Transcription of Local Truncation Error
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Local Truncation Error
R. Mark ProsserCS 370 Local Truncation ErrorThelocal Truncation Error (LTE) of a numerical method is an estimate of the errorintroduced in a single iteration of the method, assuming that everything fed into the methodwas perfectly accurate. Recall thaty1,y2,..,yNrefer to the numerically computed valuesandy(t1),y(t2),..,y(tN)refer to the corresponding exact values (so thatyn y(tn)). Todetermine the Local Truncation Error , analyse a general iteration of a method where the valueyn+1is computed. We want to determine the difference,LTE=y(tn+1) yn+1based on the assumption thatyn+1is determined from exact information. That is, if we havea method of the formyn+1= (tn,yn,f,h)where represents the formula for the numerical method, then we are going to assume thatyn=y(tn), +1= (tn,y(tn),f,h)and, having made this assumption, we are going to examine the differencey(tn+1) yn+1which we call the Local Truncation :1.
+O(h) h2 2 +O(h3) = y(t n)+y0(t n)h+(y0(t n+1) y0(t n)) h 2 +O(h3)+O(h3) = y(t n)+ h 2 (y 0(t n)+y(t n+1))+O(h3) Finally, since y0(t) = f(t;y(t)), we have y(t n+1) = y(t n)+ h 2 (f(t n;y(t n))+f(t n+1;y(t n+1)))+ O(h3) (1) from which we get the Trapezoidal method. The terms on the right-hand side, other than the O(h3) term, are what you would ...
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