}
+% ------------------------------------
+% Step 57
+% ------------------------------------
+
+@article{Benzi2006,
+ author = { Benzi, Michele and Olshanskii, Maxim A. },
+ title = { An Augmented Lagrangian‐Based Approach to the Oseen Problem },
+ journal = { SIAM J. Sci. Comput. },
+ year = { 2006 },
+ volume = { 28 },
+ issue = { 6 },
+ pages = { 2095--2113 },
+ doi = {10.1137/050646421},
+ url = {http://doi.org/10.1137/050646421},
+}
+
+@article{HeisterRapin2013,
+ author = { Heister, Timo and Rapin, Gerd },
+ title = { Efficient augmented Lagrangian-type preconditioning for the Oseen problem using Grad-Div stabilization },
+ journal = { Int. J. Numer. Meth. Fluids },
+ year = { 2013 },
+ volume = { 71 },
+ issue = { 1 },
+ pages = { 118--134 },
+ doi = {10.1002/fld.3654},
+ url = {http://doi.org/10.1002/fld.3654},
+}
+
+
+@article{Ghia1982,
+ author = { Ghia, U and Ghia, K.N and Shin, C.T },
+ title = { High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method },
+ journal = { Journal of Computational Physics },
+ year = { 1982 },
+ volume = { 48 },
+ issue = { 3 },
+ pages = { 387--411 },
+ doi = {10.1016/0021-9991(82)90058-4},
+ url = {http://doi.org/10.1016/0021-9991(82)90058-4},
+}
+
+@article{Erturk2005,
+ author = { Erturk, E. and Corke, T. C. and G\"ok\c{c}\"ol, C. },
+ title = { Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers },
+ journal = { Int. J. Numer. Meth. Fluids },
+ year = { 2005 },
+ volume = { 48 },
+ issue = { 7 },
+ pages = { 747--774 },
+ doi = {10.1002/fld.953},
+ url = {http://doi.org/10.1002/fld.953},
+}
+
+@article{Yang1998,
+ author = { Yang, Jaw-Yen and Yang, Shih-Chang and Chen, Yih-Nan and Hsu, Chiang-An },
+ title = { Implicit Weighted ENO Schemes for the Three-Dimensional Incompressible Navier--Stokes Equations },
+ journal = { Journal of Computational Physics },
+ year = { 1998 },
+ volume = { 146 },
+ issue = { 1 },
+ pages = { 464--487 },
+ doi = {10.1006/jcph.1998.6062},
+ url = {http://doi.org/10.1006/jcph.1998.6062},
+}
+
+@article{Bruneau2006,
+ author = { Bruneau, Charles-Henri and Saad, Mazen },
+ title = { The 2D lid-driven cavity problem revisited },
+ journal = { Computers \& Fluids },
+ year = { 2006 },
+ volume = { 35 },
+ issue = { 3 },
+ pages = { 326--348 },
+ doi = {10.1016/j.compfluid.2004.12.004},
+ url = {http://doi.org/10.1016/j.compfluid.2004.12.004},
+}
+
+
+
% ------------------------------------
% Step 58
% ------------------------------------
@f}
with a parameter $\gamma$ and an invertible matrix $W$. Here
-$\gamma B^TW^{-1}B$ is the Augmented Lagrangian term; see [1] for details.
+$\gamma B^TW^{-1}B$ is the Augmented Lagrangian term; see @cite Benzi2006 for details.
Denoting the system matrix of the new system by $G$ and the right-hand
side by $b$, we solve it iteratively with right preconditioning
\tilde{S}^{-1} \approx -(\nu+\gamma)M_p^{-1}.
@f}
-See [1] for details.
+See @cite Benzi2006 for details.
We decompose $P^{-1}$ as
@f{eqnarray*}
@f}
Here two inexact solvers will be needed for $\tilde{A}^{-1}$ and
-$\tilde{S}^{-1}$, respectively (see [1]). Since the pressure mass
+$\tilde{S}^{-1}$, respectively (see @cite Benzi2006). Since the pressure mass
matrix is symmetric and positive definite,
CG with ILU as a preconditioner is appropriate to use for $\tilde{S}^{-1}$. For simplicity, we use
the direct solver UMFPACK for $\tilde{A}^{-1}$. The last ingredient is a sparse
matrix-vector product with $B^T$. Instead of computing the matrix product
in the augmented Lagrangian term in $\tilde{A}$, we assemble Grad-Div stabilization
$(\nabla \cdot \phi _{i}, \nabla \cdot \phi _{j}) \approx (B^T
-M_p^{-1}B)_{ij}$, as explained in [2].
+M_p^{-1}B)_{ij}$, as explained in @cite HeisterRapin2013.
<h3> Test Case </h3>
-We use the lid driven cavity flow as our test case; see [3] for details.
