Development of the generalized MacCormack scheme and its extension to low Mach number flows

Summary The low Mach number performance of the MacCormack scheme is examined. The inherent dissipation in the scheme is found to suffer from the degradation in accuracy observed with traditional, density‐based methods for compressible flows. Two specific modifications are proposed, leading to the formation of the generalized MacCormack scheme within a dual‐time framework (called GMC‐PC). The first modification involves reformulating the flux by splitting it into particle convection and acoustic parts, with the former terms treated using the traditional MacCormack discretization and the latter terms augmented by the addition of a pressure‐based artificial dissipation. The second modification involves a reformulation of the traditional nonlinear fix introduced by MacCormack in 1971, which is found to be necessary to suppress pressure oscillations at low Mach numbers. The new scheme is demonstrated to have superior performance, independent of Mach number, compared with standard MacCormack implementations using several canonical test problems. Copyright © 2017 John Wiley & Sons, Ltd. The traditional MacCormack scheme is found to give unsatisfactory results for low Mach number flows. This paper proposes a new version of the scheme, the generalized MacCormack scheme with preconditioning, to address the limitations found in the original scheme. This new scheme exhibits superior accuracy and efficiency for low Mach number flows with minimal numerical dissipation. Several canonical flows illustrate the improvements in the new scheme relative to the traditional scheme.