Stokes equation

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Description

This benchmark presents the two-dimensional instationary Stokes equation, which models flow of an incompressible fluid in a domain ,,,,. The associated partial differential equation system is given by:

$\begin{align} \frac{\partial v}{\partial t} &= \Delta v - \nabla p + f, \qquad (x,t) \in \Omega \times (0,T] \\ 0 &= \operatorname{div} v, \\ v &= 0, \qquad \qquad \qquad \quad \; (x,t) \in \partial \Omega \times (0,T] \end{align}$

with velocity variable $v(x,t)$ and pressure variable $\rho(x,t)$, on a spatial domain $\Omega = [0,1] \times [0,1] \subset \mathbb{R}^2$, and an external forcing term $f$. The boundary conditions are no-slip.

A finite volume discretization on a uniform, staggered grid yields the descriptor system:

$\begin{align} \begin{bmatrix} E_{11} & 0 \\ 0 & 0 \end{bmatrix} \begin{pmatrix} \dot{v}_h(t) \\ 0 \end{pmatrix} &= \begin{bmatrix} A_{11} & A_{12} \\ A_{12}^\intercal & 0 \end{bmatrix} \begin{pmatrix} v_h(t) \\ \rho_h(t) \end{pmatrix} + \begin{bmatrix} B_1 \\ B_2 \end{bmatrix} u(t) \\ y(t)\quad & = \begin{bmatrix} C_1 \,\;&\;\, C_2 \end{bmatrix} \begin{pmatrix} v_h(t) \\ \rho_h(t) \end{pmatrix} \end{align}$

The matrix $A_{11}$ matrix is the discretized Laplace operator, while $A_{12}$ corresponds to the discrete gradient and divergence operators. For this benchmark the compound discretization of the boundary values and external forcing $[B_1 \; B_2]^\intercal \in \mathbb{R}^{N \times 1}$ is chosen (uniformly) randomly, whereas the output matrix $[C_1 \; C_2] \in \mathbb{R}^{1 \times N}$ is set to:

$\begin{align} \begin{bmatrix} C_1 & C_2 \end{bmatrix} = \begin{bmatrix} 1 & 0 & \dots & 0 \end{bmatrix}. \end{align}$

Data

This is a procedural benchmark. A MATLAB m-file to generate $E, A, B, C$ matrices can be found as part of the M.E.S.S project, under:

 DEMOS/models/stokes/stokes_ind2.m

Dimensions

System structure:

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

System dimensions:

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

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, Stokes equation. MORwiki - Model Order Reduction Wiki, 2018. https://modelreduction.org/morwiki/Stokes_equation

@MISC{morwiki_stokes,
  author =       {{The MORwiki Community}},
  title =        {Stokes equation},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {https://modelreduction.org/morwiki/Stokes_equation},
  year =         {20XX}
}

• For the background on the benchmark:

@PHDTHESIS{Sch07,
  author =       {M.Schmidt},
  title =        {Systematic discretization of input/output maps and other contributions to the control of distributed parameter systems},
  school =       {TU Berlin},
  year =         {2007},
  doi =          {10.14279/depositonce-1600}
}

Contact

Christian Himpe