# Analysis of Carleman Linearization of Lattice Boltzmann

^{1}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Early Attempts for Quantum Simulation of Fluids

#### 1.2. Carleman Linearization

## 2. Lattice Boltzmann

#### Nonlinearity Ratio

## 3. Carleman Linearization for Lattice Boltzmann

#### 3.1. Number of Variables

#### 3.2. Carleman Linearization of Collision Step

#### 3.3. Carleman Linearization of Streaming Step

#### 3.4. Error Bound

## 4. Numerical Results

#### 4.1. Logistic Equation

#### 4.2. D1Q3

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\overrightarrow{v}$ | Continuum particle velocity |

${\overrightarrow{c}}_{i}$ | Discrete velocity in the ith direction |

Q | Number of discrete velocities number of modes at each lattice site, indexed by i |

D | Number of dimensions of the lattice |

x | Dimensions of the lattice, indexed by d, independent position vector variable |

N | Number of Carleman variables |

${N}_{{x}_{d}}$ | Number of sites across the dth dimension ${x}_{d}$ of the lattice |

G | Volume of the lattice in the units obtained by the product of the number of sites in each dimension ${\mathsf{\Pi}}_{d=1}^{D}{N}_{{x}_{d}}$ |

${f}_{i}$ | Discrete density distribution weight |

$\mathsf{\Omega}$ | Collision operator defined by $\frac{d\overrightarrow{f}}{dt}=\mathsf{\Omega}\left(\overrightarrow{f}\right)$ |

${w}_{i}$ | Weight of the ith discrete density |

${O}_{c}$ | Truncation order in Carleman linearization |

R | Ratio providing measure of nonlinearity parametrizing the error bound of the Carleman technique |

${F}_{0}$ | Coefficient vector of zero-order terms in a quadratic ODE |

${F}_{1}$ | Coefficient matrix of first-order terms in a quadratic ODE |

${F}_{2}$ | Coefficient matrix of second-order terms in a quadratic ODE |

t | Independent time variable |

$\overrightarrow{u}$ | Flow velocity |

$\rho $ | Local fluid density in lattice units |

c | Lattice speed |

C | Carleman linearization matrix |

$\epsilon $ | Norm of the solution error |

${f}_{C}$ | Approximated solution of the system |

$\Delta t$ | Discrete timestep |

V | Vector of Carleman variables |

$\overrightarrow{e}$ | Lattice vectors |

$Ma$ | Mach number |

K | Scaling factor of the logistic equation |

T | Total integration time |

p | Order of the polynomial describing the driving function $\mathsf{\Omega}$ |

${\overrightarrow{P}}^{\left(k\right)}$ | a vector of polynomial functions of kth order in Carlemann variables |

$Re$ | Reynold’s number |

L | Characteristic length scale defined in units of $\Delta x$ |

U | Characteristic velocity in lattice units |

$\nu $ | Kinematic viscosity in lattice units |

${c}_{s}$ | Speed of sound in lattice units |

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**Figure 1.**An illustration of the D1Q3 lattice Boltzmann scheme showing how the collision step is a relaxation of the discrete densities towards their equilibrium values whereas streaming involves assigning their values to neighboring cells in the respective directions.

**Figure 3.**Numbering convention used for the discrete densities of the D1Q3 lattice Boltzmann scheme. In 1D, the lattice vectors correspond to the respective scalars $-1,0,1$.

**Figure 4.**Illustration of how the streaming step preserves uniformity under periodic boundaries only, using D1Q2. The uniform initial flow with periodic boundaries coincides with the error-free case of collision without streaming while errors form at the boundaries when non-periodic boundary conditions are applied. We see that with the same initial conditions and periodic boundary, local and neighboring information of discrete densities are interchangeable. (

**a**) Uniform initial flow field, (

**b**) Periodic boundary conditions, (

**c**) Non-periodic boundary conditions.

**Figure 5.**Illustration of the propagation of Carleman linearization error with timestep in a two-dimensional domain with uniform initial flow and non-periodic boundaries. Lattice cells with no fill have discrete exact discrete densities whereas ones with red fill have discrete densities with error. (

**a**) Initial condition, at $t=0$, (

**b**) After one timestep, at $t=\Delta t$. (

**c**) After two timesteps, at $t=2\Delta t$.

**Figure 6.**The analytical (

**left**) (analytical integration) and numerical (

**right**) (discrete time-stepping) solutions of the Carleman-linearized logistic equation are shown with their corresponding errors (bottom) as a function of time, varying initial conditions and Carleman linearization orders. The predicted time of validity is shown as a vertical asymptote in each plot.

**Figure 7.**The solution of the discrete densities of the fluid in D1Q3 for successive collisions is shown for different $\frac{\Delta t}{\tau}$, for the exact and Carleman-linearized formulations as a function of time and Carleman linearization order. The bottom figures show the normalized errors for each discrete density. Note that the solution is exact beyond the first linearization order.

**Figure 8.**Visualization of the sparsity of the Carleman matrix for the collision term at various orders.

**Table 1.**Example of D1Q3 expanded to a second order truncation in Carleman linearization, with $N=9$ Carleman variables. Note that the dummy variable ${V}_{1}=1$ is defined to simplify the form of Equation (23).

Carleman Variables | Lattice Variables |
---|---|

${V}_{1}$ | 1 |

${V}_{2}$ | ${f}_{1}^{2}$ |

${V}_{3}$ | ${f}_{3}^{2}$ |

${V}_{4}$ | ${f}_{1}{f}_{2}$ |

${V}_{5}$ | ${f}_{1}{f}_{3}$ |

${V}_{6}$ | ${f}_{2}{f}_{3}$ |

${V}_{7}$ | ${f}_{1}$ |

${V}_{8}$ | ${f}_{2}$ |

${V}_{9}$ | ${f}_{3}$ |

**Table 2.**The maximum endtime validity for Carleman linearization of the logistic equation as predicted by Equation (41).

$\mathit{f}(\mathit{t}=0)$ | T |
---|---|

0 | ∞ |

0.1 | 2.40 |

0.2 | 1.79 |

0.3 | 1.47 |

0.4 | 1.25 |

0.5 | 1.10 |

0.6 | 0.98 |

0.7 | 0.89 |

0.8 | 0.81 |

0.9 | 0.75 |

1 | 0.69 |

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Itani, W.; Succi, S. Analysis of Carleman Linearization of Lattice Boltzmann. *Fluids* **2022**, *7*, 24.
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**AMA Style**

Itani W, Succi S. Analysis of Carleman Linearization of Lattice Boltzmann. *Fluids*. 2022; 7(1):24.
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**Chicago/Turabian Style**

Itani, Wael, and Sauro Succi. 2022. "Analysis of Carleman Linearization of Lattice Boltzmann" *Fluids* 7, no. 1: 24.
https://doi.org/10.3390/fluids7010024