In this paper, one-dimensional and two-dimensional pure-positivity-preserving (PPP) methods are proposed for fifth-order finite volume multi-resolution WENO (MR-WENO) schemes to solve some extreme problems on structured meshes. The MR-WENO spatial reconstruction procedures only require one five-cell, one three-cell, and one one-cell stencils for achieving uniform fifth-order accuracy in smooth regions and keeping essentially non-oscillatory property in non-smooth regions in one dimension. One redefines five new cell averages vectors after performing such spatial reconstructions and design one quartic polynomials vector and three quadratic polynomials vectors based on them. After that, a new detective process is used to examine the positivity of density and pressure of three quadratic polynomials vectors inside the whole target cell. If the negativity happens, a new compression limiter is carried out to enable the positivity of density and pressure of three quadratic polynomials vectors over the whole target cell and the positivity of density and pressure of one quartic polynomials vector at the midpoint of the target cell. It is a new way to design the positivity-preserving methods to keep fifth-order accuracy and the positivity over the target cell instead of only at some discrete Gauss-Lobatto quadrature points, since the precise minimum values of the density and pressure are now available. Then a theoretically proof is given to increase the optimal sufficient CFL number from 1/12 to 1/6 for the fifth-order WENO schemes. This methodology can be expanded to multi-dimensions easily. Unlike some classical positivity-preserving methods, the PPP methods could apply a special four-point Gauss-Lobatto quadrature formula or any other quadrature formulas on condition that their numerical precision is no smaller than four. Since the optimal CFL number of 1/6 is a sufficient but not necessary condition, the novelty PPP methods for fifth-order finite volume MR-WENO schemes with a larger practical CFL number of 0.6 are also available and robust enough when simulating some extreme problems without timely halving its value.

中文翻译：

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具有最佳 CFL 数的纯正性守恒方法，用于结构化网格上的五阶 MR-WENO 方案


该文针对五阶有限体积多分辨率WENO（MR-WENO）方案提出了一维和二维纯正正守恒（PPP）方法，以解决结构化网格上的一些极端问题。MR-WENO 空间重建程序只需要一个五单元、一个三单元和一个单单元模板，即可在平滑区域实现统一的五阶精度，并在一维的非平滑区域中保持基本非振荡特性。在执行此类空间重建后，人们重新定义了五个新的单元平均向量，并基于它们设计了一个四次多项式向量和三个二次多项式向量。之后，使用一个新的检测过程来检查整个目标单元内三个二次多项式向量的密度和压力的正性。如果发生负性，则执行新的压缩限制器，以启用整个目标单元上三个二次多项式向量的密度和压力的正性，以及目标单元中点处的一个四次多项式向量的密度和压力的正性。这是一种设计正性保持方法的新方法，以保持目标单元的五阶精度和正性，而不仅仅是在一些离散的 Gauss-Lobatto 正交点处，因为现在可以获得密度和压力的精确最小值。然后，给出了一个理论证明，将五阶 WENO 方案的最佳足够 CFL 数从 1/12 增加到 1/6。这种方法可以很容易地扩展到多维度。 与一些经典的正性守恒方法不同，PPP 方法可以应用特殊的四点 Gauss-Lobatto 求积公式或任何其他求积公式，条件是它们的数值精度不小于 4。由于最佳 CFL 数 1/6 是充分但非必要的条件，因此在模拟一些极端问题而不及时减半其值的情况下，具有更大实际 0.6 的五阶有限体积 MR-WENO 方案的新 PPP 方法也可用且足够稳健。