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Perfectly transparent boundary conditions and wave propagation in lattice Boltzmann schemes

Thomas Bellotti

TL;DR

This work presents a systematic way of developing perfectly transparent boundary conditions for lattice Boltzmann schemes tackling linear problems in one and two space dimensions, and insists on asymptotics for these coefficients in the spirit of analytic combinatorics.

Abstract

Systems of N = 1, 2, . . . first-order hyperbolic conservation laws feature N undamped waves propagating at finite speeds. On their own hand, multi-step Finite Difference and lattice Boltzmann schemes with q = N + 1, N + 2, . . . unknowns involve N ''physical'' waves, which are aimed at being as closely-looking as possible to the ones of the PDEs, and q-N ''numerical-spurious-parasitic'' waves, which are subject to their own speed of propagation, and either damped or undamped. The whole picture is even more complicated in the discrete setting-as numerical schemes act as dispersive media, thus propagate different harmonics at different phase (and group) velocities. For compelling practical reasons, simulations must always be conducted on bounded domains, even when the target problem is unbounded in space. The importance of transparent boundary conditions, preventing artificial boundaries from acting as mirrors producing polluting ricochets, naturally follows. This work presents, building on Besse, Coulombel, and Noble [ESAIM: M2AN, 55 (2021)], a systematic way of developing perfectly transparent boundary conditions for lattice Boltzmann schemes tackling linear problems in one and two space dimensions. Our boundary conditions are ''perfectly'' transparent, at least for 1D problems, as they absorb both physical and spurious waves regardless of their frequency. After presenting, in a simple framework, several approaches to handle the fact that q > N , we elect the so-called ''scalar'' approach (which despite its name, also works when N > 1) as method of choice for more involved problems. This method solely relies on computing the coefficients of the Laurent series at infinity of the roots of the dispersion relation of the bulk scheme. We insist on asymptotics for these coefficients in the spirit of analytic combinatorics. The reason is two-fold: asymptotics guide truncation of boundary conditions to make them depending on a fixed number of past time-steps, and make it clearduring the process of computing coefficients-whether intermediate quantities can be safely stored using floating-point arithmetic or not. Numerous numerical investigations in 1D and 2D with N = 1 and 2 are carried out, and show the effectiveness of the proposed boundary conditions.

Perfectly transparent boundary conditions and wave propagation in lattice Boltzmann schemes

TL;DR

This work presents a systematic way of developing perfectly transparent boundary conditions for lattice Boltzmann schemes tackling linear problems in one and two space dimensions, and insists on asymptotics for these coefficients in the spirit of analytic combinatorics.

Abstract

Systems of N = 1, 2, . . . first-order hyperbolic conservation laws feature N undamped waves propagating at finite speeds. On their own hand, multi-step Finite Difference and lattice Boltzmann schemes with q = N + 1, N + 2, . . . unknowns involve N ''physical'' waves, which are aimed at being as closely-looking as possible to the ones of the PDEs, and q-N ''numerical-spurious-parasitic'' waves, which are subject to their own speed of propagation, and either damped or undamped. The whole picture is even more complicated in the discrete setting-as numerical schemes act as dispersive media, thus propagate different harmonics at different phase (and group) velocities. For compelling practical reasons, simulations must always be conducted on bounded domains, even when the target problem is unbounded in space. The importance of transparent boundary conditions, preventing artificial boundaries from acting as mirrors producing polluting ricochets, naturally follows. This work presents, building on Besse, Coulombel, and Noble [ESAIM: M2AN, 55 (2021)], a systematic way of developing perfectly transparent boundary conditions for lattice Boltzmann schemes tackling linear problems in one and two space dimensions. Our boundary conditions are ''perfectly'' transparent, at least for 1D problems, as they absorb both physical and spurious waves regardless of their frequency. After presenting, in a simple framework, several approaches to handle the fact that q > N , we elect the so-called ''scalar'' approach (which despite its name, also works when N > 1) as method of choice for more involved problems. This method solely relies on computing the coefficients of the Laurent series at infinity of the roots of the dispersion relation of the bulk scheme. We insist on asymptotics for these coefficients in the spirit of analytic combinatorics. The reason is two-fold: asymptotics guide truncation of boundary conditions to make them depending on a fixed number of past time-steps, and make it clearduring the process of computing coefficients-whether intermediate quantities can be safely stored using floating-point arithmetic or not. Numerous numerical investigations in 1D and 2D with N = 1 and 2 are carried out, and show the effectiveness of the proposed boundary conditions.

