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Computer simulations

We performed numerical simulations of these models using the method of lines. In this, we model the integrodifferential equation using a set of 200 ordinary differential equations. In the cases presented here, the connection polarities are taken to be $\psi\equiv 0$ , the components of f and of K are cosines, and we solved the system

$\begin{displaymath}\dot\theta_j=\omega_j+\frac{1}{200} \sum_{i=1}^{200}K_{i,j}\cos(\theta_j-\theta_i) \end{displaymath}$

for $j=1,\cdots,200$ .

A series of computations was done for the single and double layer cases. In the two examples presented here, the initial data had the form

$\begin{displaymath}\theta_j(0)=j2\pi/200+R_j \end{displaymath}$

for $j=1,\cdots,200$ , where R_j is a uniformly distributed random variable having $\vert R_j\vert\le 2\pi/10$ .

We calculate the solutions to time $t=1000\pi$ using an adaptive step Runge-Kutta scheme of orders 7 and 8 [12], and by that time the solution had achieved the form $\theta\approx s+c t$ and the wave speed (c) is estimated by evaluating $\theta_1(1000\pi)/1000\pi$ .

The results of the single layer simulation are shown in the Figure 1. In this, $\omega = 1, N=1,L=\pi$ and the predicted wave speed is $1+\pi$ . In Figure 2 we plot the numerical error, which is deviation of our calculated solution from the theoretical solution derived above

$\begin{displaymath}\left\vert 1-\frac{\theta_{computed}(s,T)-\theta(0,T)}{s}\right\vert \end{displaymath}$

at $T=1000 \pi$ . We see there that the error is O(10^-6)for this simulation.

**Figure:** Initial data (top) and calculated solution $\theta _j(1000\pi )$ (bottom). In this case, $c_{obs}\approx 4.14$ and $c_{calc}=1+\pi$ .
$\begin{figure}\centerline{\psfig{figure=fig1_a.ps,width= 5cm}} \centerline{\psfig{figure=fig1_b.ps,width= 5cm}} \end{figure}$

**Figure 2:** Error of our computer simulation of the single layer model shown in Figure 1.
$\begin{figure}\centerline{\psfig{figure=error.ps,width=5cm}} \end{figure}$

In the case of two layers, we take similar initial data for each of the two layers as described above, but we take $\omega_1/\omega_2=0.01$ . The results are shown in Figure 3 This simulation demonstrates that a striated structure can support different wave speeds.

**Figure:** Plotted here are $\cos\theta_{1,j}(1000\pi)$ (top) and $\cos\theta_{2,j}$ (bottom). The estimated wave speeds in these cases are c₁ = 5.01 and c₂ =0.264.
$\begin{figure}\centerline{\psfig{figure=fig2_a.ps,width= 5cm}} \centerline{\psfig{figure=fig2_b.ps,width= 5cm}} \end{figure}$

Next: Conclusions Up: No Title Previous: Steady Progressing Wave Solutions

Hans Mittelmann
2000-04-04