Interval Versions of Central-Difference Method for Solving the Poisson Equation in Proper and Directed Interval Arithmetic

To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h² +k²), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained.