Bound Preserving Lax-Wendroff Flux Reconstruction Method for Special Relativistic Hydrodynamics

TL;DR

The paper advances numerical relativistic hydrodynamics by delivering a Jacobian-free Lax-Wendroff flux reconstruction method that achieves high-order accuracy on Cartesian grids while preserving physical admissibility. It blends the LWFR scheme with a first-order FV limiter on sub-cells and introduces a new Legendre-based discontinuity indicator to guide adaptive blending, ensuring oscillation control near discontinuities without sacrificing sub-cell information. Admissibility is enforced through a convex reformulation of the admissible set and a scaling limiter, enabling robust performance even for extreme velocities and strong shocks. Numerical verifications in 1D and 2D demonstrate superior accuracy, stability, and shock-capturing capabilities across smooth flows, Riemann problems, jets, and KH instabilities, highlighting the method’s potential for high-fidelity RHD simulations. The approach combines efficiency (single-stage time update), robustness (bound-preserving limiter), and general applicability to relativistic flows with an ideal EOS.

Abstract

Lax-Wendroff flux reconstruction (LWFR) schemes have high order of accuracy in both space and time despite having a single internal time step. Here, we design a Jacobian-free LWFR type scheme to solve the special relativistic hydrodynamics equations on Cartesian grids. We then blend the scheme with a first-order finite volume scheme to control the oscillations near discontinuities. We also use a scaling limiter to preserve the physical admissibility of the solution after ensuring the scheme is admissible in means. A particular focus is given to designing a discontinuity indicator model to detect the local non-smoothness in the solution of the highly non-linear relativistic hydrodynamics equations. Finally, we present numerical results for a wide range of test cases to show the robustness and efficiency of the proposed scheme.

Figures (23)

• Figure 1: Division of the element $\Omega_{pq}$ into $4\times 4$ sub-elements. (The dotted points inside the elements denote the solution points.)
• Figure 2: Isentropic smooth flow problem in 1D: Plot of fluid density using the scheme with degrees $N=3,4$ and $50$ cells.
• Figure 3: First Riemann problem in 1D: Plot of fluid density, pressure, and velocity using the scheme with degrees $N=3,4$ with $200$ and $500$ cells.
• Figure 4: Second Riemann problem in 1D: Plot of fluid density, pressure, and velocity using the scheme with polynomial degrees $N=3,4$ with $500$ and $1500$ cells.
• Figure 5: Third Riemann problem in 1D: Plot of fluid density, pressure, and velocity using the scheme with polynomial degrees $N=3,4$ with $200$ and $500$ cells.

Theorems & Definitions (14)

• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 6.1
• Definition 6.2
• Remark 6.1
• Theorem 6.1
• proof