G. Berikelashvili, M. M. Gupta, and M. Mirianashvili

On the Choice of Initial Conditions of Difference Schemes for Parabolic Equations

abstract:
We consider the first initial-boundary value problem for linear heat conductivity equation with constant coefficient in $\Omega\times(0,T]$, where $\Omega$ is a unit square. A high order accuracy ADI two level difference scheme is constructed on a 18-point stencil using Steklov averaging operators. We prove that the finite difference scheme converges in the discrete $L_2$-norm with the convergence rate $O(h^s+\tau^{s/2})$, when the exact solution belongs to the anisotropic Sobolev space $W_2^{s,s/2}$, $s\in(2,\,4]$.

Mathematics Subject Classification: 65M06, 65M12, 65M15

Key words and phrases: Heat equation, ADI difference scheme, high order convergence rate