Partially characteristic-decomposition-free high-order non-oscillatory scheme for compressible Euler equations

Building upon the alternative finite difference weighted essentially non-oscillatory (WENO) formulation introduced in prior works [Shu and Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,” J. Comput. Phys. 77, 439–471 (1988)], Xu and Shu [“Local characteristic decomposition–free high-order finite difference WENO schemes for hyperbolic systems endowed with a coordinate system of Riemann invariants,” SIAM J. Sci. Comput. 46, A1352–A1372 (2024)] developed a local characteristic decomposition (LCD) free WENO scheme for the shallow water equations. By leveraging the interpolation within the coordinate system of Riemann invariants, their approach circumvents the computationally expensive LCD required in traditional WENO schemes. This work advances the methodology in two key directions: (1) Extension to the Compressible Euler Equations: For systems involving contact, rarefaction, and shock waves, a targeted essentially non-oscillatory (TENO)-based discontinuity indicator is employed to classify wave types dynamically. An adaptive variable selection strategy is then proposed to optimize the choice of interpolation variables (e.g., conserved variables for smooth regions and corresponding Riemann invariants for contact and rarefaction waves), thereby significantly reducing the reliance on both LCD and nonlinear shock-capturing schemes. Numerical benchmarks confirm that interpolating Riemann invariants in contact and rarefaction waves effectively suppresses spurious oscillations. A detailed analysis of wave-type distributions in test cases underscores the efficiency of the adaptive strategy. (2) Hybrid TENO-I-avs Framework: The alternative WENO formulation is extended to a new hybrid method that integrates the interpolation-based TENO framework, the high-order linear scheme, and the adaptive variable selection, termed hybrid TENO-I-avs. Here, a TENO voting strategy governs discontinuity classification, interpolation of selected wave-relevant variables, and higher-order central differencing terms, enhancing robustness while preserving low numerical dissipation. Comparative studies against existing limiters [Christlieb et al., “A high-order finite difference Weno scheme for ideal magnetohydrodynamics on curvilinear meshes,” SIAM J. Sci. Comput. 40, A2631–A2666 (2018)] and approximate dispersion relation analysis validate the superior dissipation characteristics and stability of the scheme. A systematic analysis of the growth factors of nonlinear schemes under fixed Courant–Friedrichs–Lewy conditions is also conducted.