High Order Bound- and Positivity-Preserving Finite Difference Affine-Invariant AWENO Scheme for the Five-Equation Model of Two-Medium Flows (Communications in Computational Physics, 2024)
Numerical study on compressible two-medium flows has been a hot issuein recent decades. In this study, we design quasi-conservative finite difference alternative weighted essentially non-oscillatory (AWENO) schemes up to the ninth orderfor the five-equation model with the stiffened gas equation of state. We propose uniformly high-order flux-based bound- and positivity-preserving (BP-P) limiters for theAWENO schemes while preserving the equilibrium solutions simultaneously. Thoughthe BP-P limiters are used, the numerical solutions have the tendency to generate oscillations especially near strong shock and/or rarefaction waves, due to the sudden drastic scale transition of the density, pressure, etc. To resolve fine structures and the transition of different scales, the latest affine-invariant WENO (Ai-WENO) interpolation isadopted and generalized up to the ninth order. In addition, we will systematically derive CFL conditions when the Lax-Friedrichs numerical flux is applied. Moreover, weshow the potential of the BP-P limiters for a variant of the five-equation model, usually suggested in finite volume and discontinuous Galerkin methods. For illustrationpurposes, we adopt the AWENO schemes and derive the corresponding CFL conditions. A variety of one- and two-dimensional test problems illustrate the high order ofaccuracy, effectiveness, and robustness of the proposed BP-P Ai-AWENO schemes.