Subsections

• Formulation
• Performance

7 Flow in a Channel

Analyze the flow of a fluid during injection into a long vertical channel, assuming that the flow is modeled by the boundary value problem

$$\begin{array}{l} u'''' = R\left(u'u''-u u'''\right),\qquad ... ...q 1, \\ u(0)=0,\quad u(1)=1,\quad u'(0)= u'(1)=0, \end{array}$$ (4)

where $u$ is the potential function, $u'$ is the tangential velocity of the fluid, and $R$ is the Reynolds number.

Formulation

We use a $k$-stage collocation method to formulate this problem as an optimization problem with a constant merit function and equality constraints representing the solution of . We use a uniform partition with $n_h$ subintervals of $[0,1]$, and the standard [2, pages 247-249] basis representation,

$$u (t) = \sum_{j=1}^{m}\frac{(t-t_i)^{j-1}}{(j-1)!}v_{ij} + ... ...+m-1}}{(j+m-1)!\ h^{j-1}}w_{ij}, \qquad t \in [t_i,t_{i+1}] ,$$

for $u$. Note that $u \in C^{m-1}[0,1]$, where $m = 4$ is the order of the differential equation.

The constraints in the optimization problem are the initial conditions in , the continuity conditions, and the collocation equations. There are $m = 4$ initial conditions. The continuity equations are a set of $m (n_h -1)$ linear equations. The collocation equations are a set of $k \, n_h$ nonlinear equations obtained by requiring that $u$ satisfy at the collocation points $\xi_{ij} = t_i + h \rho_j$ for $i=1,\ldots,n_h$ and $j=1,\ldots, k$. The collocation points $\rho_j$ are the roots of the $k$th degree Legendre polynomial. The parameters in the optimization problem are the $(m + k ) n_h$ parameters $v_{ij}$ and $w_{ij}$ in the representation of $u$. Data for this problem appears in Table 7.1.

Table 7.1: Flow in a channel problem data

Variables	$(k+4) n_h$
Constraints	$(k+4) n_h$
Bounds	0
Linear equality constraints	$4 n_h$
Linear inequality constraints	0
Nonlinear equality constraints	$k n_h$
Nonlinear inequality constraints	0
Nonzeros in $\nabla ^2 f(x)$	0
Nonzeros in $c'(x)$	$k(k+8) n_h$

Performance

Results for the AMPL implementation with $k=4$ and $R = 10$ are summarized in Table 7.2. The starting point is the function $t^2 (3 - 2t)$ evaluated at the mesh points. Solutions for several $R$ are shown in Figure 7.1. This problem is easy to solve for small Reynolds numbers but becomes increasingly difficult to solve as $R$ increases.

Table 7.2: Performance on flow in channel problem

Solver	$n_h=50$	$n_h=100$	$n_h=200$	$n_h=400$
LANCELOT	$\ddagger$	$\ddagger$	$\ddagger$	$\ddagger$
$f$	$\ddagger$	$\ddagger$	$\ddagger$	$\ddagger$
$c$ violation	$\ddagger$	$\ddagger$	$\ddagger$	$\ddagger$
iterations	$\ddagger$	$\ddagger$	$\ddagger$	$\ddagger$
LOQO	1.55 s	2.59 s	7.03 s	22.54 s
$f$	1.00000e+00	1.00000e+00	1.00000e+00	1.00000e+00
$c$ violation	5.1e-12	1.1e-11	2.9e-11	1.9e-11
iterations	32	25	29	35
MINOS	1.09 s	3.25 s	11.15 s	32.65 s
$f$	1.00000e+00	1.00000e+00	1.00000e+00	1.00000e+00
$c$ violation	3.8e-13	2.4e-13	2.0e-13	3.8e-07
iterations	5	5	5	5
SNOPT	2.14 s	7.47 s	27.91 s	98.5 s
$f$	1.00000e+00	1.00000e+00	1.00000e+00	1.00000e+00
$c$ violation	6.1e-05	4.6e-05	4.1e-05	3.9e-05
iterations	3	3	3	3
$\dagger$ Errors or warnings. $\ddagger$ Timed out.

LANCELOT is unable to solve even simple versions of the problem, advancing very slowly toward the solution (as judged from the value of the merit function).

Figure 7.1: Tangential velocity $u'$ for Reynolds numbers $R=0, 10^2, 10^4$