Estimation of Discretization Errors Using the Method of Nearby Problems

Abstract

The method of problems is developed as an approach for estimating numerical errors due to insufficient mesh resolution. A key aspect of this approach is the generation of accurate, analytic curve fits to an underlying numerical solution. Accurate fits are demonstrated using fifth-order Hermite splines that provide for solution continuity up to the third derivative, which is recommended for second-order differential equations. This approach relies on the generation of a new problem (and corresponding exact solution) that is nearby the original problem of interest, and the nearness requirements are discussed. The method of problems is demonstrated as an accurate discretization error estimator for steady-state Burgers's equation for a viscous shock wave at Reynolds numbers of 8 and 64. A key advantage of using the method of problems as an error estimator is that it requires only one additional solution on the same mesh, as opposed to multiple mesh solutions required for extrapolation-based error estimators. Furthermore, the present results suggest that the method of problems can produce better error estimates than other methods in the preasymptotic regime. The method of problems is also shown to provide a useful framework for evaluating other discretization error estimators. This framework is demonstrated by the generation of exact solutions to problems Burgers's equation as well as a form of Burgers's equation with a nonlinear viscosity variation.