An Interval Version of the Crank-Nicolson Method – The First Approach

To study the heat or diffusion equation, the Crank-Nicolson method is often used. This method is unconditionally stable and has the order of convergence

O

(

k

2

 + 

h

2

), where

k

and

h

are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including not only the method error, but also representation and rounding errors. Therefore, we propose an interval version of the Crank-Nicolson method from which we would like to obtain solutions including the discretization error. Applying such a method in interval floating-point arithmetic allows one to obtain solutions including all possible numerical errors. Unfortunately, for the proposed interval version of Crank-Nicolson method, we are not able to prove that the exact solution belongs to the interval solutions obtained. Thus, the presented method should be modified in the nearest future to fulfil this necessary condition. A numerical example is presented. Although in this example the exact solution belongs to the interval solutions obtained, but the so-called wrapping effect significantly increases the widths of these intervals.