Distinct-positivity-preserving methods for fifth-order finite volume ALW-MR-WENO schemes with a bigger sufficient CFL number for extreme problems

In this paper, the new distinct-positivity-preserving (DPP) methods are proposed for fifth-order finite volume multi-resolution WENO schemes with adaptive linear weights (which are termed as the ALW-MR-WENO schemes) [J. Comput. Phys., 493 (2023), 112471] for solving some extreme problems on structured meshes. Associated WENO spatial reconstruction procedures only require two hierarchical unequal-sized central spatial stencils for achieving fifth-order accuracy in smooth regions and keeping the essentially non-oscillatory property in non-smooth regions. One redefines five cell averages after performing such spatial reconstructions and then designs one quartic polynomial and two cubic reconstruction polynomials based on them in one dimension. After that, a new detective process is used to examine the positivity of the point values of such cubic reconstruction polynomials at some checking points inside the target cell. If the negativity happens, a new over-determined compression limiter is carried out to ensure that the compressed polynomials can achieve the positivity in the whole target cell instead of only at some discrete points inside it. The quartic polynomial could use its four quadrature point values by using a four-point Gauss-Lobatto quadrature formula when performing numerical integrations. Since it is easy to rewrite the point values of one quartic polynomial with the point values of two cubic polynomials timing with associated linear weights, both cubic polynomials could use a four-point Gauss-Lobatto quadrature formula or a three-point Simpson's quadrature formula without losing any algebra precisions. By doing this novelty switching methodology between two quadrature formulas, it is the first time to theoretically prove and increase a sufficient CFL number from 1/12 to 1/7.2 for the fifth-order WENO schemes in one dimension. This methodology can be easily extended to multi-dimensions. Finally, the novelty DPP methods for fifth-order ALW-MR-WENO schemes with a larger practical CFL number of 0.6, instead of the sufficient CFL number of 1/7.2, are also simple and robust enough for some extreme problems without timely halving the CFL number to avoid the appearance of negative density and negative pressure.