Stokes equation: Difference between revisions

Help

Revision as of 08:08, 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{aligned} \frac{\partial v}{\partial t} & = Δ v - \nabla p + f, (x, t) \in Ω \times (0, T] \\ 0 & = div v, \\ v & = 0, (x, t) \in \partial Ω \times (0, T] \end{aligned}

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

A finite difference discretization yields the descriptor system:

\begin{aligned} [\begin{matrix} E_{11} & 0 \\ 0 & 0 \end{matrix}] (\begin{matrix} {\dot{v}}_{h} (t) \\ 0 \end{matrix}) & = [\begin{matrix} A_{11} & A_{12} \\ A_{12}^{⊺} & 0 \end{matrix}] (\begin{matrix} v_{h} (t) \\ ρ_{h} (t) \end{matrix}) + [\begin{matrix} B_{1} \\ B_{2} \end{matrix}] u (t) \\ y (t) & = [\begin{matrix} C_{1} & C_{2} \end{matrix}] (\begin{matrix} v_{h} (t) \\ ρ_{h} (t) \end{matrix}) \end{aligned}

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 $[B_{1} B_{2}]^{⊺} \in ℝ^{N \times 1}$ is chosen (uniformly) randomly, whereas the output matrix $[C_{1} C_{2}] \in ℝ^{1 \times N}$ is set to:

\begin{aligned} [\begin{matrix} C_{1} & C_{2} \end{matrix}] = [\begin{matrix} 1 & 0 & \dots & 0 \end{matrix}] . \end{aligned}

Origin

Data

Dimensions

Citation

References

Contact

User:Himpe

@@ Line 1: / Line 1: @@
 {{preliminary}} <!-- Do not remove -->
+[[Category:benchmark]]
+[[Category:procedural]]
+[[Category:SISO]]
 [[Category:linear]]
+[[Category:Sparse]]
 ==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] \\
+&= \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==