README.md

2026RCC

Codex/ChatGPT (July 2026)

Summary

The paper develops an efficient modal discontinuous Galerkin method for multicomponent compressible Euler flows. Conservative variables are advanced in time, while the spatial residual is evaluated from the $L^2$ projection of entropy variables onto the discontinuous Galerkin space. This entropy-projection formulation avoids the elementwise matrix inversions required when entropy variables are evolved directly. A Direct Enforcement of Entropy Balance correction supplies semidiscrete entropy conservation or stability without costly over-integration, and directional artificial diffusion controls oscillations near shocks. Manufactured-solution tests verify high-order accuracy in one and two dimensions. Compared with an over-integrated entropy-variable solver, the proposed method reduces CPU time by 30--78% while retaining the designed entropy behavior. Multicomponent Riemann problems confirm robust shock treatment, and two-dimensional implosion and shock--inhomogeneity tests show sharp resolution of interacting waves, material-interface deformation, vortex formation, and small-scale interfacial instabilities.

Contributions

1. Evolved conservative variables while constructing residuals from projected entropy variables.
2. Enforced entropy conservation or stability without numerical over-integration.
3. Added directional shock capturing for discontinuity-induced oscillations.
4. Verified high-order convergence and 30--78% lower cost than the comparison method.
5. Resolved multicomponent waves and interfaces in demanding one- and two-dimensional tests.