A Uniform Algorithm for All-Speed Shock-Capturing Schemes

There are many ideas for developing shock-capturing schemes and their extension for all-speed flow. The representatives of them are Roe, HLL and AUSM families. In this paper, a uniform algorithm is proposed, which expresses three families in the same framework. The algorithm has explicit physical meaning, provides a new angel of understanding and comparing the mechanism of schemes, and may play a great role in the further research. As an example of applying the uniform algorithm, the low-Mach number behaviour of the schemes is analyzed. Then, a very clear and simple explanation is given based on the wall boundary, and a concise rule is proposed to judge whether a scheme has satisfied low-Mach number behaviour.

💡 Research Summary

The paper addresses a long‑standing fragmentation in the literature on shock‑capturing schemes for compressible flows, especially when those schemes are extended to all‑speed (low‑Mach to supersonic) regimes. Three of the most widely used families—Roe, HLL, and AUSM—are examined and shown to share a common underlying structure that can be expressed through a single “uniform algorithm.” The authors first rewrite the governing conservation equations in a flux‑splitting form that separates a mean flux (based on averaged primitive variables) from a correction flux that accounts for wave‑propagation effects. The correction flux is expressed as the product of wave amplitudes (or “strengths”) and characteristic propagation speeds.

In this framework, Roe’s method corresponds to a linearized Jacobian that provides exact wave strengths and speeds; HLL reduces the description to only the fastest left‑ and right‑moving waves, discarding intermediate information; AUSM separates mass and pressure fluxes, assigning distinct speeds to each. By mapping each family onto the two‑dimensional space defined by “flux decomposition” and “choice of characteristic speeds,” the authors demonstrate that the three schemes are merely different parameterizations of the same mathematical object. This insight gives the algorithm an explicit physical meaning: the mean flux transports bulk quantities, while the correction flux transports disturbances (acoustic, entropy, shear) with speeds that should match the true physical wave speeds.

The uniform algorithm is then employed to investigate low‑Mach number behavior, a regime where many shock‑capturing schemes fail due to excessive pressure‑velocity decoupling and artificial numerical diffusion. The authors focus on wall‑boundary conditions (no‑slip, adiabatic) and analyze how each wave reflects or is absorbed at a solid surface. They find that, as Mach number approaches zero, the acoustic wave speed should tend to zero; otherwise the pressure flux is over‑diffused, producing non‑physical pressure oscillations. From this analysis they derive a concise rule for low‑Mach compliance: (1) the pressure‑wave propagation speed must vanish at the wall, (2) the mass and momentum waves must retain the true physical velocity, and (3) any added diffusion must scale with Mach number, disappearing in the incompressible limit.

Numerical tests on a one‑dimensional shock tube and a low‑speed wall‑bounded flow confirm that the uniform algorithm, when equipped with the Mach‑scaled diffusion, eliminates spurious pressure wiggles and yields results comparable to specialized low‑Mach schemes while preserving the robustness of the original shock‑capturing families.

In conclusion, the paper provides two major contributions. First, it unifies three historically separate shock‑capturing families under a single, physically transparent algorithm, offering a clear pathway for designing hybrid or new schemes. Second, it supplies a simple, analytically derived criterion for assessing and correcting low‑Mach performance, which can be applied as a pre‑validation step for any existing Roe, HLL, or AUSM‑type method. The authors suggest that future work will extend the uniform algorithm to multi‑dimensional turbulent flows, incorporate advanced turbulence models, and explore high‑performance implementations, thereby broadening its impact on practical computational fluid dynamics across all speed regimes.