Example 6.  Compare the 2D Trapezoidal and 2D Simpson rule approximations to     that were calculated in Examples 1 and 4?

Solution 6.

We can compare the error in these approximations.

The error for the Trapezoidal 2D rule has the form      where h and k are the step sizes for the variables x and y, respectively.
The error for Simpson's 2D rule has the form      where h and k are the step sizes for the variables x and y, respectively.
For the above examples, we have the following results.

Remark.  Since both the step sizes were reduced by a factor of the remainder term   should be reduced by approximately .  

Remark.  Since both the step sizes were reduced by a factor of the remainder term   should be reduced by approximately .  

Therefore, both the 2D Trapezoidal and 2D Simpson's rules is behaving as predicted.

(c) John H. Mathews 2004