17.3.1999 Mikko Kokkonen, 33015p
The image below shows the solutions of the boundary value problem. The blue curve is the true solution, the green curve is the numerical solution obtained with scheme 1 and the red curve is the solution obtained with scheme 2. Here n = 19.
Estimation of the rate of convergence (using the technique of Project 1.1 that has been implemented in a Matlab-script):
n h E_h1 E_h2 R_1 R_2 4.0000 0.2000 0.0946 0.0054 9.0000 0.1000 0.0486 0.0014 0.9592 1.9987 19.0000 0.0500 0.0247 0.0003 0.9800 1.9997 49.0000 0.0200 0.0099 0.0001 0.9910 1.9999 99.0000 0.0100 0.0050 0.0000 0.9961 2.0000The scheme 2 converges faster because the two-sided finite difference approximation of the first derivative is more accurate than the one-sided (backward) approximation used in the scheme 1.
True Error h error estimate 0.1000 0.0527 1.0417 0.0500 0.0137 0.2604 0.0250 0.0035 0.0651 0.0125 0.0009 0.0163 0.0062 0.0002 0.0041The lesson of this exercise is that if the f'' has a large maximum value compared to the average value of the function, then Theorem 2.2 can largely overestimate the actual error.
n h e_1 e_5 e_100 R_1 R_5 R_100 4 0.2000 0.0002 0.0140 0.0025 9 0.1000 0.0000 0.0037 0.0098 1.9946 1.9100 -1.9721 19 0.0500 0.0000 0.0010 0.0304 1.9986 1.9758 -1.6368 49 0.0200 0.0000 0.0002 0.0362 1.9997 1.9948 -0.1935 99 0.0100 0.0000 0.0000 0.0141 1.9993 1.9990 1.3632 199 0.0050 0.0000 0.0000 0.0037 2.0000 1.9998 1.9105 399 0.0025 0.0000 0.0000 0.0010 1.9999 1.9999 1.9759