The Cauchy problem for the parabolic equation $$\frac{∂u}{∂t} =\frac{∂}{∂x} (k(x,t) \frac{∂u}{∂x}) +
f(u,x,,t), x \in R, t > 0,$$ $$u(x,0) = u_0(x), x\in R,$$ is considered. Under conditions $u(x, t) = X(x)T_1(t) + T_2 (t)$, $\frac{∂u}{∂x} \neq 0$, $k(x,t)=k_1(x)k_2(t)$, $f(u,x,t) = f_1(x,t)f_2(u)$, it is shown that the above problem is
equivalent to a system of two first-order ordinary differential
equations for which exact difference schemes with special Steklov
averaging and difference schemes with any order of approximation are
constructed on the moving mesh. On the basis of this approach, the exact
difference schemes are constructed also for boundary-value problems and
multi-dimensional problems. Presented numerical experiments confirm the
theoretical results investigated in the paper.