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.

Figures (6)

• Figure 1: Grid-dependence study for the DRV reconstruction on $N=200,400,800,1600$. Left: the maximum refined wave-location error $E_{\max}^{\mathrm{ref}}$. Sod stays at roundoff, the dominant visible residual is the boundary-controlled LeBlanc rarefaction head, and the severe-expansion errors remain small but nonzero for the one-step closure. Right: the $10$--$90$ density contact width $W_\rho$. The dashed reference line is $\Delta x$, showing that the reconstructed contact width remains essentially one cell across the tested resolutions.
• Figure 2: Sod shock tube on $N=600$ cells at $t=0.15$. Dashed: exact density. Dotted black: baseline WENO--5+HLLC density. Solid: reconstructed density. The one-step closure aligns the refined wave locations with the exact self-similar Riemann solution to plotting accuracy.
• Figure 3: Severe expansion shock tube on $N=1000$ cells at $t=0.12$ (logarithmic vertical scale). Dashed: exact internal energy. Dotted black: baseline WENO--5+HLLC internal energy. Solid: reconstructed internal energy. The one-step closure removes the contact-layer defect and aligns the wave locations to within a few $10^{-4}$ to $10^{-3}$.
• Figure 4: LeBlanc shock tube on $N=1000$ cells at $t=0.5$. Dashed: exact internal energy. Dotted black: baseline WENO--5+HLLC internal energy. Solid: reconstructed internal energy. The contact and shock are much better aligned after the one-step closure, and the contact-layer thermodynamic contamination is strongly reduced.
• Figure 5: Independent comparison with the two jump-like non-DRV reconstructions at common resolution $N=600$ cells. Top: Sod density. Middle: severe-expansion internal energy near the contact. Bottom: LeBlanc internal energy near the contact. At the plotting scales shown here, the one-step and exact-closure DRV curves are visually indistinguishable; in either case the DRV reconstruction is visibly sharper and is the only one among the three reconstructions that eliminates the positive LeBlanc contact overshoot while keeping the contact nearly discontinuous.

Theorems & Definitions (19)

• Lemma 2.1: First-moment transport for a regularized differentiated shock layer
• proof
• Proposition 3.1: Obstruction to post-differentiation diagonalization
• proof
• Proposition 3.2: Local orthogonality at a pure contact
• proof
• Lemma 3.3: Exact algebraic form of the smooth DRVs
• proof
• Proposition 3.4: Family selectivity for the 1-wave and 2-wave detectors
• proof