Difference schemes of arbitrary order of accuracy for semilinear parabolic equations

The Cauchy problem for a semilinear parabolic equation is considered. Under the conditions u(x, t) = X(x)T1(t) + T2(t), ∂u ∂x = 0, it is shown that the problem is equivalent to the system of two ordinary differential equations for which exact difference scheme (EDS) with special Steklov averaging and difference schemes with arbitrary order of accuracy (ADS) are constructed on the moving mesh. The special attention is paid to investigating approximation, stability and convergence of the ADS. The convergence of the iteration method is also considered. The presented numerical examples illustrate theoretical results investigated in the paper.