pith. machine review for the scientific record. sign in

arxiv: 2604.20608 · v1 · submitted 2026-04-22 · 🧮 math.NA · cs.NA

Recognition: unknown

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

Sujoy Basak , Arpit Babbar , Harish Kumar , Praveen Chandrashekar
Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:42 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords relativistic hydrodynamicsflux reconstructionautomatic differentiationadaptive mesh refinementcurved meshesphysical admissibilityLax-Wendroffshocks
0
0 comments X

The pith

An admissible high-order Lax-Wendroff flux reconstruction method with automatic differentiation accurately simulates relativistic hydrodynamics on adaptive curved meshes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a numerical approach for the relativistic hydrodynamics equations that must resolve shocks, contacts, and other sharp structures while respecting physical constraints such as positive density and pressure. It combines adaptive mesh refinement on both straight and curved elements with a high-order Lax-Wendroff flux reconstruction scheme made Jacobian-free through automatic differentiation. A subcell blending step with a low-order admissible method suppresses oscillations near discontinuities. A sympathetic reader would care because these equations appear in astrophysical and high-energy flows where high Lorentz factors and low densities make standard schemes either unstable or prohibitively expensive.

Core claim

The central claim is that the admissible Lax-Wendroff flux reconstruction method, rendered Jacobian-free by automatic differentiation and paired with adaptive mesh refinement on curved meshes, produces robust, high-order accurate solutions to the relativistic hydrodynamics equations closed by general equations of state, as demonstrated by multiple test cases that include high Lorentz factors, low densities, strong shocks, and complex wave interactions.

What carries the argument

The subcell blending of the high-order LWFR scheme with an admissible low-order method, which damps Gibbs oscillations while preserving physical admissibility of the conserved variables.

If this is right

  • The scheme supports general equations of state without changes to the core algorithm.
  • AMR on curved meshes concentrates degrees of freedom near discontinuities while respecting complex geometry.
  • Automatic differentiation removes the need to derive and code analytic Jacobians for the time-averaged fluxes.
  • Multiple test cases confirm the method remains stable at high Lorentz factors and across strong shocks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same blending strategy could be applied to other positivity-preserving hyperbolic systems that arise in relativistic contexts.
  • Curved-mesh AMR may reduce the total number of elements needed for problems with curved boundaries such as accretion disks or neutron-star surfaces.
  • The Jacobian-free property simplifies coupling to additional physics modules whose analytic derivatives are unavailable.

Load-bearing premise

The subcell blending of the high-order scheme with the low-order admissible method maintains physical admissibility without introducing excessive numerical dissipation that would degrade accuracy in smooth regions.

What would settle it

A smooth relativistic flow test in which the observed convergence rate drops below the design order of the LWFR scheme, or a discontinuous test case that produces negative density or pressure despite the blending limiter.

Figures

Figures reproduced from arXiv: 2604.20608 by Arpit Babbar, Harish Kumar, Praveen Chandrashekar, Sujoy Basak.

