This paper is concerned with the numerical simulation of two-dimensional flame balls. We describe a Galerkin-type discretization in an adaptive basis of orthogonal wavelets. The solution is represented by means of a reduced basis being adapted in each time step. This algorithm is applied to compute the evolution of circular and elliptic thermodiffusive flames. In particular, we study the influence of the Lewis number, the strength of radiative losses and of the initial radius. The results show different scenarios. We find repeated splitting of the flame ball which is studied in some detail, extinction after excessive growth and also obtain quasi-steady flame balls.