Scale-Invariant Multi-resolution Alternative WENO Scheme for the Euler Equations

The finite difference multi-resolution alternative weighted essentially non-oscillatory (MR-AWENO) scheme has been designed to solve hyperbolic conservation laws (Wang et al. in Comput Methods Appl Mech Eng 382:113853, 2021). However, the scheme is not scale-invariant and generates numerical oscillations near strong shocks. To overcome this issue, we design the scale-invariant Si-weights and the MR-AWENO-Si operator by including the new global smoothness indicator and the descaler. The resulting scale-invariant MR-AWENO-Si scheme captures discontinuity of any scale in the essentially non-oscillatory (ENO) way efficiently and robustly. We also demonstrate an interesting application of the scale-invariant scheme to achieve the well-balanced property for the compressible Euler equations under a gravitational potential field by modifying the numerical fluxes, reformulating the source terms, and enforcing the MR-AWENO-Si operator. A theoretical proof is given, and extensive one- and two-dimensional classical examples are used to verify the performance of the MR-AWENO-Si scheme in terms of accuracy, robustness, and well-balanced property.