Intermittent Sub-grid Wave Correction from Differentiated Riemann Variables

Abstract

We introduce a low-cost every-$K$-step correction for one-dimensional Euler computations. The correction uses differentiated Riemann variables (DRVs) -- characteristic derivatives that isolate the left acoustic wave, the contact, and the right acoustic wave -- to locate the current wave packet, sample the surrounding constant states, perform a short Newton update for the intermediate pressure and contact speed, and conservatively remap a sharpened profile back onto the grid. The ingredients are elementary -- filtered centered differences, local state sampling, a single Newton step, and conservative cell averaging -- yet the effect on accuracy is disproportionate. On a long-time severe-expansion benchmark ($N=900$, $t=0.4$), intermittent correction drives the intermediate-state errors from $O(10^{-2})$ to $O(10^{-13})$, i.e. to machine precision. On a long-time LeBlanc benchmark ($N=800$, $t=1$), the method crosses a qualitative threshold: one-shot final-time reconstruction fails entirely (shock location error $2.7\times 10^{-1}$), whereas correction every three steps recovers an almost exact sharp solution with contact and shock positions accurate to machine precision. The same detector-and-Newton mechanism handles two-shock and two-rarefaction packets without case-specific logic, with plateau improvements of four to sixteen orders of magnitude. In an unoptimized Python prototype the wall-clock overhead is below a factor of two even on the most aggressively corrected benchmark. To our knowledge, no comparably lightweight fixed-grid add-on has been shown to recover this level of coarse-grid accuracy on the long-time LeBlanc and related near-vacuum problems.

Figures (6)

• Figure 1: Long-time severe expansion on $N=900$ cells at $t=0.4$. Top row: resolved plateau velocity and pressure for the uncorrected run and the intermittently corrected run ($K=50$), together with the exact star-state values. Bottom row: final sharp density and internal-energy profiles. Intermittent DRV correction removes the plateau drift visible in the uncorrected run, producing a final sharp reconstruction that is effectively exact.
• Figure 2: Long-time LeBlanc on $N=800$ cells at $t=1$. Top row: resolved plateau velocity and pressure for the uncorrected run and the intermittently corrected run ($K=3$), together with the exact star-state values. Bottom row: final sharp density and internal-energy profiles. This is the benchmark on which one-shot final-time DRV reconstruction fails and intermittent correction becomes essential.
• Figure 3: Long-time two-shock collision ($1$-S/$2$-C/$3$-S) on $N=800$ cells at $t=0.07$. Top row: resolved plateau velocity and pressure. The uncorrected run (blue dashed) exhibits wall-heating oscillations; intermittent correction $K=10$ (orange dash-dot) maintains the exact star values. Bottom row: final sharp density and internal-energy profiles, both effectively exact.
• Figure 4: Long-time two-rarefaction expansion ($1$-R/$2$-C/$3$-R) on $N=800$ cells at $t=0.3$. Top row: the uncorrected velocity plateau (blue dashed) drifts linearly from $-0.075$ to $+0.075$ around the exact $u_\ast=0$; intermittent correction $K=20$ (orange dash-dot) holds $u=0$ to machine precision. The pressure plateau is similarly corrected. Bottom row: final sharp profiles, both overlaying the exact solution.
• Figure 5: Double-Sod on $N=1200$ cells at $t=0.1$. The initial data contain two interfaces at $x=\pm 0.4$, so the global solution is not self-similar. Top row: pointwise $|u-u_\ast|$ errors on the raw plateau for the uncorrected run and the intermittently corrected run ($K=20$), shown separately for the left and right star regions. Bottom row: final sharp density around the left and right interfaces. The intermittent correction can be applied independently on the two half-domains because the two local wave packets remain disjoint up to $t=0.1$.

Theorems & Definitions (1)

• Remark 2.1