We discuss a new approach for the numerical computation of
self-similar blow-up solutions of certain nonlinear partial
differential equations. These solutions become unbounded in
finite time at a single point at which there is a growing and
increasingly narrow peak. Our main focus is on a quasilinear
parabolic problem in one space dimension, but our approach can
also be applied to other problems featuring blow-up solutions.
For the model we consider here, the problem of the computation of
the self-similar solution profile reduces to a nonlinear,
second-order ordinary differential equation on an unbounded
domain, which is given in implicit form. We demonstrate that a
transformation of the independent variable to the interval
[0,1] yields a singular problem which facilitates a stable
numerical solution. To this end, we implemented a collocation code
which is designed especially for implicit second order problems.
This approach is compared with the numerical solution by standard
methods from the literature and by well-established numerical
solvers for ODEs. It turns out that the new solution method compares favorably
with previous approaches in its stability and efficiency.
Finally, we comment on the applicability of our method to other
classes of nonlinear PDEs with blow-up solutions.