(init Stokes) |
(Stokes description init) |
||
| Line 1: | Line 1: | ||
{{preliminary}} <!-- Do not remove --> |
{{preliminary}} <!-- Do not remove --> |
||
| + | |||
| + | [[Category:benchmark]] |
||
| + | [[Category:procedural]] |
||
| + | [[Category:SISO]] |
||
[[Category:linear]] |
[[Category:linear]] |
||
| + | [[Category:Sparse]] |
||
==Description== |
==Description== |
||
| + | This benchmark presents the two-dimensional instationary [[wikipedia:Stokes_flow|Stokes equation]], |
||
| + | which models flow of an incompressible fluid in a domain. |
||
| + | The associated partial differential equation system is given by: |
||
| + | :<math> |
||
| + | \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} |
||
| + | </math> |
||
| + | with velocity variable <math>v(x,t)</math> and pressure variable <math>\rho(x,t)</math>, |
||
| + | on a spatial domain <math>\Omega = [0,1] \times [0,1] \subset \mathbb{R}^2</math>, |
||
| + | and an external forcing term <math>f</math>. |
||
| + | The boundary conditions are no-slip. |
||
| + | A finite difference discretization yields the descriptor system: |
||
| − | |||
| + | :<math> |
||
| + | \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} |
||
| + | </math> |
||
| + | The matrix <math>A_{11}</math> matrix is the discretized Laplace operator, |
||
| + | while <math>A_{12}</math> corresponds to the discrete gradient and divergence operators. |
||
| + | For this benchmark the compound discretization of the boundary values and external forcing <math>[B_1 \; B_2]^\intercal \in \mathbb{R}^{N \times 1}</math> is chosen (uniformly) randomly, |
||
| + | whereas the output matrix <math>[C_1 \; C_2] \in \mathbb{R}^{1 \times N}</math> is set to: |
||
| + | :<math> |
||
| + | \begin{align} |
||
| + | \begin{bmatrix} C_1 & C_2 \end{bmatrix} = \begin{bmatrix} 1 & 0 & \dots & 0 \end{bmatrix}. |
||
| + | \end{align} |
||
| + | </math> |
||
==Origin== |
==Origin== |
||
Revision as of 09:08, 26 June 2019
Note: This page has not been verified by our editors.
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 difference discretization 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: \[ \begin{align} \begin{bmatrix} C_1 & C_2 \end{bmatrix} = \begin{bmatrix} 1 & 0 & \dots & 0 \end{bmatrix}. \end{align} \]