and implicit schemes for different mean free paths, highlighting the method's potential for Boltzmann-type simulations.

Abstract

This paper extends our recent results on multi-dimensional discrete-velocity models to the numerical level. By adopting an operator splitting scheme and introducing a suitable discrete Lyapunov function, we derive numerical control laws that ensure the corresponding numerical solutions decay exponentially in time. To handle the stiff source term, we also use an implicit scheme for the collision part and prove the stability of the resulting schemes. The theoretical results are validated through three numerical simulations for the two-dimensional coplanar model.

Numerical boundary control of multi-dimensional discrete-velocity kinetic models

TL;DR

The paper addresses numerical boundary control for multi-D discrete-velocity kinetic systems with stiff collision terms. It develops an operator-splitting scheme that combines upwind discretization for advection with forward-Euler collision steps and introduces a discrete Lyapunov function to prove exponential decay of the numerical solution under appropriate boundary conditions. A semi-implicit variant is proposed to handle stiffness, with stability proven independent of the stiffness parameter at the cost of growing auxiliary parameters as the stiffness increases. Validations on the 2-D coplanar model demonstrate feasible numerical boundary controls, rapid decay, and the practical trade-offs between explicit and implicit schemes for different mean free paths, highlighting the method's potential for Boltzmann-type simulations.

Abstract

This paper extends our recent results on multi-dimensional discrete-velocity models to the numerical level. By adopting an operator splitting scheme and introducing a suitable discrete Lyapunov function, we derive numerical control laws that ensure the corresponding numerical solutions decay exponentially in time. To handle the stiff source term, we also use an implicit scheme for the collision part and prove the stability of the resulting schemes. The theoretical results are validated through three numerical simulations for the two-dimensional coplanar model.
Paper Structure (10 sections, 5 theorems, 88 equations, 5 figures)

This paper contains 10 sections, 5 theorems, 88 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Boundary control of multi-D discrete-velocity models
  3. Main results
  4. Numerical schemes
  5. Numerical boundary stabilization
  6. Semi-implicit numercial schemes
  7. Numerical simulations for the coplanar model
  8. Numerical boundary conditions
  9. Numerical simulation
  10. Concluding remarks

Key Result

Lemma 2.1

There exists an invertible matrix $P$ and a diagonal positive definite matrix $\Lambda_0=\hbox{diag}\{\lambda_{10},\cdots,\lambda_{K0}\}$ such that and where $\Lambda \in \mathbb{R}^{r\times r}$ is a diagonal matrix with positive entries and $r \leq K.$

Figures (5)

  • Figure 1: Spatial discretization of the square domain $\Omega = (0,1)^2$ with $N=4$.
  • Figure 2: Illustration of the notation $f_{\boldsymbol{j},k}^n$.
  • Figure 3: Simulation I, Time evolution of logarithm of $\ell^2$-norm of the solution with four different spatial grids.
  • Figure 4: Simulation II, Time evolution of logarithm of $\ell^2$-norm of the solution with three different numercial boundary conditions.
  • Figure 5: Simulation III, Time evolution of logarithm of $\ell^2$-norm of different $\sigma$ with implicit or explicit schemes.

Theorems & Definitions (14)

  • Lemma 2.1
  • Remark 3.1
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Lemma 3.6
  • proof
  • Theorem 3.7
  • ...and 4 more