A Characteristic-wise Alternative WENO-Z Finite Difference Scheme for Solving the Compressible Multicomponent Non-reactive Flows in the Overestimated Quasi-conservative Form

Abstract

The fifth, seventh and ninth order characteristic-wise alternative weighted essentially non-oscillatory (AWENO) finite difference schemes are applied to the fully conservative (FC) form and the overestimated quasi-conservative (OQC) form of the compressible multicomponent flows. Several linear and nonlinear numerical operators such as the linear Lax–Friedrichs operator and linearized nonlinear WENO operator and their mathematical properties are defined in order to build a general mathematical (numerical) framework for identifying the necessary and sufficient conditions required in maintaining the equilibriums of certain physical relevant properties discretely. In the case of OQC form, the AWENO scheme with the modified flux can be rigorously proved to maintain the equilibriums of velocity, pressure and temperature. Furthermore, we also show that the FC form cannot maintain the equilibriums without an additional advection equation of auxiliary variable involving the specific heat ratio. Extensive one- and two-dimensional classical benchmark problems, such as the moving material interface problem, multifluid shock-density interaction problem and shock-R22-bubble interaction problem, verify the theoretical results and also show that the AWENO schemes demonstrate less dissipation error and higher resolution than the classical WENO-Z scheme in the splitting form (Nonomura and Fujii in J Comput Phys 340:358–388, 2017).