Discretization: Fixed Domain with Uniform Mesh

The infinite spatial domain $-\infty < x < \infty$ is truncated to $\underline{x} \le x \le \overline{x}$ with Neumann boundary conditions

$$\frac{\partial u}{\partial x}(\underline{x},t) = 0 \qquad \text{and} \qquad \frac{\partial u}{\partial x}(\overline{x},t) = 0,$$

applied at the left and right boundaries. We set $\underline{x}=-50$ and $\overline{x}=750$.

The resulting initial boundary-value problem for the Nagumo equation is discretized using a method of lines approach based on a cubic Hermite collocation method on a uniform spatial grid $x_j = \underline{x} + j\Delta x$ for $j=0,1,\dots,N$. For large N (e.g., N=800) the solution to the truncated initial-boundary value problem closely approximates the Nagumo initial-value problem solution over a long time range $t \in [0,525].$ Note that 525 is the approximate number of time units required for a wavefront of speed c=4/3 to advance 700 spatial units towards the right boundary. Note that the approximation of this travelling wave evolution problem over an arbitrarily long time interval becomes prohibitively expensive because for the size of the fixed frame of reference $\overline{x}-\underline{x}$ must be increased if the wavefront location is to remain in the computational spatial domain.

Unfortunately, as the spatial domain size is increased, the number of mesh points N must be increased. If N is too small relative to the domain length $\overline{x}-\underline{x}$, then over long time intervals, discretization errors cause the wavefront to propagate too quickly. For example, with N=80 and $\Delta x=10$ discretization results in a wavefront which propagates at a constant speed which is 20 per cent faster than the exact travelling wave solution. See Figure 1. The first twelve frames illustrate how the growth of propagation errors which result when a coarse mesh is used. Ultimately the coarse discretization results in an oscillation which produces a spurious asymptotic state connecting the two stable steady solutions u=1 and u=-1/2.

  

Figure: Fixed Uniform Mesh $x \in [-50,750]$, $\Delta x=10$. 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

If adaptive methods are applied to this problem, a natural choice of mesh is one which places more points in regions where the solution is changing, i.e., along the wavefront connecting the u=1 and u=0 states. Also it is important to place points in regions where the solution is unstable, i.e., where $u \approx 0$. In terms of equidistribution with monitor functions, [Li et al., 1999] reported poor results with the arclength monitor function (applied to the Fisher equation). Better results with adaptive equidistribution on a fixed mesh can be obtained, but only by using non-standard monitor functions, such as the curvature monitor function
$$\begin{align*}M(u,u_{xx}) = \sqrt{1 + [ 1.5(1-u) ]^2 + \biggl[ 1000(1.015-u) \frac{\partial^2 u}{\partial x^2} \biggr]^2} \end{align*}$$
weighted to favour points where $u(x_j) \approx 0,$which was defined by [Qiu & Sloan, 1998].

However, the numerical approximation of travelling wave evolution over long time intervals is better accomplished with a moving coordinate system, which we consider in the following section: