In the same way as in Problem 1, compare the performance of the central difference scheme (4.10) and the fourth-order scheme (4.15).

Problem 1

Consider the function \(u=\sin x\). Apply the forward difference (4.4), backward difference (4.9), and central difference (4.10) to evaluate the derivative \(d u / d x\) at \(x=1.0\). Use the uniform grids with steps \(\Delta x=\) \(1.0,0.5,0.1\), and 0.01 . Compare the results with the exact value of the derivative. Determine which grid steps give accurate results (e.g. with the error less than 0.01) for each scheme. Explain why one scheme is more accurate than the two others.