Sharp error bounds for the Crank-Nicolson and Saulyev difference scheme in connection with an initial boundary value problem for the inhomogeneous heat equation
The standard heat equation with one space variable and standard initial-boundary conditions on the strip parallel to the \(t\)-axis in the \(t-x\) plane is considered. For this model problem, two finite difference schemes are investigated: the Crank-Nicolson method: \[ u_i^{n+1}= u^n_i+ {\textstyle {k \over {2h^2}}} (u^n_{i-1}- 2u^n_i+ u^n_{i+1}+ u^{n+1}_{i-1}- 2u_i^{n+1}+ u^{n+1}_{i+1})+ {\textstyle {k\over 2}} (\varphi^n_i+ \varphi_i^{n+1}) \] and the Saulyev method: \[ u_i^{n+1}= u^n_i+ {\textstyle {k\over h^2}} (u^{n+1}_{i-1}- u_i^{n+1}+ u^n_{i+1}- u^n_i)+ k\varphi_i^{n+1}; \] \(k\) and \(h\) are \(t\)- and \(x\)-steps, respectively. The Crank-Nicolson scheme is of order two in the space \(C^{(2,2)}\), while the Saulyev scheme is second order in the larger space \(C^{(2,1)}\) (first variable is \(x!\)), and is only conditionally stable. It is proved that the above mentioned bounds are sharp in the sense that it is always possible to find such an element in an appropriate space for which the residual is large -- \(O(h^2)\) and is not small -- \(o(h^2)\). The authors apply the method of proving this fact: they use the Green function of the finite difference operator. This hint replaces the positivity argument currently used in the elliptic case, which in the present situation can not be applied directly. In the proof, the generalized version of the uniform boundedness theorem, for the case of sublinear bounded functionals is involved. In fact, the result concerns general abstract moduli of continuity.
scientific article; zbMATH DE number 826956