The paper shows how to combine together the Lattice Boltzmann Methods with the time splitting and the grid refinement techniques, in order to solve reaction-diffusion processes including very fast reaction dynamics, i.e. with time and length scales that vary in a wide range of values. The method is applied to the reaction prototype problem: M
← M + L
ML with semi-infinite diffusion conditions and in presence of an electrode where Nernst + flux balance conditions are considered. Two important geometries are considered, planar and spherical, and off-lattice boundary conditions are set up, for general irregular and curved boundaries. We discuss the need, for some cases, of applying the time splitting and the grid refinement approach to have a numerical scheme more easily handled and to substantially reduce the computational time. Furthermore, we point out the physico-chemical conditions to apply the time splitting and the grid refinement to optimise accuracy and performance. In particular, we stress: a) the range of values of the relaxation BGK parameter to have the best performance in solving the pure diffusive scheme and b) the best values of the grid refinement factor to preserve a good accuracy and drastically reduce the time of computation and the memory usage.