Numerical analysis of radially nonsymmetric blow-up solutions of a nonlinear parabolic problem

The process of combustion for a nonlinear heat conducting medium with a nonlinear volume source is considered. Blow-up self-similar solutions which describe the evolution of radially nonsymmetric waves — with complex symmetry and spiral waves — are realized numerically. Their asymptotic behaviour is analysed and their metastability is established. To solve the self-similar nonlinear problem the continuous analog of the Newton method and the finite element method are used. The semidiscrete Galerkin finite element method and an explicit difference scheme are used to solve the nonlinear parabolic problem. Special adaptive grids, consistent with the structure of the self-similar solutions, are utilized.