In this interesting paper the authors deal with the following two- dimensional nonlinear parabolic differential equation \(\nu_ 1u_{xx}+\nu_ 2u_{yy}=f(x,y,t,u,u_ x,u_ y,u_ t)\) for \(0\leq x,y\leq 1\), \(t\geq 0\), where \(\nu_ 1\), \(\nu_ 2\) are positive constants. The unknown function \(u\) satisfies the initial condition \(u(x,y,0)=u_ 0(x,y)\) and Dirichlet boundary conditions. Let \(k\) and \(h\) be temporal and spatial (both for \(x\) and \(y\)) step sizes, respectively. The authors construct a two-level implicit finite-difference scheme of order of accuracy \(O(k^ 2+kh^ 2+h^ 4).\) The scheme is based on a ninth- spatial points pattern.
For the linear diffusion-convection equation one has \(f=u_ t+\bar u u_ x+\bar\nu u_ y\), where \(\bar u\), \(\bar\nu\) are appropriate constants. In this case the scheme under consideration is of order of accuracy \(O(k^ 2+h^ 4)\) and possesses unconditional stability. The usefulness of the above difference scheme is illustrated with high efficiency by several numerical examples including Burger’s equation.