Carleman Linearization of lattice Boltzmann

This is a repository of MATLAB codes accompanying "Analysis of Carleman Linearization of Lattice Boltzmann". If you find this work useful in your research, please cite:

Itani, W. & Succi, S. (2021). Analysis of Carleman Linearization of Lattice Boltzmann.

LB Code

This folder contains the variants of the code used for studying and visualizing the Carleman linearization of the lattice Boltzmann scheme.

Collision

1. carlemannBoltzmann.m: It sets up the linearized lattice Boltzmann problem with number of discrete velocites Q, number of dimensions D, lattice weights w, lattice speed c, incompressible density rho, lattice vectors e, relaxation time T, number of timesteps time, ensemble size for the study K_max, Carleman linearization order order, initial flow velocity uu, initial discrete density values f_0, timestep size dt. It then calls the latticeBoltzmann, carlemannLinearize, timeStep, computeResultsArray and plotSolReg functions in this order.
2. latticeBoltzmann.m: It returns the collision term omega for a single phase fluid with BGK equilibrium function for the parameters listed in carlemannBoltzmann.
3. carlemannLinearize.m: It returns the Carlemann variables V and matrices A for the different linearization orders of a given function. They are calculated recursively in increasing order to avoid computing terms that do not appear in the equations.
4. timeStep.m: It returns the function and linearized matrix to compute the next value in time for a first order equation of f driven by omega. Both implicit and explicit Euler schemes are available.
5. computeResultsArray.m: It returns cell arrays containing the values of the original variables f and Carleman variables *V at different times for given initial conditions, relaxation times, and linearization orders.
6. plotSolReg.m: It plots the exact and Carleman-linearized solutions of each discrete density f_i with the respective normalized error.

postProcessing

The folder contains different post-processing tools to visualize the sparsity of the Carleman matrix, and calculate the norms used in the nonlinearity ratio on which the analytically-derived Carleman linearization error depends. The nonlinear driving functions omega1 and omega2 are assumed to pre-calculated.

Random Initial

The folder contains a variation of the code in Collision with a post-processing function tool in dataTable.m to visualize the root sum of squares of the error with respect to the ratio of timestep to relaxation time, linearization order and distance of the initial condition to the weights vector.

Logistic Code

The folder contains a variant of LB Code\Collision modified for a single population, namely the logistic equation with a single variable. Unlike the lattice Boltzmann carlemannLinearize.m, the function in this folder precalculates the Carlemann variables by taking the different orders of the original variable, and then computes the Carlemann linearization matrices for all chosen orders.