Linear Advection

Description

The Linear Advection benchmark is an input-output system based on the one-dimensional linear advection equation (also known as linear transport equation):

$\begin{align} \partial_t z = -a \, \partial_x z, \end{align}$

which is a first order hyperbolic partial differential equation.

For this benchmark, the left-hand-side boundary of the domain is considered to be the input $u(t)$,

$\begin{align} z(0,t) = u(t), \end{align}$

while the right-hand-side boundary is selected as output $y(t)$:

$\begin{align} y(t) = z(1,t), \end{align}$

whereas the domain is assumed to be of unit length and $a > 0$ describes the transport velocity.

A spatial discretization, here with a simple upwind finite difference scheme leads to the standard input-output system form:

$\begin{align} \dot{x}(t) &= Ax(t) + Bu(t), \\ y(t) &= Cx(t), \end{align}$

with matrices:

$\begin{align} A = -\frac{a}{\Delta x} \begin{pmatrix} -1 & & & 0 \\ 1 & -1 \\ & \ddots & \ddots \\ 0 & & 1 & -1 \end{pmatrix}, \quad B = \frac{1}{\Delta x} \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix}. \end{align}$

The challenge of this SISO system benchmark lies in its hyperbolicity. Even though this system seems simple, its numerical simulation and especially order reduction can be complex due to the transport phenomenon, which is known in control theoretic terms as dead time. Projection-based model reduction is generally not well suited for systems derived from hyperbolic PDEs, especially, energy-based method tend to produce unstable reduced order models due to the non-normal system matrix. This system can be used to model flame-acoustic interaction.

Data

To procedurally instantiate the state-space components for this benchmark model, it is assumed the reciprocal spatial resolution $N := \frac{1}{\Delta x} \in \mathbb{N}$ is given, which determines the order of the system, alongside the transport velocity $a$.

function [A,B,C] = lte(a,N)

    A = -(abs(a) * N) * spdiags(ones(N,1) * [-1,1],[-1,0],N,N);
    B = N * sparse(1,1,1,N,1);
    C = sparse(1,N,1,1,N);
end

Dimensions

System structure:

$\begin{align} \dot{x}(t) &= Ax(t) + Bu(t) \\ y(t) &= Cx(t) \end{align}$

System dimensions:

$A \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times 1}$, $C \in \mathbb{R}^{1 \times N}$.

Citation

To cite this benchmark and its data:

The MORwiki Community, Linear Advection. hosted at MORwiki - Model Order Reduction Wiki, 2019. http://modelreduction.org/index.php/Linear_Advection

@MISC{morwiki_linad,
  author =       {{The MORwiki Community}},
  title =        {Linear Advection},
  howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
  url =          {http://modelreduction.org/index.php/Linear_Advection},
  year =         2019
}

Contact

Christian Himpe