From nonlinear PDEs to singular ODEs

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.