Riding a Spiral Wave: Numerical Simulation of Spiral Waves in a Co-Moving Frame of Reference

TL;DR

This work describes an approach to numerical simulation of spiral waves dynamics of large spatial extent, using small computational grids, with good results on both the horizontal and the vertical.

Abstract

We describe an approach to numerical simulation of spiral waves dynamics of large spatial extent, using small computational grids.

Figures (14)

• Figure 1: (Color online) Sketch of skew-product decomposition of an equivariant flow using a Representative Manifold $\mathcal{M}$, which has exactly one transversal intersection with evey group orbit $g\in\mathcal{G}$ within the relevant stratum of the phase space ${\cal{B}}$ and is diffeomorphic to the orbit manifold. Trajectory $(\mathbf{U},\mathbf{U}',\mathbf{U}")$ of an equivariant flow in ${\cal{B}}$ is a relative periodic orbit, since it projects onto the trajectory $(\mathbf{V},\mathbf{V}',\mathbf{V}"=\mathbf{V})$ on $\mathcal{M}$ which is periodic. The flow on $\mathcal{M}$ is devoid of symmetry $\mathcal{G}$.
• Figure 2: (Color online) Non-uniqueness of the revised tip pinning condition.
• Figure 3: (Color online) Three consecutive runs of Barkley model, $a=0.52$, $b=0.05$, $c=0.02$, $L=20$, $\Delta_x=1/5$, $\Delta_t=1/2000$, ${\vec{r}}_{\textrm{inc}}=(2,0)$. The runs $t\in[0,11]$ and $t\in[22,33]$ are direct simulations. The run $t\in[11,22]$ is a quotient system simulation, the pinning points are indicated by small white crosses. The third pinning condition is engaged at $t\approx16.5$.
• Figure 4: (Color online) Three consecutive runs of FHN model, $\alpha=0.2$, $\beta=0.7$, $\gamma=0.5$, $L=30$, $\Delta_x=1/3$, $\Delta_t=1/720$, ${\vec{r}}_{\textrm{inc}}=(20/3,0)$. The runs $t\in[0,22]$ and $t\in[44,66]$ are direct simulations. The run $t\in[22,44]$ is a quotient system simulation, the pinning points are indicated by small white crosses. The third pinning condition is engaged at $t\approx27.5$.
• Figure 5: (Color online) (a,b) Reconstructed tip trajectories from (a) simulation shown in fig. \ref{['fig:bkl-movie']} and (b) simulation shown in fig. \ref{['fig:fhn-movie']}. The pieces labelled 1 are trajectories obtained in direct simulations in the laboratory FoR. The pieces labelled 2 are trajectories obtained via co-translating simulations, with first two pinning conditions engaged. The pieces labelled 3 correspond to co-moving (co-translating and co-rotating) simulations with all three pinning conditions engaged. The final pieces labelled 4 correspond to direct simulations in a non-moving FoR, which has been displaced with respect to the laboratory FoR during the quotient system simulations. (c) Same as (a), with $\Delta_x=1/10$, $\Delta_t=1/4000$. (d) Same as (b), with $\Delta_x=1/10$, $\Delta_t=1/4000$.