Approaches to the Numerical Estimates of Grid Convergence of NSE in the Presence of Singularities

Abstract

Abstract Evaluating the order of accuracy (order) is an integral part of the development and application of numerical algorithms. Apart from theoretical functional analysis to place bounds on error estimates, numerical experiments are often essential for nonlinear problems to validate the estimates in a reliable answer. The common workflow is to apply the algorithm using successively finer temporal/spatial grid resolutions δ i ${\delta _i}$ , measure the error \isin i ${\isin _i}$ in each solution against the exact solution, the order is then obtained as the slope of the line that fits ( log \isin i , log δ i ) $(\log {\isin _i}, \log {\delta _i})$ . We show that if the problem has singularities like divergence to infinity or discontinuous jump, this common workflow underestimates the order if solution at regions around the singularity is used. Several numerical examples with different levels of complexity are explored. A simple one-dimensional theoretical model shows it is impossible to numerically evaluate the order close to singularity on uniform grids.