Table 2.

Errors and convergence rates of the scheme (3.18)–(3.22)

Grid sizes	1282–2562		2562–5122		5122–10242
Error	$1.09\times {10}^{-2}$		$2.87\times {10}^{-3}$		$7.25\times {10}^{-4}$
Rate		1.93		1.98

Parameters are given in the text and the initial data is given by (4.1)–(4.2). Four tests with a refinement step size $s=C{h}^2$ are presented. The error here is the global error between two nearby tests. The rate here suggests second order convergence rate in $h$ is attained, i.e., $e=\mathcal{O}\left(h^2\right)$