Page - 271 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.5 RateofConvergence 271 midpoint methods), the error was a single number for each choice of n, i.e., the number of sub-intervals. With eachΔti now, however, there is basically one error for each point in the mesh (remember:we computean approximation to a function now, not a single number, as we did with integrals)! What we would like to have, is a single number Ei that we could refer to as “the error” of the corresponding experiment. TherearedifferentwaystoarriveatEi,andwemightreasonasfollows.Foreach of our experiments, at each point in the mesh, there will be a difference (generally notzero)betweenthetruesolutionandourcomputedapproximation.Thecollection ofall these differencesmakesupan errormesh function,havingvaluesat the mesh points only. Also, with a total of Nt time steps, there will be Nt + 1 points in the time mesh (Nt increases when Δt decreases, for a fixed time interval [0,T]). Thus, for each Δti, the error mesh function comprises Nt + 1 function values en, n = 0,1,.. .,Nt. Now, one simple way to get to Ei, is to use the maximum of all en values,9 comparing absolute values. In fact, we did that previously in test_ode_FE_exact_linear.py(Sect. 8.2.7). Other alternatives utilize a constructed error function e(t), being a continuous functionof time. The functione(t)may be generated from the error mesh function by simply connecting successive en values with straight lines. With e(t) in place, onemaychoose to use theL2 norm (read“L-twonorm”)ofe(t), ‖e‖L2 = √∫ T 0 e(t)2dt , (8.85) for Ei. The L2 norm has nice mathematical properties and is much used. When computing the integralof theL2 norm,we may use the trapezoidalmethod,and let the integration points coincide with the mesh points. Because of the straight lines composing e(t), the integral computation will then become exact (within machine precision).Thus, assuminga uniformmesh,wecanproceed towrite (8.85)as ‖e‖L2 ≈ √√ √ √Δt ( 1 2 (e0)+ 1 2 ( eNt )+ Nt−1∑ n=1 (en) 2 ) , (8.86) Finally, we make yet another approximation, by simply disregarding the contribu- tions from e0 and eNt . This is acceptable, since these contributions go to zero as Δt→0.The resultingexpression is called thediscreteL2 norm,and isdenotedby l2. In thisway,we get thefinal andsimplerexpression10 as ‖en‖l2 = √√ √ √Δt Nt−1∑ n=1 (en) 2 . (8.87) 9 This is referred to as the discrete (L∞) norm (read “L-infinity norm”, but often called the “max norm”) for the error mesh functionen. 10 We should add that, in this expression,Δt may be switched with T Nt , followed by droppingT , since thiscommonscalingfactor is independentof thevectorvalues.Finally, it isusuallypreferred touse the length of the vector, i.e.Nt +1 instead ofNt.