Unified formulation for compressible and incompressible flows by using multi-integrated moments I: one-dimensional inviscid compressible flow

This paper presents a novel numerical formulation for computing flows of any Mach number. The following features make the present scheme different from the existing works: (1) The spatial discretizations are constructed by using two kinds of moments which are defined as the volume integrated average (VIA) and the surface integrated average (SIA), both of which are computed as the prognostic variables; (2) the CIP-CSL3 (constrained interpolation profile-conservative semi-lagrangian with third-order polynomial function) scheme is used for the advection transport; (3) a simple artificial compression can be devised by modifying the slope parameter in the CIP-CSL3 scheme. Moreover, the conservation is exactly assured for the VIA quantities. As the first of the series, this paper mainly focuses on the computations of inviscid compressible flows. Our numerical results show that the present method gives correct shock speed, well-resolved shock front, contact discontinuity and rarefaction waves for a wide spectrum of test problems.