Completed repeated Richardson extrapolation for compressible fluid flows

Abstract

Abstract Richardson extrapolation is a powerful approach for reducing the spatial discretization errors and increasing, in this way, the accuracy of the computed solution obtained by using many numerical methods for solving different scientific and engineering problems. This approach has been used in a variety of computational fluid dynamics problems to reduce the numerical error, but it has been mainly restricted to the computation of incompressible fluid flows and on grids with coincident nodes. The purpose of this work is to present a completed repeated Richardson extrapolation (CRRE) procedure for a more generic type of grids not necessarily with coincident nodes, and test it on compressible fluid flows. Three tests are performed for one- and quasi-one-dimensional Euler equations, i.e. (i) the Rayleigh flow, (ii) the isentropic flow, and (iii) the adiabatic flow through a nozzle. The last test presents a normal shock wave. In order to build a simple CRRE solver, these problems are solved using a first-order upwind-type finite difference method as the base scheme. The normal shock wave problem is also solved using a high-order weighted essentially non-oscillatory (WENO) scheme to compare it with the CRRE procedure. The procedure we have proposed can increase the achieved accuracy and significantly decrease the magnitude of the spatial error in all three tests. Its performance can be best demonstrated in the Rayleigh flow test, where the spatial discretization error is reduced by seven orders of magnitude and the achieved accuracy is increased from 0.998 to 6.62 on the grid with 10240 nodes. Similar performance can be observed for the isentropic flow, for which the spatial discretization error is reduced by nine orders of magnitude and the achieved accuracy is again increased from 0.996 to 6.73 on the grid with 10240 nodes. Finally, in the adiabatic flow with a normal shock wave, the procedure can reduce the spatial discretization error both upstream and downstream of the shock. We do remark, however, that the more expensive high-order WENO scheme has errors of lower magnitude upstream of the shock and has a sharper shock transition for this shocked test case.