Sub-cell Wave Reconstruction from Differentiated Riemann Variables

Abstract

We introduce a postprocessing procedure that recovers sub-cell wave geometry from a standard one-dimensional Euler shock-capturing computation using differentiated Riemann variables (DRVs) -- characteristic derivatives that separate the three wave families into distinct localized spikes. Filtered DRV surrogates detect the waves, plateau sampling extracts the local states, and a pressure-wave-function Newton closure completes the geometry. The entire pipeline adds less than $0.25\%$ to the cost of a baseline WENO--5/HLLC solve. For Sod, a severe-expansion problem, and the LeBlanc shock tube, wave locations are recovered to within roundoff or $O(10^{-4})$ and the contact is sharpened to one cell width; a pattern-agnostic extension handles all four Riemann configurations with errors at the $10^{-6}$--$10^{-8}$ level. Direct comparison with MUSCL--THINC--BVD and WENO-Z--THINC--BVD shows that neither reproduces the combination of sharp contacts, small contact-window internal-energy error, and elimination of the LeBlanc positive overshoot achieved by the DRV reconstruction.