A metal rod is cooled by a bath at both ends and its interior temperature is found by…
PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Research software practice, verification and validation · Research software practice, verification and validation
Scientific Computing for Physicists, Ch Are they the right equations? Solution verification, benchmarks and validation
Problem
A metal rod is cooled by a bath at both ends and its interior temperature is found by an explicit finite-difference scheme, with the diffusion number α held fixed on every grid. The temperature at the rod's centre at a fixed time is computed on N=25, 50 and 100 intervals (so the refinement ratio between successive grids is r=2): T(N=25)=310.0222K,T(N=50)=310.0158K,T(N=100)=310.0142K. Someone argues: "the value changed by only 0.0016 K (about 5 parts per million) between N=50 and N=100, so the N=100 grid is converged; take 310.0142 K as the exact answer, with no further numerical uncertainty to report."
((a)) Compute the observed order p^, the Richardson-extrapolated value fext, and the grid convergence index at the fine grid, using a safety factor Fs=1.25.
((b)) State what kind of evidence this three-grid study actually provides, and identify the specific class of coding error that even a clean, monotone, second-order-convergent three-grid study of this kind cannot detect.