Sharpening the estimate of the stability constant in the maximum-norm of the Crank–Nicolson scheme for the one-dimensional heat equation Original Research Article

This paper is concerned with the stability constant C∞ in the maximum-norm of the Crank–Nicolson scheme applied to the one-dimensional heat equation. A well known result due to S.J. Serdyukova is that C∞<23. In the present paper, by using a sharp resolvent estimate for the discrete Laplacian together with the Cauchy formula, it is shown that 3⩽C∞<4.325. This bound also holds when the heat equation is considered on a bounded interval along with Dirichlet or Neumann boundary conditions.