Nd solve vector norm read

Map[First[(1 - x[100 π]) /. NDSolve[{x''[t] + x[t] == 0, x[0] == 1, x'[0] == 0}, x, {t, 0, 100 π}, Method -> ExplicitRungeKutta, NormFunction -> #]]&, {1, 2, ∞}]

{0.005493283411851402, 0.005493283411851402, 0.005493283411851402}

h = 2π / 16.;
grid = h Range[16];
err16 = Sin[grid] - ListCorrelate[{1, -1} / h, Cos[grid], {1, 1}]

{-0.16932384605030953, -0.11903049272323196, -0.05061582589314606, 0.025504641595567312, 0.0977422586015072, 0.15509950277114803, 0.188844253624376, 0.193839178740714, 0.16932384605030942, 0.11903049272323096, 0.05061582589314717, -0.025504641595566313, -0.09774225860150798, -0.1550995027711488, -0.18884425362437562, -0.19383917874071413}

h = 2π / 32.;
grid = h Range[32];
err32 = Sin[grid] - ListCorrelate[{1, -1} / h, Cos[grid], {1, 1}]

{-0.09472827213888085, -0.08795642657727393, -0.07780446486883275, -0.06466252120873017, -0.04903563312173598, -0.031524333153372885, -0.012801570740975432, 0.006413148855794248, 0.025381414738570696, 0.043374287087006635, 0.05970030990726649, 0.07373208329811132, 0.08493037407723936, 0.09286483821008079, 0.09723055868970995, 0.09785976332641878, 0.09472827213888074, 0.08795642657727226, 0.0778044648688343, 0.0646625212087315, 0.04903563312173265, 0.031524333153373885, 0.012801570740976653, -0.006413148855793138, -0.02538141473856992, -0.043374287087008745, -0.05970030990726605, -0.07373208329811187, -0.08493037407724024, -0.0928648382100804, -0.09723055868970917, -0.09785976332641891}

{Norm[err16, 2], Norm[err32, 2]}

{0.5529854533912648, 0.392278712275836}

{Norm[err16, 2] / Sqrt[16], Norm[err32, 2] / Sqrt[32]}

{0.1382463633478162, 0.06934573439134255}