Paper Structure

This paper contains 46 sections, 15 theorems, 90 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Notations and $z$-transform
  3. Scalar problems
  4. 1D setting
  5. Target partial differential equation
  6. Space and time discretization
  7. Numerical schemes and their properties
  8. $\textnormal{D}_{1}\textnormal{Q}_{2}$ scheme
  9. Fourth-order $\textnormal{D}_{1}\textnormal{Q}_{3}$ scheme
  10. Transparent boundary conditions for the $\textnormal{D}_{1}\textnormal{Q}_{2}$ scheme
  11. Spatial roots of the characteristic equation
  12. Systemic approach
  13. Scalar approach
  14. Back to asymptotic expansions using analytic combinatorics
  15. Numerical experiments
  16. ...and 31 more sections

Key Result

proposition thmcounterproposition

The $\textnormal{D}_{1}\textnormal{Q}_{2}$ scheme is such that there exists a unique eigenvalue $z_{\textnormal{phy}}(\xi\Delta x)$ of $\bm{{E}}(e^{i\xi\Delta x})$ with $\xi\Delta x\in[-\pi, \pi]$ (equivalently, root of eq:charEquation with $\kappa=e^{i\xi\Delta x}$) such that This entails that the scheme is first-order accurate when $\omega\in(0, 2)$ and second-order accurate when $\omega = 2$.

Figures (16)

  • Figure 1: Schematic description of the $\textnormal{D}_{1}\textnormal{Q}_{2}$ algorithm. Cold colors indicate moments, with blue for the conserved moment. Warm colors indicate distribution functions.
  • Figure 2: Symbols $z_{\textnormal{phy}}(\xi\Delta x)$ and $z_{\textnormal{spu}}(\xi\Delta x)$ parametrized globally and continuously for the $\textnormal{D}_{1}\textnormal{Q}_{2}$ with $\mathscr{C} = \tfrac{5}{6}$.
  • Figure 3: Solutions for the $\textnormal{D}_{1}\textnormal{Q}_{2}$ using $\omega = 2$, comparing the systemic approach and the scalar approach.
  • Figure 7: Plot of $\vartheta_{\textnormal{I}}$ and $\vartheta_{\textnormal{II}}$ given by \ref{['eq:thetaI']} and \ref{['eq:thetaII']} as functions of $\mathscr{C}$.
  • Figure 8: Values of $|\beta_{n}|$ versus their asymptotics given by the leading order of the right-hand side of \ref{['eq:asymptoticsD1Q3']}.
  • ...and 11 more figures

Theorems & Definitions (28)

  • remark thmcounterremark: How \ref{['eq:charEquation']} could be already known
  • proposition thmcounterproposition: Consistency and stability of the $\textnormal{D}_{1}\textnormal{Q}_{2}$ scheme on $\mathbb{Z}$
  • remark thmcounterremark: Damping versus dissipation
  • remark thmcounterremark: Dissipation at low frequencies
  • lemma thmcounterlemma: Damping in the $\textnormal{D}_{1}\textnormal{Q}_{2}$ scheme
  • proof
  • remark thmcounterremark: A taste of $\ell^{\infty}$ stability
  • remark thmcounterremark: The "role" of boundary conditions
  • proposition thmcounterproposition: Consistency and stability of the fourth-order $\textnormal{D}_{1}\textnormal{Q}_{3}$ scheme on $\mathbb{Z}$
  • lemma thmcounterlemma: Absence of damping in the fourth-order $\textnormal{D}_{1}\textnormal{Q}_{3}$ scheme
  • ...and 18 more