Completed repeated Richardson extrapolation for compressible fluid flows

•The original completed Richardson extrapolation suffers from order limitation when used repeatedly.•A completed repeated Richardson extrapolation procedure for a more generic type of grid is presented.•The completed repeated Richardson extrapolation works with compressible fluid flow problems.•The achieved accuracy of first-order smooth solutions can be greatly improved.•The error magnitude of first-order solutions can be greatly reduced, even in solutions with shocks, except for the shock region. Richardson extrapolation is a powerful approach for reducing spatial discretization errors and increasing, in this way, the accuracy of the computed solution obtained by use of 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 numerical errors, but its use has been restricted mainly to the computation of incompressible fluid flows and on grids with coincident nodes. In this work we present a completed repeated Richardson extrapolation (CRRE) procedure for a more generic type of grid not necessarily with coincident nodes, and test it on compressible fluid flows. Three tests are performed for one-dimensional and quasi-one-dimensional Euler equations: (i) Rayleigh flow, (ii) isentropic flow, and (iii) adiabatic flow through a nozzle. The last test involves a normal shock wave. To build a simple solver, these problems are solved by a first-order upwind-type finite difference method as the base scheme. The normal shock wave problem is also solved with a high-order weighted essentially nonoscillatory (WENO) scheme to compare it with the CRRE procedure. The procedure we propose can increase the achieved accuracy and significantly decrease the magnitude of the spatial error in all three tests. Its performance is 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 a grid with 10,240 nodes. Similar performance is observed for isentropic flow, for which the spatial discretization error is reduced by nine orders of magnitude and the achieved accuracy is increased from 0.996 to 6.73 on a grid with 10,240 nodes. Finally, in adiabatic flow with a normal shock wave, the procedure can reduce the spatial discretization error both upstream and downstream of the shock. However, the more expensive high-order WENO scheme results in