Multi-dimensional uniform-positivity-preserving methods for high-order finite-volume multi-resolution weighted essentially non-oscillatory schemes with adaptive linear weights

Multi-dimensional uniform-positivity-preserving (UPP) methods are proposed for high-order finite-volume multi-resolution weighted essentially non-oscillatory schemes with adaptive linear weights (ALW-MR-WENO schemes) when solving the Euler equations on structured meshes. By utilizing two central spatial stencils, the ALW-MR-WENO schemes generate a polynomial vector and maintain high-order accuracy in smooth domains while preserving the non-oscillatory property near strong discontinuities. The cell averages are reformulated to construct the auxiliary polynomial vectors for the UPP methods. By applying a nested bisection method, the positivity of the density polynomial and the polynomial on the numerator of pressure rational function are detected over the computational cell instead of at some Gauss–Lobatto quadrature points as usual. Once the negativity occurs, a novel nonlinear compression limiter is imposed to suppress the positivity of density and pressure functions throughout the computational cell while preserving the high-order accuracy. Furthermore, this work introduces the Hu–Tan–Zhu formulas for algebraic polynomials of corresponding degrees that require only three points and ensure a larger boundary quadrature weight of 1/6, where all quadrature weights remain positive and their summation is one. These new formulas enable a rigorous proof of uniformly sufficient Courant–Friedrichs–Lewy number of 1/6 for arbitrary high-order UPP ALW-MR-WENO schemes, marking a significant improvement over the classical positivity-preserving (PP) methods. The numerical results indicate that the UPP methods reduce approximately 5%–65% computational time compared to the classical PP methods with their sufficient Courant–Friedrichs–Lewy numbers are 1/12, 1/20, or 1/30 as the ALW-MR-WENO schemes increase their precisions from the fifth- to seventh- or ninth-order accuracy, respectively.