Sharpening the estimate of the stability constant in the maximum-norm of the Crank-Nicolson scheme for the one-dimensional heat equation (Q1612453)

This paper is concerned with the stability constant \(C_{\infty}\) in the maximum norm of the Crank-Nicolson scheme applied to the one dimensional heat equation. By using a sharp resolvent estimate for the discrete Laplacian together with the Cauchy formula, it is shown that \(3\leq C_{\infty}<4.325.\) This bound also holds when the heat equation is considered on a bounded interval along with Dirichlet or Neumann boundary conditions.