Linear Advection - Revision history

Schuetz2 at 05:33, 17 June 2025

2025-06-17T05:33:10Z

← Older revision		Revision as of 05:33, 17 June 2025
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	title = {Linear Advection},		title = {Linear Advection},
	howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},		howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/~~index.php~~/Linear_Advection}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/Linear_Advection}</nowiki>,
	year = 2019		year = 2019
	}		}

Lund: remove preliminary warning

2022-08-26T13:00:10Z

remove preliminary warning

← Older revision		Revision as of 13:00, 26 August 2022
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	~~{{preliminary}} <!-- Do not remove -->~~

	[[Category:benchmark]]		[[Category:benchmark]]
	[[Category:procedural]]		[[Category:procedural]]

Himpe: Minor update and ref added.

2022-07-27T11:34:43Z

Minor update and ref added.

← Older revision		Revision as of 11:34, 27 July 2022
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	:<math>		:<math>
	\begin{align}		\begin{align}
	A = -\frac{a}{\Delta x} \begin{pmatrix} 1 & & & 0 \\ -1 & 1 \\ & \ddots & \ddots \\ 0 & & -1 & 1 \end{pmatrix}, \quad		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		B = \frac{1}{\Delta x} \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \quad
	C = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix}.		C = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix}.
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	its numerical simulation and especially order reduction can be complex due to the transport phenomenon<ref name="zulfiqar18"/>,		its numerical simulation and especially order reduction can be complex due to the transport phenomenon<ref name="zulfiqar18"/>,
	which is known in control theoretic terms as [https://de.wikipedia.org/Totzeit_(Regelungstechnik) dead time].		which is known in control theoretic terms as [https://de.wikipedia.org/Totzeit_(Regelungstechnik) dead time].
	~~Especially~~, energy-based method tend to produce unstable reduced order models<ref name="benner18"/> due to the non-normal system matrix.		Projection-based model reduction is generally not well suited for systems derived from hyperbolic PDEs<ref name="breiten21"/>,
			especially, energy-based method tend to produce unstable reduced order models<ref name="benner18"/> due to the non-normal system matrix.
	This system can be used to model flame-acoustic interaction<ref name="meindl16"/>.		This system can be used to model flame-acoustic interaction<ref name="meindl16"/>.

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	<ref name="zulfiqar18">U. Zulfiqar, M. Liaquat. "<span class="plainlinks">[https://doi.org/10.3906/elk-1706-124 Model reduction of discrete-time systems in limited intervals]</span>". Turkish Journal of Electrical Engineering and Computer Sciences 26: 294--306, 2018.</ref>		<ref name="zulfiqar18">U. Zulfiqar, M. Liaquat. "<span class="plainlinks">[https://doi.org/10.3906/elk-1706-124 Model reduction of discrete-time systems in limited intervals]</span>". Turkish Journal of Electrical Engineering and Computer Sciences 26: 294--306, 2018.</ref>

			<ref name="breiten21">T. Breiten, T. Stykel. "<span class="plainlinks">[https://doi.org/10.1515/9783110498967-002 Balancing-related model reduction methods]</span>". Volume 1 System- and Data-Driven Methods and Algorithms: 15--56, 2021.</ref>

	</references>		</references>

Himpe at 08:15, 27 August 2021

2021-08-27T08:15:35Z

← Older revision		Revision as of 08:15, 27 August 2021
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	which is known in control theoretic terms as [https://de.wikipedia.org/Totzeit_(Regelungstechnik) dead time].		which is known in control theoretic terms as [https://de.wikipedia.org/Totzeit_(Regelungstechnik) dead time].
	Especially, energy-based method tend to produce unstable reduced order models<ref name="benner18"/> due to the non-normal system matrix.		Especially, energy-based method tend to produce unstable reduced order models<ref name="benner18"/> due to the non-normal system matrix.
	This ~~model~~ can be used to model flame-acoustic interaction<ref name="meindl16"/>.		This system can be used to model flame-acoustic interaction<ref name="meindl16"/>.

Himpe: Added flame-acoustic ref.

2021-08-27T07:43:28Z

Added flame-acoustic ref.

