Computer simulation of nonlinear dynamics of a domain wall in a magnetically uniaxial unbounded medium is carried out using a dynamically adaptive numerical scheme based on the method of wavelet expansions. The structure of the wavelet algorithm is presented in detail. A comparative testing of the numerical scheme of the new type is performed and is shown that this scheme is very fast as compared to the traditional method of finite-difference approximations. Results of the simulation of the dynamics in fields with a piece-wise-constant dependence on time are presented. The efficiency of the wavelet approach upon numerical simulation of nonlinear dynamics of well-localized objects with internal structure is discussed.