Discretization: Moving Domain

In a moving frame of reference, we truncate the infinite spatial domain $-\infty < x < \infty$ to $\underline{x}(t) \le x \le \overline{x}(t)$and set

$$\frac{d\underline{x}}{dt}=\frac{d\overline{x}}{dt}=c_0 = 4/3.$$

Letting z=x-c₀ t denote the moving coordinate variable, we set $\underline{x}(0)=\underline{z}=-50$and $\overline{x}(0)=\overline{z}=50$.

A Neumann boundary condition

$$\frac{\partial u}{\partial z}(\underline{z},t) = 0$$

is applied at the left moving boundary where the initial condition evolves to the back end of a wave near the stable equilibrium solution u=1.

A Robin boundary condition

$$\frac{\partial u}{\partial z}(\overline{z},t) + \beta_0 u(\overline{z},t)= 0$$

is applied at the right moving boundary. This boundary condition preserves the asymptotic behaviour of the evolving wavefront at the unstable front end of the wave (near u=0) and seems to be more accurate than either Neumann or Dirichlet boundary conditions¹ in the moving frame of reference.

  

Figure: Moving Uniform Mesh $x \in [-50+4t/3,50+4t/3]$, $\Delta x=1.25$. Exact solution in red. Collocation solution in green with uniformly spaced mesh points in blue.

t=0	t=10	t=20
t=30	t=75	t=120
t=165	t=210	t=255
t=300	t=345	t=390
t=435	t=480	t=525

Using the method of lines with Hermite collocation for the spatial discretization we numerically integrate the Nagumo initial-boundary value problem in the speed c₀=4/3 moving frame of reference. A uniform moving spatial mesh with N=80 nodes and $\Delta x=[\overline{z}-\underline{z}]/N=1.25$ is used, to facilitate comparison with the fixed domain results. Figure 2 shows that the discrete (blue) solution closely approximates the the exact solution (in red) to the Nagumo travelling wave initial value problem.

The advantage of the moving frame of reference over the fixed frame of reference is that using a smaller computational domain allows a smaller spatial discretization step size $\Delta x$ to be used, which gives a closer approximation² to the continuous problem. If the wavespeed and decay rate parameters c₀ and $\beta_0$are not known a priori, this close relationship between the continuous and discrete problem can be used to compute numerical approximations for these parameters.