Convection-Diffusion - Revision history

Mlinaric: Add categories, small fixes

2023-09-01T18:07:14Z

Add categories, small fixes

← Older revision		Revision as of 18:07, 1 September 2023
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	[[Category:benchmark]]		[[Category:benchmark]]
	[[Category:procedural]]		[[Category:procedural]]
			[[Category:linear]]
			[[Category:time invariant]]
			[[Category:first differential order]]
	[[Category:SISO]]		[[Category:SISO]]
	[[Category:Sparse]]		[[Category:Sparse]]
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	==Description==		==Description==

	The '''Convection-Diffusion''' benchmark is given by a finite-difference discretization of a two-dimensional dynamic convection-diffusion heat equation on a unit square domain, which was introduced as a example in <ref name="lyapack"/>:		The '''Convection-Diffusion''' benchmark is given by a finite-difference discretization of a two-dimensional dynamic convection-diffusion heat equation on a unit square domain, which was introduced as an example in <ref name="lyapack"/>:

	:<math>		:<math>
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	</math>		</math>

	For this procedural partial differential equation benchmark Dirichlet boundary conditions are used.		For this procedural partial differential equation benchmark, Dirichlet boundary conditions are used.
	The setup of this benchmark is rather generic, in that it allows based on the configuration of <math>f_x</math>, <math>f_y</math>, <math>g</math> and <math>f</math> to tune the influence of the diffusive, convective and reactive differential operators.		The setup of this benchmark is rather generic, in that it allows based on the configuration of <math>f_x</math>, <math>f_y</math>, <math>g</math>, and <math>f</math> to tune the influence of the diffusive, convective and reactive differential operators.

	==Origin==		==Origin==
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	In the folder <tt>examples</tt> of LYAPCK the MATLAB functions <tt>fdm_2d_matrix</tt> and <tt>fdm_2d_vector</tt> are utilized to generate the system, input matrices:		In the folder <tt>examples</tt> of LYAPCK the MATLAB functions <tt>fdm_2d_matrix</tt> and <tt>fdm_2d_vector</tt> are utilized to generate the system, input matrices:

	A = fdm_2d_matrix(n0,fx_str,fy_str,g_str);		:<syntaxhighlight lang="matlab">
	B = fdm_2d_vector(n0,f_str);		A = fdm_2d_matrix(n0, fx_str, fy_str, g_str);
			B = fdm_2d_vector(n0, f_str);
			</syntaxhighlight>

	The no argument determines the common spatial resolution, hence the discretized problem has dimension <math>N = n_0^2</math>.		The <tt>n0</tt> argument determines the common spatial resolution, hence the discretized problem has dimension <math>N = n_0^2</math>.
	The arguments <tt>fx_str</tt>, <tt>fy_str</tt>, <tt>g_str</tt>, and <tt>f_str</tt> are strings (see documentation) describing the convection in the first dimension, second dimension, the reaction coefficient and the source term respectively.		The arguments <tt>fx_str</tt>, <tt>fy_str</tt>, <tt>g_str</tt>, and <tt>f_str</tt> are strings (see documentation) describing the convection in the first dimension, the second dimension, the reaction coefficient, and the source term, respectively.
	The output matrix can be generated in the same ~~fashaion~~ as the input matrix.		The output matrix can be generated in the same fashion as the input matrix.

	For example, in <ref name="lyapack"/>, the examples are constructed with:		For example, in <ref name="lyapack"/>, the examples are constructed with:

	A = fdm_2d_matrix(20,'10x','100y','0');		:<syntaxhighlight lang="matlab">
	B = fdm_2d_vector(20,'.1<x<=.3');		A = fdm_2d_matrix(20, '10x', '100y', '0');
	C = fdm_2d_vector(20,'.7<x<=.9')';		B = fdm_2d_vector(20, '.1<x<=.3');
			C = fdm_2d_vector(20, '.7<x<=.9')';
			</syntaxhighlight>

	resulting in a SISO system of order <math>400</math>.		resulting in a SISO system of order <math>400</math>.
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	:<math>		:<math>
	\begin{align}		\begin{align}
	\dot{x}(t) &= Ax(t) + Bu(t) \\		\dot{x}(t) &= A x(t) + B u(t) \\
	y(t) &= Cx(t)		y(t) &= C x(t)
	\end{align}		\end{align}
	</math>		</math>
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	System dimensions:		System dimensions:

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

	==Citation==		==Citation==


	To cite this benchmark, use the following references:		To cite this benchmark, use the following references:

Lund: remove preliminary warning

2022-08-26T13:04:19Z

remove preliminary warning

← Older revision		Revision as of 13:04, 26 August 2022
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	~~{{preliminary}} <!-- Do not remove -->~~

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

Himpe: init convection diffusion

2018-11-22T09:34:22Z

init convection diffusion

New page

{{preliminary}} 

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

==Description==

The '''Convection-Diffusion''' benchmark is given by a finite-difference discretization of a two-dimensional dynamic convection-diffusion heat equation on a unit square domain, which was introduced as a example in <ref name="lyapack"/>:

:<math>
\begin{align}
\Delta u - f_x \partial_x u - f_y \partial_y u - g u &= f \\
u &= 0
\end{align}
</math>

For this procedural partial differential equation benchmark Dirichlet boundary conditions are used.
The setup of this benchmark is rather generic, in that it allows based on the configuration of <math>f_x</math>, <math>f_y</math>, <math>g</math> and <math>f</math> to tune the influence of the diffusive, convective and reactive differential operators.

==Origin==

This benchmark is part of LYAPACK <ref name="lyapack"/>.

==Data==

In the folder <tt>examples</tt> of LYAPCK the MATLAB functions <tt>fdm_2d_matrix</tt> and <tt>fdm_2d_vector</tt> are utilized to generate the system, input matrices:

A = fdm_2d_matrix(n0,fx_str,fy_str,g_str);
B = fdm_2d_vector(n0,f_str);

The no argument determines the common spatial resolution, hence the discretized problem has dimension <math>N = n_0^2</math>.
The arguments <tt>fx_str</tt>, <tt>fy_str</tt>, <tt>g_str</tt>, and <tt>f_str</tt> are strings (see documentation) describing the convection in the first dimension, second dimension, the reaction coefficient and the source term respectively.
The output matrix can be generated in the same fashaion as the input matrix.

For example, in <ref name="lyapack"/>, the examples are constructed with:

A = fdm_2d_matrix(20,'10*x','100*y','0');
B = fdm_2d_vector(20,'.1<x<=.3');
C = fdm_2d_vector(20,'.7<x<=.9')';

resulting in a SISO system of order <math>400</math>.

==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, use the following references:

* For the benchmark:
::The MORwiki Community, '''LYAPACK - A MATLAB Toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problems, and Linear–Quadratic Optimal Control Problems'''. Version 1.0, netlib, 1999. http://www.netlib.org/lyapack

@MISC{lyapack,
author = {T. Penzl},
title = {LYAPACK - A MATLAB Toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problems, and Linear–Quadratic Optimal Control Problems},
howpublished = {netlib},
note = {Version 1.0},
url = <nowiki>{http://www.netlib.org/lyapack}</nowiki>,
year = {1999}
}

==References==

<references>

<ref name="lyapack">T. Penzl. <span class="plainlinks">[http://www.netlib.org/lyapack/README LYAPACK - A MATLAB Toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problems, and Linear-Quadratic Optimal Control Problems]</span>, Version 1.0, netlib: 1999. [http://www.netlib.org/lyapack/README netlib.org/lyapack/README]</ref>

</references>