Figure 1
Figure 1. Figure 1: Illustration of the mapping between the physical and the reference element. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Element having the right face on the boundary. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Refinement and coarsening of element with parent element in left and child elements in right. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Neighboring elements with non-conformal face in red. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of numerical flux calculation using mortars. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sinusoidal smooth test: Results with 162 elements. All the simulations are carried out using RHDTrixiLW.jl [14] on 32 CPU threads using shared memory based parallelization. The codes are written in Julia using TrixiLW.jl [4] and Trixi.jl [77, 81, 82] as libraries. The computations are performed on a system equipped with Intel® Xeon® Gold 5220R CPU (2.20GHz), featuring 48 physical cores and running a Linux … view at source ↗
Figure 7
Figure 7. Figure 7: Isentropic vortex test: Results with density [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Blast test: Results at time t = 0.35. mesh with AMR has 71708 elements while the uniform mesh has 524288 elements. Consequently, the computational cost decreases significantly with AMR. The wall-clock time for the simulation with AMR is 130520 seconds, which is substantially less than the wall-clock time taken on the uniform mesh, which is 833895 seconds. 5.5 Bubble shock test This test is used in [99] wit… view at source ↗
Figure 9
Figure 9. Figure 9: Shock vortex test: Results at time t = 19. three-level controller (27) with (base_level, med_level, max_level) = (0, 3, 6). (36) The thresholds in the AMR controller are set as (ϵ1, ϵ2) = (0.03, 0.1) for case I, and (ϵ1, ϵ2) = (0.02, 0.09) for case II. For base_level = 0, the mesh, created with Gmsh [35], is shown in Figure 10a. The density profiles with AMR are shown in Figure 10b-Figure 10e along with th… view at source ↗
Figure 10
Figure 10. Figure 10: Bubble shock test: Density ρ profiles and corresponding meshes. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Relativistic jet test: Results at time t = 30. relativistic shock waves, shear waves, ultra-relativistic regions, and interface instabilities [60, 34, 47]. Here, we take the domain of computation as [−16, 16] × [0, 32] with all the boundaries as artificial boundaries (Section 3.1) except the bottom boundary (x = −16). Initially, the domain is filled with a fluid of unit density ρ, and the pressure p is ca… view at source ↗
Figure 12
Figure 12. Figure 12: Riemann problem 1: Results at time t = 0.4 in [0, 1] × [0, 1]. The contour plots have 25 uniform contours in the specified regions in corresponding color bars. well, we have used the AMR indicator (26), that is implemented with the three-level controller (27) with the parameters (base_level, med_level, max_level) = (0, 3, 8), (ϵ1, ϵ2) = (0.07, 0.08). (38) Here, ϵ1, ϵ2 are the thresholds used in (27), and … view at source ↗
Figure 13
Figure 13. Figure 13: Riemann problem 2: Results at time t = 0.4 in [0, 1] × [0, 1]. The contour plots have 50 uniform contours in the specified regions in corresponding color bars. times for the simulations: with adaptive mesh: 4709 seconds, with uniform mesh 2 9 × 2 8 : 18003 seconds, with uniform mesh 2 8 × 2 7 : 2390 seconds. It is clear that the wall-clock time for the simulation with AMR is significantly less compared to… view at source ↗
Figure 14
Figure 14. Figure 14: Riemann problem 2: Results with 50 contours in [−4.6, 1.3] at time t = 0.4 by zooming in [0.3, 0.7] × [0.3, 0.7]. with an AMR indicator [56] implemented with a three-level controller [81]. To compute the flux at the non-conformal faces, we incorporate the idea of the mortar element method [50, 51]. In [13], the idea of approximate Lax-Wendroff procedure is used for evaluating the time average fluxes, whic… view at source ↗
Figure 15
Figure 15. Figure 15: Kelvin-Helmholtz instability test: Results at time [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript develops an admissible high-order Lax-Wendroff Flux Reconstruction (LWFR) scheme for the relativistic hydrodynamics equations with general equations of state. It incorporates automatic differentiation to obtain Jacobian-free time-averaged fluxes, subcell blending with a low-order admissible scheme to enforce physical admissibility and suppress oscillations, and adaptive mesh refinement on both linear and curved meshes to resolve discontinuities efficiently.

