Re: extremely simple problem... can you solve it properly?
 

I totally agree with you.

I do not have to repeat that Numerical methods by definition are approximate primarily due to discretization which is the first step to go on solving any differential equation. Accuracy (i.e. solution that approaches the exact solution) can be increased by optimizing the spatial resolution and temporal resolution (for transient. Now why one needs to get to analytically correct solution after first time step or first iteration. However if one is interested in a solution after time t1 why not divide time period (t1-0) into smaller dt's. The solution at t1 will then be closer to anlytical solution. This is just euqal to increasing the spatial and temporal reslotuion to cope with the abrupt change in the variables from t=0 to t=1. Any numerical analyst knows that you get to solution insteps