magic fix of the numerical method in the forward euler schemeunc un nc un we can replace un in the last equation by the recently computed value unc from the first equationunc un nc nc ( ( before justifying this fix more mathematicallylet us try it on the previous example the results appear in fig we see that the amplitude does not growbut the phase is not entirely correct after periods (fig rightwe see significant difference between the numerical and the exact solution decreasing decreases the error for examplewith intervals per periodwe only see small phase error even after , periods (!we can safely conclude that the fix results in an excellent numerical methodlet us interpret the adjusted scheme mathematically first we order ( )-( such that the difference approximations to derivatives become transparentunc un vn nc unc ( ( we interpret ( as the differential equation sampled at mesh point tn because we have on the right-hand side the left-hand side is then forward difference or forward euler approximation to the derivative see fig on the other handwe interpret ( as the differential equation sampled at mesh point tnc since we fig adjusted methodfirst three periods (leftand period - (right