Linear Advection

LinearAdvection implements coupled linear transport with constant matrices:

\[\frac{\partial U}{\partial t} + \sum_{d=1}^D A_d \frac{\partial U}{\partial x_d} = 0,\]

where $U$ has $M$ components and every $A_d$ is an $M \times M$ matrix.

Specifying the velocities

velocities has one entry per dimension, each describing $A_d$. The dimension D is the number of entries.

LinearAdvection(((1.0,),), Float64)               # 1D scalar: u_t + u_x = 0
LinearAdvection(((1.0,), (0.5,)), Float64)        # 2D scalar, velocity (1, 0.5)
LinearAdvection(((1.0, -1.0),), Float64)          # 1D, two variables: diag(1, -1)

A tuple of $M$ numbers is read as the diagonal matrix $\operatorname{diag}(a_1, \dots, a_M)$. This is the format produced by generate_relax_params for kinetic systems. Other formats are converted by param2vel.

In a configuration: :PDE_name => :linear, :PDE_velocities => ((1.0,), (1.0,)).

Implementation

• Wave speeds. The constructor precomputes the spectral radius of every $A_d$; max_eigenvalue returns it without any work at run time.
• Variables. prim2cons and cons2prim are the identity.
• Moving grids. For scalar equations (M = 1), fluid_velocity returns the advection velocity; a Lagrangian grid then carries the solution exactly.
• Flux. $F_d(U) = A_d U$; velocity(eq, U, d) returns $A_d$.
• Kinetic use. kinetic_wave_speed(eq, d, k) returns the diagonal entry $(A_d)_{kk}$. It is meaningful for diagonal systems, which is how kinetic relaxation systems are built.

Exact solution

The exact solution works with any initial condition and any dimension:

1. Commutation check. For $D > 1$ the matrices must commute ($A_i A_j = A_j A_i$). Otherwise the closure throws an error.
2. Characteristic variables. It diagonalizes $\sum_d A_d = R \Lambda R^{-1}$. The speed of characteristic $m$ in direction $d$ is $\lambda_{m,d} = (R^{-1} A_d R)_{mm}$.
3. Tracing. Each characteristic variable is the initial condition at $x - \lambda_m t$, projected with $R^{-1}$. Periodic directions are wrapped into the domain. The physical state is reassembled with $R$.

API

LinearAdvection{D, M, T, L} <: HyperbolicPDE{D, M, T}