Numerical Study of Dissipative Weak Solutions for the Euler Equations of Gas Dynamics

We study dissipative weak (DW) solutions of the Euler equations of gas dynamics using the first-, second-, third-, fifth-, seventh-, and ninth-order local characteristic decomposition-based central-upwind (LCDCU), low-dissipation central-upwind (LDCU), and viscous finite volume (VFV) methods, whose higher-order extensions are obtained via the framework of the alternative weighted essentially non-oscillatory (A-WENO) schemes. These methods are applied to several benchmark problems, including several two-dimensional Riemann problems and a Kelvin-Helmholtz instability test. The numerical results demonstrate that for methods converging only weakly in space and time, the limiting solutions are generalized DW solutions, approximated in the sense of ${\cal K}$-convergence and dependent on the numerical scheme. For all of the studied methods, we compute the associated Young measures and compare the DW solutions using entropy production and energy defect criteria.

💡 Research Summary

This paper investigates dissipative weak (DW) solutions of the multidimensional compressible Euler equations by means of a suite of high‑order finite‑volume and finite‑difference schemes. Three families of numerical fluxes are considered: the local characteristic decomposition‑based central‑upwind (LCDCU) flux, the low‑dissipation central‑upwind (LDCU) flux, and the viscous finite‑volume (VFV) flux. All three are embedded in the alternative weighted essentially non‑oscillatory (A‑WENO) framework, yielding schemes of order 1, 2, 3, 5, 7 and 9. The authors apply these methods to a set of benchmark problems: several two‑dimensional Riemann problems and a Kelvin‑Helmholtz (KH) instability test.

The theoretical backbone rests on the concepts of consistent approximations, K‑convergence, and Young measures. A sequence of numerical solutions generated on refined meshes is shown to converge only weakly in the usual Lebesgue spaces; strong convergence is absent for the raw solutions. However, when the solutions are averaged in space and time (Cesàro averaging), the sequence exhibits K‑convergence to a limit that can be represented by a Young measure. This limit is precisely a DW solution: it satisfies the Euler equations in the distributional sense, respects the entropy inequality, and includes a Reynolds‑stress defect and an energy defect that quantify the unresolved sub‑grid oscillations.

The numerical experiments confirm the theoretical predictions. For each test case, the authors compute density, velocity and pressure fields for all six orders and three flux families. They observe that, even with identical initial and boundary data, the limiting DW solution depends on the choice of numerical method and its order. Higher‑order schemes produce richer fine‑scale structures, which manifest as broader support in the associated Young measures. The paper quantifies these differences by constructing the Young measures in selected sub‑domains and visualising their probability distributions.

A central contribution is the systematic assessment of two physical selection criteria for DW solutions: entropy production and energy defect. Entropy production is measured by integrating the entropy inequality residual; energy defect is obtained from the balance between kinetic‑plus‑internal energy and the work of the Reynolds‑stress term. The results show that low‑dissipation schemes (LDCU, high‑order A‑WENO) generate smaller entropy production, while the VFV scheme, by virtue of its explicit viscous term, can directly control the energy defect. Consequently, the authors propose a hierarchy for selecting the most physically admissible DW solution: prioritize minimal entropy production, then verify that the energy defect lies within the bounds dictated by the theory (i.e., between 2 min{1, γ‑1} E and 2 max{1, γ‑1} E).

The paper concludes with a discussion of limitations and future work. The current study is restricted to uniform Cartesian grids and a limited set of initial configurations; extending the methodology to unstructured meshes, complex geometries, and realistic boundary conditions remains an open challenge. Moreover, while the numerical construction of Young measures is demonstrated, a rigorous analytical link between these measures and the underlying turbulence statistics is still to be established. Nonetheless, the work provides a clear roadmap for using high‑order, low‑dissipation schemes to obtain convergent approximations of DW solutions, and it offers concrete criteria—entropy production and energy defect—for selecting among the multiple admissible weak limits that arise in multidimensional gas dynamics simulations.