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

摘要

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 delta(i), measure the error epsilon(i) in each solution against the exact solution, the order is then obtained as the slope of the line that fits (log epsilon(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.