A Comparative Study of Low-Dissipation Numerical Schemes for Hyperbolic Conservation Laws

This work provides a comparative assessment of several low-dissipation numerical schemes for hyperbolic conservation laws, highlighting their performance relative to the classical Harten-Lax-van Leer (HLL) schemes. The schemes under consideration include the classical Harten-Lax-van Leer-Contact (HLLC), the recently proposed TV flux splitting, the low-dissipation Central-Upwind (LDCU), and the local characteristic decomposition-based Central-Upwind (LCDCU) schemes. These methods are extended to higher orders of accuracy, up to the fifth order, within both finite-volume and finite-difference frameworks. A series of numerical experiments for the one- and two-dimensional Euler equations of gas dynamics are performed to evaluate the accuracy, robustness, and computational efficiency of the studied schemes. The comparison highlights the trade-offs between resolution of contact and shear waves, robustness in the presence of shocks, and computational cost. The investigated low-dissipation schemes show comparable levels of numerical dissipation, with only subtle differences appearing in selected benchmark problems. The results provide practical guidance for selecting efficient low-dissipation solvers for the simulation of complex compressible flows.

💡 Research Summary

The paper presents a systematic comparative study of four low‑dissipation numerical schemes for hyperbolic conservation laws—Harten‑Lax‑van Leer‑Contact (HLLC), TV flux‑splitting, Low‑Dissipation Central‑Upwind (LDCU), and Local‑Characteristic‑Decomposition Central‑Upwind (LCDCU)—and benchmarks them against the classical HLL solver. The authors first review the limitations of the original HLL method, namely its inability to represent contact and shear waves explicitly, which leads to excessive numerical diffusion. They then describe how each of the four alternative schemes mitigates this drawback: HLLC restores the missing contact wave by introducing a three‑wave Riemann fan; TV splitting decomposes the flux into advective and pressure components aligned with characteristic families, preserving stationary contacts; LDCU augments a central‑upwind framework with a sub‑cell projection step and an anti‑diffusion term that sharply resolves contact/shear layers while maintaining non‑oscillatory behavior; LCDCU further reduces diffusion by applying a local characteristic decomposition to the diffusion term, achieving especially low dissipation in high‑speed shock regions.

All schemes are extended to first, second, third, and fifth order of accuracy. Second‑order accuracy is obtained via a piecewise‑linear reconstruction with a generalized minmod limiter (θ = 1 in the tests). Third‑ and fifth‑order extensions are built within the finite‑difference A‑WENO framework, adding second‑order (Fₓₓ) and fourth‑order (Fₓₓₓₓ) correction terms to the base finite‑volume flux. Non‑linear weighting is performed on characteristic variables to avoid spurious oscillations. The paper details the computation of wave speeds, the formulation of the anti‑diffusion term in LDCU, and the characteristic‑based diffusion in LCDCU.

A comprehensive suite of numerical experiments is carried out for the one‑dimensional and two‑dimensional Euler equations. One‑dimensional tests include the Sod shock tube, Lax problem, Shu‑Osher interaction, and a double‑shock collision, which probe the schemes’ ability to capture contact discontinuities, shear layers, and high‑frequency acoustic waves. Two‑dimensional tests comprise the double‑Mach reflection, a Mach‑reflection problem with strong oblique shocks, the Kelvin‑Helmholtz instability, and a vortex‑shear interaction, all of which involve complex wave interactions and multidimensional effects.

The results reveal clear trade‑offs. HLLC and TV splitting deliver the sharpest resolution of contact and shear waves, with TV splitting showing slightly better robustness in the presence of strong shocks. LDCU provides a balanced performance: it is robust across all test cases, resolves contact/shear features adequately, and incurs only a modest increase in computational cost due to the sub‑cell projection and anti‑diffusion term. LCDCU achieves the lowest overall numerical diffusion, especially at fifth‑order accuracy, and maintains non‑oscillatory solutions even in the most demanding shock‑interaction problems, but it is the most expensive scheme because of the characteristic decomposition and higher‑order correction terms. In terms of raw CPU time, HLLC and TV splitting are the cheapest, followed by LDCU, with LCDCU being the costliest.

Overall, the four low‑dissipation schemes all outperform the baseline HLL solver, confirming that the additional algorithmic complexity translates into tangible gains in accuracy and wave resolution. The authors conclude that the choice among HLLC, TV splitting, LDCU, and LCDCU should be guided by the specific physics of the problem (importance of contact/shear resolution versus shock robustness) and the available computational budget. For problems where contact and shear fidelity dominate, HLLC or TV splitting are recommended; for simulations requiring both robustness and moderate computational overhead, LDCU offers an attractive compromise; and for the highest‑accuracy demands, especially at fifth order, LCDCU is the preferred option. The paper thus provides practical guidance for selecting efficient low‑dissipation solvers in complex compressible‑flow simulations.