The Error of the Crank-Nicolson Method for Linear Parabolic Equations with a Derivative Boundary Condition

Abstract

The accuracy of finite difference formulae for partial differential equations is usually characterized by the approximation order of the local truncation error as determined by Taylor expansions. Thus the local truncation error of the Crank-Nicolson formula is second order in the two step sizes, h and k, and this carries over to the global error in case of Dirichlet boundary conditions, but not if a derivative boundary condition is approximated by a first order formula. We study the pointwise global error on the form
h c(t,x) + k d(t,x) + h^2 f(t,x) + k^2 g(t,x) + ...
and show that the auxiliary functions, c, d, f, and g are solutions to related parabolic problems. We also present a method based on the computed numbers on refined grids to determine the order of the error and the dominant terms in the error expansion. The techniques are demonstrated on two examples.