High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations

This paper deals with the development of accurate one-step schemes for the numerical simulation of unsteady compressible flows. Pursuing our work in Daru and Tenaud [V. Daru, C. Tenaud, Comput. Fluids 30 (2001) 89] where third-order schemes were considered, we follow the Lax–Wendroff approach to develop high order TVD combined time–space schemes by correcting the successive modified equations. In the scalar case, TVD schemes accurate up to seventh order (OSTVD7) in time and space are obtained (in smooth regions and away from extrema). To avoid the clipping and the loss of accuracy that is common to the TVD schemes near extrema, we develop monotonicity-preserving (MP) conditions derived from Suresh and Huynh [A. Suresh, H.T. Huynh, J. Comput. Phys. 136 (1997) 83] to locally relax the TVD limitation for this family of one-step schemes. Numerical results for long time integration in the scalar case show that the MP one-step approach gives the best results compared to several multistage schemes, including WENO schemes. The extension to systems and to the multidimensional case is done in a simplified way which does not preserve the scalar order of accuracy. However we show that the resulting schemes have a very low level of error. For validation, the present algorithm has been checked on several classical one-dimensional and multidimensional test cases, including both viscous and inviscid flows: a moving shock wave interacting with a sine wave, the Lax shock tube problem, the 2D inviscid double Mach reflection and the 2D viscous shock wave–vortex interaction. By computing these various test cases, we demonstrate that very accurate results can be obtained by using the one-step MP approach which is very competitive compared to multistage high order schemes.