The present note is concerned with the numerical simulation of two--dimensional homogeneous decaying turbulence. We employ a Galerkin--type discretization in an adaptive basis of orthogonal wavelets. Therewith the solution is represented by means of a reduced basis being adapted at each time step. It is one of the first times that wavelet decomposition of a turbulent flow field is not only used for a posteriori analysis but directly for the computation of the flow in a reduced basis. A classical Fourier method furnishes a suitable reference solution. 1 Introduction The direct numerical simulation (DNS) of two--dimensional turbulence has a long tradition. Although it can be shown that from the physical point of view its properties are different with respect to three--dimensional turbulence [9], the two--dimensional setting furnishes a convenient environment to test diverse approaches and new techniques. The standard setting is determined by statistically homogeneous flows with periodic...