Exact Difference Schemes for Parabolic Equations

Abstract

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.