Admissible Lax-Wendroff Flux Reconstruction Method with Automatic Differentiation on Adaptive Curved Meshes for Relativistic Hydrodynamics

TL;DR

This paper presents an adaptive high-order Lax-Wendroff flux reconstruction method with automatic differentiation for relativistic hydrodynamics, effectively handling shocks and complex geometries with high accuracy.

Contribution

It introduces a Jacobian-free, adaptive mesh refinement scheme combining high-order LWFR with automatic differentiation and low-order blending for stable, accurate RHD simulations.

Findings

Successfully captures shocks and discontinuities in RHD.

Demonstrates robustness and accuracy on complex, curved meshes.

Efficiently handles high Lorentz factors and low-density flows.

Abstract

The relativistic hydrodynamics (RHD) equations can give rise to solutions which have shocks, contact discontinuities, and other sharp structures, which interact and evolve over time. Capturing these sharp waves effectively requires a mesh with high resolution, making the scheme computationally expensive. In this work, adaptive mesh refinement is used with the high-order Lax-Wendroff flux reconstruction (LWFR) method to solve the system of RHD equations, which is closed with general equations of state. To make the scheme Jacobian-free, the idea of automatic differentiation is incorporated for computing the temporal derivatives in the time average flux approximations. The high-order method is blended with an admissible low-order method at the subcell level to control the Gibbs oscillations and maintain the physical admissibility of the solution. Finally, several test cases involving high Lorentz factors, low densities, low pressures, strong shock waves, and other discontinuities are used to demonstrate the robustness, accuracy, and effectiveness of the proposed method. These simulations are performed with AMR using various linear and curved meshes to show the scheme's efficiency and ability to handle complex geometries.