  We propose a threshold-type algorithm to the $L^2$-gradient flow of the
Canham-Helfrich functional generalized to $\mathbb{R}^N$. The algorithm to the
Willmore flow is derived as a special case in $\mathbb{R}^2$ or $\mathbb{R}^3$.
This algorithm is constructed based on an asymptotic expansion of the solution
to the initial value problem for a fourth order linear parabolic partial
differential equation whose initial data is the indicator function on the
compact set $\Omega_0$. The crucial points are to prove that the boundary
$\partial\Omega_1$ of the new set $\Omega_1$ generated by our algorithm is
included in $O(t)$-neighborhood from $\partial\Omega_0$ for small time $t>0$
and to show that the derivative of the threshold function in the normal
direction for $\partial\Omega_0$ is far from zero in the small time interval.
Finally, numerical examples of planar curves governed by the Willmore flow are
provided by using our threshold-type algorithm.
