Kinetic Relaxation

The kinetic relaxation approach replaces a nonlinear system of $N_M$ conservation laws by $N_K$ linear transport equations with a stiff source:

\[\partial_t f_k + \sum_{d=1}^D \lambda_{k,d}\,\partial_{x_d} f_k = \frac{\mathcal{M}_k(u) - f_k}{\epsilon}, \qquad u = \text{sum of the kinetic variables belonging to each macroscopic variable}.\]

As $\epsilon \to 0$ the macroscopic variables $u$ satisfy the original conservation law. The transport part is a diagonal LinearAdvection system, so all spatial schemes work on it unchanged. The source is solved implicitly.

Setting it up

Use the kinetic pipeline with an IMEX time stepper and the Kinetic_ namespace:

vels, idx = generate_relax_params(2.0, D, M)   # speeds ±2 along each axis

params[:sim_func_name]      = :run_kinetic_simulation
params[:Time_stepper]       = :ARS222
params[:Kinetic_velocities] = vels
params[:Kinetic_indices]    = idx
params[:Kinetic_epsilon]    = 1e-6

The kinetic speed must exceed the largest wave speed of the macroscopic system (subcharacteristic condition). For Burgers with $|u| \le 1$, a speed of 2.0 is safe.

Mapping kinetic to macroscopic variables (Kin2Macro)

Kin2Macro is built from a list of edges. Kin2Macro([1, 3, 6]) assigns kinetic variables 1:2 to macroscopic variable 1 and 3:5 to variable 2.

• km(f) sums each group and returns the macroscopic state.
• km(k) returns the macroscopic index of kinetic variable k.

generate_relax_params(v, D, M) produces $2D$ variables per macroscopic variable, with speeds $\pm v$ along each axis, and the matching edges.

Equilibrium and source term

For kinetic variable $k$ belonging to macroscopic variable $m$ the equilibrium is

\[\mathcal{M}_k(u) = c_m\Big(u_m + \sum_{d=1}^D F_d^m(u)\, s_{k,d}\Big), \qquad s_{k,d} = \begin{cases} \gamma / \lambda_{k,d} & \lambda_{k,d} \neq 0 \\ 0 & \text{otherwise,}\end{cases}\]

where $c_m = 1 / (\text{number of kinetic variables of } m)$, $F_d^m$ is the physical flux, and $\gamma$ is interior_factor (default $D$). The constructor precomputes the $s_{k,d}$ and $1/\epsilon$.

For the velocity sets from generate_relax_params with the default interior_factor, the equilibrium reproduces the macroscopic state and flux exactly:

\[\sum_{k \in m} \mathcal{M}_k(u) = u_m, \qquad \sum_{k \in m} \lambda_{k,d}\,\mathcal{M}_k(u) = F_d^m(u).\]

evaluate_source returns $(\mathcal{M}_k(u) - f_k)/\epsilon$.

Implicit solve

In each IMEX stage with implicit coefficient $a_{ii}$, the stage value satisfies $Y = Y^* + a_{ii}\Delta t\,(\mathcal{M}(u) - Y)/\epsilon$. The relaxation does not change the macroscopic variables ($\sum_{k\in m}(\mathcal{M}_k - f_k) = 0$), so $u$ can be taken from $Y^*$, and implicit_solve is exact and explicit in form:

\[Y_k = \frac{Y^*_k + \frac{a_{ii}\Delta t}{\epsilon}\,\mathcal{M}_k(u(Y^*))}{1 + \frac{a_{ii}\Delta t}{\epsilon}} .\]

This holds for any $\epsilon > 0$, including the stiff limit.

Initialization

set_initial_conditions! has a kinetic method. It takes the macroscopic initial condition and sets every particle to the equilibrium $\mathcal{M}(u_0(x))$, so you never write kinetic initial data by hand.

API

Kin2Macro{NM, NK}

RelaxationSourceTerm{D, NM, NK, T, MEQ} <: AbstractImplicitSourceTerm