Lecture 6: =1=Truncation Error, Consistency, and Convergence

Local truncation errorModified EulerFor the modified Euler, the amplification factor is given byQ(hλ) = 1 + hλ + (hλ)2 .2It is stable for 0 < h < 2|λ| .For consistency, we examine the local truncation error:τ n+1 (h) = (eλh − Q(λh))y nh= 1 [(1 + hλ + 1 h2! (hλ)2 + 1 )3! (hλ)3 + O(h 4 ) ..()].. − 1 + hλ + (hλ)2 y n2= 1 ( )1h 3! (hλ)3 + O(h 4 )=( 1 3! h2 λ 3 + O(h 3 ))y n ⇒ τ n+1 (h) = O(h 2 )() Truncation Error, Consistency, and Convergence 8 October 2012 20 / 54y n