Page - 149 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

6.5 RateofConvergence 149 Clearly,when h= b−a n , (n sub-intervals of equal size h for an integration interval [a,b]), an alternative expressionforE followsfrom E=Khr, (6.22) =K ( b−a n )r , (6.23) =K(b−a)r ( 1 n )r , (6.24) which,by introducinganotherconstantC=K(b−a)r, gives E=Cn−r . (6.25) Convergence Rate for the Trapezoidal and Midpoint Methods Using, for example,thetrapezoidalmethod,wemaycarryoutsomeexperimentalrunswithour test problem ∫1 0 3t 2et 3 dt, doublingn foreach run:n=4,8,16.Thecorresponding errors are then 12%, 3% and 0.78%, respectively. These numbers indicate that the error is reducedbyroughlya factor4whendoublingn.Thus, it seems that theerror converges to zero asn−2, which suggests a convergence rate r = 2. In fact, it can beshownmathematically that the trapezoidaland the midpointmethodbothhavea convergencerate r=2, i.e., theyarebothsecond-ordermethods.Soon,wewill see howthis fact (andmore)canbeexploited in the testingofcode. Remarkonthe DefinitionofConvergenceRate When we later address numerical solution methods for ordinary differential equations (Chap.8), convergencerate is essentially defined like in (6.21),we just switch (not required) the symbol h with Δt (i.e., the spacing between computedsolutionvalues). However, with iterative methods for the solving of nonlinear algebraic equations (Chap.7), convergence rate is defined differently. In that case, one usually relates the error at an iteration to the error at the previous iteration, and theconvergencerateappearsasa parameter in that relation.