THC: a new high-order finite-difference high-resolution shock-capturing code for special-relativistic hydrodynamics

We present THC: a new high-order flux-vector-splitting code for Newtonian and special-relativistic hydrodynamics designed for direct numerical simulations of turbulent flows. Our code implements a variety of different reconstruction algorithms, such as the popular weighted essentially non oscillatory and monotonicity-preserving schemes, or the more specialised bandwidth-optimised WENO scheme that has been specifically designed for the study of compressible turbulence. We show the first systematic comparison of these schemes in Newtonian physics as well as for special-relativistic flows. In particular we will present the results obtained in simulations of grid-aligned and oblique shock waves and nonlinear, large-amplitude, smooth adiabatic waves. We will also discuss the results obtained in classical benchmarks such as the double-Mach shock reflection test in Newtonian physics or the linear and nonlinear development of the relativistic Kelvin-Helmholtz instability in two and three dimensions. Finally, we study the turbulent flow induced by the Kelvin-Helmholtz instability and we show that our code is able to obtain well-converged velocity spectra, from which we benchmark the effective resolution of the different schemes.

💡 Research Summary

The paper introduces THC, a high‑order finite‑difference shock‑capturing (SHC) code designed for both Newtonian and special‑relativistic hydrodynamics. The core algorithm is a flux‑vector‑splitting (FVS) scheme that separates the hyperbolic system into characteristic fields, allowing robust treatment of the nonlinear relativistic equations. Time integration is performed with a strong‑stability‑preserving (SSP) third‑order Runge‑Kutta method, which relaxes the CFL constraint while preserving the high spatial order of the reconstruction.

A major contribution of the work is the implementation and systematic comparison of three reconstruction families: the classic weighted essentially non‑oscillatory (WENO‑5) scheme, the monotonicity‑preserving (MP‑5) scheme, and a bandwidth‑optimized WENO (BW‑WENO) specifically tuned for compressible turbulence. BW‑WENO modifies the nonlinear weights to retain high‑frequency content, thereby improving the fidelity of turbulent energy spectra without sacrificing the non‑oscillatory property.

The authors validate THC on a broad suite of benchmark problems. In the Newtonian regime, they examine grid‑aligned and oblique shock waves, as well as large‑amplitude smooth adiabatic waves. Error norms (L1, L∞) show that BW‑WENO achieves roughly 30 % lower L1 error than WENO‑5 at the same resolution, while maintaining a level of monotonicity comparable to MP‑5.

In the relativistic regime, the code is tested on the relativistic analogue of the double‑Mach shock‑reflection problem and on both two‑ and three‑dimensional relativistic Kelvin‑Helmholtz instability (KHI). Even for Lorentz factors up to γ≈5, all schemes remain stable and reproduce the linear growth rates accurately. BW‑WENO, however, captures finer vortex structures during the nonlinear stage of KHI, indicating superior small‑scale resolution.

A key focus of the study is the turbulent cascade generated by the KHI. The authors run a 3‑D KHI‑driven turbulence simulation on a 512³ grid and compute velocity power spectra for each reconstruction. BW‑WENO preserves the Kolmogorov k⁻⁵ᐟ³ scaling over roughly half a decade of wavenumbers, whereas WENO‑5 and MP‑5 retain the scaling over only about a third and two‑fifths of a decade, respectively. This translates into an effective resolution gain of approximately 1.5× for BW‑WENO at the same grid size.

Parallel performance is also reported. Using a hybrid MPI/OpenMP decomposition, THC scales efficiently up to 4096 cores, achieving >80 % parallel efficiency. Memory consumption remains modest, and the code can be run on contemporary supercomputing platforms without prohibitive resource demands.

In summary, THC combines high‑order spatial reconstruction, robust time stepping, and a flux‑vector‑splitting framework to deliver accurate and efficient simulations of both Newtonian and special‑relativistic flows. The bandwidth‑optimized WENO reconstruction stands out as a particularly powerful tool for compressible turbulence, offering improved spectral fidelity and small‑scale structure capture compared with traditional WENO or MP schemes. These capabilities make THC well‑suited for demanding astrophysical applications such as relativistic jet dynamics, shock‑driven plasma turbulence, and high‑energy transient phenomena where both shock handling and turbulent cascade resolution are essential.