Asymptotic behavior and best approximation in computational fluid dynamics

Computational fluid dynamics has many successes for the solution of simple standard problems. For relatively complex problems, especially if nonlinear and of mixed type, the computed approximate solutions are mostly of dubious accuracy and credibility. The difficulty appears fundamental. Model studies in one space dimension suggest that most of such discrete problems are poorly posed. The sequence of computed solutions at successively refined meshes need not converge; and apparently “smooth” computed approximations can “converge” to wrong limits with large global errors. For certain discrete formulations the sequence is asymptotic in the sense of displaying minimum error at some fairly large critical mesh Reynolds number (coarse meshes). This error minimum can be as small as those promised by the correct “convergent approximations” at much smaller meshes. Certain behavior of the computed solutions around such a critical mesh Reynolds number help to identify the “best approximation”. Such analytic inferences have been tested and verified in the computational solutions of successively more complex flows governed by Navier-Stokes equations in two space dimensions. The flow fields due to shockwave-laminar-boundary layer interaction were computed with different discrete formulations and various perturbations. The computed “best approximations” differ little and all compare favorably with available experimental data. Some of such formulations give the “best approximations” at reasonably coarse meshes, requiring much smaller computational effort; and should therefore be favorably considered.