  A three-point monotone difference scheme is proposed for solving a
one-dimensional non-stationary convection-diffusion-reaction equation with
variable coefficients. The scheme is based on a parabolic spline and allows to
linearly reproduce the numerical solution of the boundary value problem over
the integral segment in the form of the function which continuous with its
first derivative. The constructed difference scheme give a highly effective
tool for solving problems with a small parameter at the older derivative in a
wide range of output data of the problem. In the test case, numerical and exact
solutions of the problem are compared with the significant dominance of the
convective term of the equation over the diffusion. Numerous calculations
showed the high efficiency of the new monotonous scheme developed.
