Discretization Error Estimates Using Exact Solutions to Nearby Problems

A methodology is presented for generating exact solutions to equations that are “near” the Navier-Stokes equations. First, a highly accurate numerical solution to the Navier-Stokes equations is computed. Second, an analytic function is fit to the numerical solution by least squares optimization. Next, this analytic solution is operated on by the Navier-Stokes equations (including auxiliary relations) to obtain a small analytic source term. When the Navier-Stokes equations are perturbed by adding this source term, the analytic function is recovered as the exact solution. Approaches are presented which address the “goodness” of the curve-fitting procedure and the “nearness” of the modified set of equations to the Navier-Stokes equations. Two examples are given for compressible fluid flow: fully developed flow in a channel, and lid-driven cavity flow. The channel flow is fully captured by a third-order polynomial fit, while the driven-cavity solution is not adequately represented by polynomial curve fits up to fourth order. The generation of an exact solution to a set of equations near the Navier-Stokes equations allows for the evaluation of various discretization error estimators, without reverting to simplification of the governing equations or use of a highly refined “truth” mesh. Preliminary results for a number of extrapolation-based error estimators are also presented.