+We use the lid driven cavity flow as our test case;
+see [this page](http://www.cfd-online.com/Wiki/Lid-driven_cavity_problem) for details.
The computational domain is the unit square and the right-hand side is
$f=0$. The boundary condition is
@f{eqnarray*}
search algorithm for globalization of the Newton method.
The cavity reference values for $\mathrm{Re}=400$ and $\mathrm{Re}=7500$ are
-from [4] and [5], respectively, where $\mathrm{Re}$ is the Reynolds number and
-can be located at [8]. Here the viscosity is defined by $1/\mathrm{Re}$.
+from @cite Ghia1982 and @cite Erturk2005, respectively, where $\mathrm{Re}$ is the
+[Reynolds number](https://en.wikipedia.org/wiki/Reynolds_number).
+Here the viscosity is defined by $1/\mathrm{Re}$.
Even though we can still find a solution for $\mathrm{Re}=10000$ and the
-references contain results for comparison, we limit our discussion here to
+papers cited throughout this introduction contain results for comparison,
+we limit our discussion here to
$\mathrm{Re}=7500$. This is because the solution is no longer stationary
-starting around $\mathrm{Re}=8000$ but instead becomes periodic, see [7] for
-details.
-
-<h3> References </h3>
-<ol>
-
- <li> An Augmented Lagrangian-Based Approach to the Oseen Problem, M. Benzi and M. Olshanskii, SIAM J. SCI. COMPUT. 2006
- <li> Efficient augmented Lagrangian-type preconditioning for the Oseen problem using Grad-Div stabilization, Timo Heister and Gerd Rapin
- <li> http://www.cfd-online.com/Wiki/Lid-driven_cavity_problem
- <li> High-Re solution for incompressible flow using the Navier-Stokes Equations and a Multigrid Method, U. Ghia, K. N. Ghia, and C. T. Shin
- <li> Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, E. Erturk, T.C. Corke and C. Gokcol
- <li> Implicit Weighted ENO Schemes for the Three-Dimensional Incompressible Navier-Stokes Equations, Yang et al, 1998
- <li> The 2D lid-driven cavity problem revisited, C. Bruneau and M. Saad, 2006
- <li> https://en.wikipedia.org/wiki/Reynolds_number
-</ol>
+starting around $\mathrm{Re}=8000$ but instead becomes periodic, see
+@cite Bruneau2006 for details.
<img src="https://www.dealii.org/images/steps/developer/step-57.Re400_Streamline.png" alt="">
Then the solution is compared with a reference solution
-from [4] and the reference solution data can be found in the file "ref_2d_ghia_u.txt".
+from @cite Ghia1982 and the reference solution data can be found in the file "ref_2d_ghia_u.txt".
<img src="https://www.dealii.org/images/steps/developer/step-57.compare-Re400.svg" style="width:50%" alt="">
+
+
<h3> Test case 2: High Reynolds Number </h3>
Newton's iteration requires a good initial guess. However, the nonlinear term
</td>
</tr>
</table>
-We compare our solution with reference solution from [5].
+We compare our solution with the reference solution from @cite Erturk2005 .
<img src="https://www.dealii.org/images/steps/developer/step-57.compare-Re7500.svg" style="width:50%" alt="">
The following picture presents the graphical result.
<img src="https://www.dealii.org/images/steps/developer/step-57.Re7500_Streamline.png" alt="">
which decreases as we perform Newton iterations, and the discretization error,
which depends on the mesh size. That is why we have to refine the
mesh and repeat Newton's iteration on the next finer mesh. From the table above, we can
-see that the residual (nonlinear error) is below 1e-12 on each mesh, but the
+see that the final residual (nonlinear error) is below $10^{-12}$ on each mesh, but the
following picture shows us the difference between solutions on subsequently finer
-meshes.
+meshes:
<img src="https://www.dealii.org/images/steps/developer/step-57.converge-Re7500.svg" style="width:50%" alt="">
+
<a name="extensions"></a>
<h3>Possibilities for extensions</h3>
containing the constant pressures and it is done in step-56. More interesting
is the comparison to other state of the art preconditioners like PCD. It turns
out that the preconditioner here is very competitive, as can be seen in the
-paper [2].
+paper @cite HeisterRapin2013.
The following table shows the timing results between our iterative approach
(FGMRES) compared to a direct solver (UMFPACK) for the whole system
<h4>3d computations</h4>
The code is set up to also run in 3d. Of course the reference values are
-different, see [6] for example. High resolution computations are not doable
+different, see @cite Yang1998 for example. High resolution computations are not doable
with this example as is, because a direct solver for the velocity block does
not work well in 3d. Rather, a parallel solver based on algebraic or geometric
-multigrid is needed. See below.
+multigrid is needed -- see below.
<h4>Parallelization</h4>
projects / dealii.git / commitdiff