We present Variable Eddington Tensor (VET)-closed Transport on Adaptive Meshes (VETTAM), a new algorithm to solve the equations of radiation hydrodynamics (RHD) with support for adaptive mesh refinement (AMR) in a frequency-integrated, two-moment formulation. The method is based on a non-local VET closure computed with a hybrid characteristics scheme for ray tracing. We use a Godunov method for the hyperbolic transport of radiation with an implicit backwards-Euler temporal update to avoid the explicit time-step constraint imposed by the light-crossing time, and a fixed-point Picard iteration scheme to handle the nonlinear gas-radiation exchange term, with the two implicit update stages jointly iterated to convergence. We also develop a modified wave-speed correction method for AMR, which we find to be crucial for obtaining accurate results in the diffusion regime. We demonstrate the robustness of our scheme with a suite of pure radiation and RHD tests, and show that it successfully captures the streaming, static diffusion, and dynamic diffusion regimes and the spatial transitions between them, casts sharp shadows, and yields accurate results for rates of momentum and energy exchange between radiation and gas. A comparison between different closures for the radiation moment equations, with the Eddington approximation (0th-moment closure) and the M1 approximation (1st-moment closure), demonstrates the advantages of the VET method (2nd-moment closure) over the simpler closure schemes. VETTAM has been coupled to the AMR FLASH (magneto-)hydrodynamics code and we summarize by reporting performance features and bottlenecks of our implementation.
Here we see how VETTAM works as an optically thick gas cloud (located in the centre of the domain) casts a shadow when irradiated by an extended light source. Our radiation hydrodynamics scheme is able to capture this challenging configuration of multiple converging rays. We also note that subtle shadow features such as the umbra, penumbra and antumbra are noticeable, and is a testament to the ability of our scheme to handle nontrivial geometrical distributions of radiation sources.
The animation below shows the evolution of the gas temperature as two point-like sources of light irradiate a dense spherical gas cloud. The evolution obtained with VETTAM (right panel) is compared with that obtained with more commonly used, but less accurate closures, such as the Flux Limited Diffusion (FLD) and Moment-1 (M1) methods. This is the test described in Section 3.8.2 of the paper. In contrast to the other methods, only VETTAM produces a reasonable solution in this scenario.
S. H. M would like to thank Yan-Fei Jiang and Shane W. Davis for discussions that assisted in the progress of the project. S. H. M would also like to thank M. Aaron Skinner and Eve Ostriker for discussions on their implementation of the M1 method in Athena, and Anna L. Rosen for information on the shadow test. C. F. acknowledges funding provided by the Australian Research Council through Future Fellowship FT180100495, and the Australia-Germany Joint Research Cooperation Scheme (UA-DAAD). M. R. K. acknowledges funding from the Australian Research Council through its Discovery Projects and Future Fellowship funding schemes, awards DP190101258 and FT180100375. RK acknowledges financial support via the Emmy Noether and Heisenberg Research Grants funded by the German Research Foundation (DFG) under grant no. KU 2849/3 and 2849/9. We further acknowledge high-performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grants pr32lo, pn73fi, and GCS Large-scale project 22542), and the Australian National Computational Infrastructure (grants ek9 and jh2) in the framework of the National Computational Merit Allocation Scheme and the ANU Merit Allocation Scheme.