The Wilson Green's function approach and, alternatively, Feynman's diffusion equation and the Hori representation have been used to derive an exact functional RG equation (EFRGE) that in the course of the RG flow interpolates between the interaction part of the lattice Ginzburg-Landau Hamiltonian and the logarithm of the generating functional of the S-matrix. Because the S-matrix vertices coincide with the amputated correlation functions of the fluctuating field, it has been suggested that in the critical region the amputation of the long-range tails makes the S-matrix functional more localized and thus more suitable to the local potential approximation (LPA) than the renormalized free energy functional used in Wilson's EFRGE. It has been shown that by means of a Legendre transform the S-matrix EFRGE can be converted into an EFRGE for the effective action (EA) thus establishing their formal equivalence. However, despite close similarity, an important difference has been established between the EA EFRGE derived in the present paper and those obtained within the quantum field-theoretic approaches. The S-matrix equation has been found to be simply connected with a generalized functional Burgers' equation. This has made possible to establish direct correspondence between the first order phase transitions and the shock wave solutions of the RG equation. The transparent semi-group structure of the S-matrix RG equation allows one to use different RG techniques at different stages of the RG flow to improve the LPA solution.