Result: Spatial-Temporal Adaptive-Order Positivity-Preserving WENO Finite Difference Scheme with Relaxed CFL Condition for Euler Equations with Extreme Conditions: Spatial-temporal adaptive-order positivity-preserving WENO finite difference scheme with relaxed CFL condition for Euler equations with extreme conditions

Summary: In extreme scenarios, classical high-order WENO schemes may result in non-physical states. The Positivity-Preserving Limiter (PP-Limiter) is often used to ensure positivity if \(\mathrm{CFL} \leq 0.5\) with a third-order TVD Runge-Kunta (RK3) scheme. This study proposes two novel conservative WENO-Z methods: AT-PP and AO-PP to improve efficiency with \(0.5 < \mathrm{CFL} < 1\) if desired. The AT-PP method detects negative cells after each RK3 stage posteriori and computes a new solution with the PP-Limiter (\(\mathrm{CFL} < 0.5\)) for that step. The AO-PP method progressively lowers the WENO operator's order and terminates with the first-order HLLC solver, proven positivity-preserving with \(\mathrm{CFL} < 1\), only at negative cells at that RK3 stage. A single numerical flux enforces conservation at neighboring interfaces. Extensive 1D and 2D shock-tube problems were conducted to illustrate the performance of AT-PP and AO-PP with \(\mathrm{CFL} = 0.9\). Both methods outperformed the classical PP-Limiter in accuracy and resolution, while AO-PP performed better computationally in some cases. The AO-PP method is globally conservative and accurate, adaptiveness, and robustness while resolving fine-scale structures in smooth regions, capturing strong shocks and gradients with ENO-property, improving computational efficiency, and preserving the positivity, all without imposing a restrictive limit on the CFL condition.