John Crank is best known for his joint work with Phyllis Nicolson on the heat equation, where a continuous solution u(x, t) is required which satisfies the second order partial differential equation
ut - uxx = 0for t > 0, subject to an initial condition of the form u(x, 0) = f (x) for all real x. They considered numerical methods which find an approximate solution on a grid of values of x and t, replacing ut(x, t) and uxx(x, t) by finite difference approximations. One of the simplest such replacements was proposed by L F Richardson in 1910. Richardson's method yielded a numerical solution which was very easy to compute, but alas was numerically unstable and thus useless. The instability was not recognised until lengthy numerical computations were carried out by Crank, Nicolson and others. Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level.
Article by: G M Phillips, St Andrews