A table of the reduced van der Waals pressure along the isotherm
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists
Scientific Computing for Physicists, Ch The numerical toolkit closed: backward error and the audit of a library routine
Problem
A table of the reduced van der Waals pressure p=P/Pc along the isotherm T/Tc=0.9, corrected by the Maxwell construction, has grid lines at v=V/Vc=0.50,0.75,1.00,…,4.00, a spacing h=0.25. Along this line of a bilinear table, interpolation reduces to linear interpolation in v. The raw isotherm is p=8(0.9)/(3v−1)−3/v2, and the Maxwell construction replaces it by the constant ps=0.6470 for 0.6034<v<2.3488.
((a)) For each of the 14 cells, find the largest error of the linear interpolant anywhere in the cell. Then halve the spacing, with a new grid line at each cell's midpoint, and find the largest error in the same cell again. Give the ratio of the two errors for every cell where the error is not zero.
((b)) Explain why six cells have zero error, and say which cells the smooth bound does not cover.
((c)) Two cells, [2.25,2.50] and [2.50,2.75], both give ratios well below 4. Halve each three more times and use the sequence of ratios to decide which one contains the kink.