In this paper a new adaptive algorithm for multi-rate circuit simulation encountered in the design of RF circuits based on spline wavelets is presented. The ordinary circuit differential equations are first rewritten by a system of (multi-rate) partial differential equations (MPDEs) in order to decouple the different time scales. Second, a semi-discretization by Rothe's method of the MPDEs results in a system of differential algebraic equations DAEs with periodic boundary conditions. These boundary value problems are solved by a Galerkin discretization using spline functions. An adaptive spline grid is generated, using spline wavelets for non-uniform grids. Moreover the instantaneous frequency is chosen adaptively to guarantee a smooth envelope resulting in large time steps and therefore high run time efficiency. Numerical tests on circuits exhibiting multi-rate behavior including mixers and PLL conclude the paper.