  The renormalization group (RG) equation in the self-consistent local
potential approximation (SC-LPA) suggested earlier for the description of
continuous phase transitions in lattice models of the Landau-Ginzburg type has
been applied to the solution of the spin-1 Blume-Capel model on the simple
cubic lattice. The calculated transition temperatures of both continuous and
the first-order phase transitions (FOPTs) in zero external field have been
found to be in excellent agreement with the best available estimates. It has
been argued that the SC-LPA RG equation may give more accurate and complete
description of the FOPTs than those reported in alternative approaches. It has
been shown that the SC-LPA RG equation can be cast in the form of the
generalized Burgers' equation (GBE). In this formulation of the RG the FOPTs
have been shown to assume the form of the shock-wave solutions of GBE in the
inviscid limit. Universality of the RG flow in the vicinity of the fixed point
describing FOPTs has been discussed.
