Intermittent Sub-grid Wave Correction from Differentiated Riemann Variables

TL;DR

This paper presents a low-cost, intermittent correction method for 1D Euler computations using differentiated Riemann variables, significantly improving long-time accuracy and shock position precision with minimal computational overhead.

Contribution

The paper introduces a novel, elementary correction technique based on DRVs that achieves near-machine precision accuracy in long-time Euler simulations with minimal overhead.

Findings

Error reduced from 10^{-2} to 10^{-13}

Almost exact shock and contact positions achieved

Method effective for complex wave interactions

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.