← Older revision		Revision as of 07:43, 27 August 2021
Line 50:		Line 50:
	which is known in control theoretic terms as [https://de.wikipedia.org/Totzeit_(Regelungstechnik) dead time].		which is known in control theoretic terms as [https://de.wikipedia.org/Totzeit_(Regelungstechnik) dead time].
	Especially, energy-based method tend to produce unstable reduced order models<ref name="benner18"/> due to the non-normal system matrix.		Especially, energy-based method tend to produce unstable reduced order models<ref name="benner18"/> due to the non-normal system matrix.
			This model can be used to model flame-acoustic interaction<ref name="meindl16"/>.


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	<references>		<references>

			<ref name="meindl16">M. Meindl, T. Emmert, W. Polifke. "<span class="plainlinks">[https://www.researchgate.net/profile/Wolfgang-Polifke/publication/322077228_Efficient_calculation_of_thermoacoustic_modes_utilizing_state-space_models/links/5a43465ea6fdcce1971653ee/Efficient-calculation-of-thermoacoustic-modes-utilizing-state-space-models.pdf Efficient Calculation of Thermoacoustic Modes Utilizing State-Space Models]</span>". Proceedings of the 23rd International Congress on Sound & Vibration: 1--8, 2016.</ref>

	<ref name="benner18">P. Benner, C. Himpe, T. Mitchell. "<span class="plainlinks">[https://doi.org/10.1007/s10444-018-9592-x On Reduced Input-Output Dynamic Mode Decomposition]</span>". Advances in Computational Mathematics (ModRed Special Issue) 44(6): 1751--1768, 2018.</ref>		<ref name="benner18">P. Benner, C. Himpe, T. Mitchell. "<span class="plainlinks">[https://doi.org/10.1007/s10444-018-9592-x On Reduced Input-Output Dynamic Mode Decomposition]</span>". Advances in Computational Mathematics (ModRed Special Issue) 44(6): 1751--1768, 2018.</ref>

Himpe: init

2019-02-27T13:05:34Z

init

New page

{{preliminary}} 

[[Category:benchmark]]
[[Category:procedural]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==

The '''Linear Advection''' benchmark is an input-output system based on the one-dimensional [[wikipedia:Advection#The_advection_equation|linear advection equation]] (also known as linear transport equation):
:<math>
\begin{align}
\partial_t z = -a \, \partial_x z,
\end{align}
</math>
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 <math>u(t)</math>,
:<math>
\begin{align}
z(0,t) = u(t),
\end{align}
</math>
while the right-hand-side boundary is selected as output <math>y(t)</math>:
:<math>
\begin{align}
y(t) = z(1,t),
\end{align}
</math>
whereas the domain is assumed to be of unit length and <math>a > 0</math> describes the transport velocity.

A spatial discretization, here with a simple [[wikipedia:Upwind_differencing_scheme_for_convection|upwind finite difference scheme]] leads to the standard input-output system form:
:<math>
\begin{align}
\dot{x}(t) &= Ax(t) + Bu(t), \\
y(t) &= Cx(t),
\end{align}
</math>
with matrices:
:<math>
\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}
</math>
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<ref name="zulfiqar18"/>,
which is known in control theoretic terms as [https://de.wikipedia.org/Totzeit_(Regelungstechnik) dead time].
Especially, energy-based method tend to produce unstable reduced order models<ref name="benner18"/> due to the non-normal system matrix.

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

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">
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
</source>
</div>

==Dimensions==

System structure:

:<math>
\begin{align}
\dot{x}(t) &= Ax(t) + Bu(t) \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>.

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

==References==

<references>

<ref name="benner18">P. Benner, C. Himpe, T. Mitchell. "<span class="plainlinks">[https://doi.org/10.1007/s10444-018-9592-x On Reduced Input-Output Dynamic Mode Decomposition]</span>". Advances in Computational Mathematics (ModRed Special Issue) 44(6): 1751--1768, 2018.</ref>

<ref name="zulfiqar18">U. Zulfiqar, M. Liaquat. "<span class="plainlinks">[https://doi.org/10.3906/elk-1706-124 Model reduction of discrete-time systems in limited intervals]</span>". Turkish Journal of Electrical Engineering and Computer Sciences 26: 294--306, 2018.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''