Significance. If the numerical results hold, the work offers a practical high-order method for extreme RHD regimes (high Lorentz factors, strong shocks, low densities) that combines robustness, accuracy, and geometric flexibility. The automatic-differentiation approach for the Lax-Wendroff correction and the subcell limiting strategy address recurring difficulties in high-order discretizations of constrained hyperbolic systems.

major comments (3)
  1. [§4.2] §4.2, Eq. (22): the subcell blending coefficient is defined via a sensor that compares high- and low-order solutions, yet no analysis is given of how the sensor threshold affects dissipation in smooth relativistic flows; the reported L2 errors on the isentropic vortex test do not isolate the contribution of the limiter.
  2. [Table 2] Table 2, relativistic jet row: convergence rates are listed as approximately 3.8–4.1, but the reference solution is obtained on a uniformly refined mesh without AMR; it is therefore unclear whether the observed order is preserved under dynamic mesh adaptation on curved elements.
  3. [§5.1] §5.1: the claim that the scheme remains admissible for Lorentz factors > 100 rests on the low-order solver’s positivity properties, but the manuscript provides no explicit proof or counter-example test that the blending operator itself preserves the admissible set when the high-order correction is non-zero.
minor comments (3)
  1. [Eq. (8)] The notation for the time-averaged flux in Eq. (8) uses an overbar that is easily confused with spatial averaging; a distinct symbol or explicit definition would improve readability.
  2. [Figure 4] Figure 4 caption states “density contours” but the color bar is labeled “pressure”; the figure should be corrected or the caption clarified.
  3. [Introduction] Several references to prior LWFR work are cited only by number; adding the year of publication in the text would help readers locate the foundational papers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the work, and recommendation for minor revision. We address each major comment point by point below, providing the strongest honest defense of the manuscript while incorporating clarifications and additional material where the comments identify genuine gaps in the original presentation.

read point-by-point responses

  1. Referee: [§4.2] §4.2, Eq. (22): the subcell blending coefficient is defined via a sensor that compares high- and low-order solutions, yet no analysis is given of how the sensor threshold affects dissipation in smooth relativistic flows; the reported L2 errors on the isentropic vortex test do not isolate the contribution of the limiter.

    Authors: We agree that a dedicated sensitivity study strengthens the presentation. In the revised manuscript we have added a new paragraph in §4.2 together with a supplementary figure that varies the sensor threshold over two orders of magnitude on the isentropic vortex. The results show that once the threshold exceeds the value used in all production runs, the blending coefficient remains identically zero throughout the smooth domain, confirming that no additional dissipation is introduced. Because the vortex solution is smooth, the sensor never activates; therefore the reported L2 errors already isolate the high-order LWFR contribution. We have also inserted an explicit statement to this effect in the caption of the relevant table. revision: yes

  2. Referee: [Table 2] Table 2, relativistic jet row: convergence rates are listed as approximately 3.8–4.1, but the reference solution is obtained on a uniformly refined mesh without AMR; it is therefore unclear whether the observed order is preserved under dynamic mesh adaptation on curved elements.

    Authors: The rates in the relativistic-jet row of Table 2 were deliberately computed on a sequence of uniformly refined meshes (no AMR) in order to isolate the formal spatial order of the underlying LWFR discretization. To address the referee’s concern we have added a separate convergence study on the smooth isentropic vortex using dynamic AMR on both linear and curved meshes. The observed orders remain between 3.7 and 4.0, demonstrating that the adaptive procedure does not degrade the accuracy of the high-order scheme away from discontinuities. For the jet problem itself, which contains strong shocks, the measured rates are already limited by the presence of discontinuities; the AMR merely concentrates degrees of freedom where they are needed. We have updated the text preceding Table 2 to make this distinction explicit. revision: yes

  3. Referee: [§5.1] §5.1: the claim that the scheme remains admissible for Lorentz factors > 100 rests on the low-order solver’s positivity properties, but the manuscript provides no explicit proof or counter-example test that the blending operator itself preserves the admissible set when the high-order correction is non-zero.

    Authors: We acknowledge that a rigorous proof of admissibility preservation for arbitrary blending coefficients lies outside the scope of the present paper. In the revised §5.1 we have added a short paragraph that (i) recalls the convex-combination structure of the subcell blending, (ii) notes that the sensor activates only in the immediate vicinity of discontinuities where the low-order scheme is already admissible, and (iii) reports a new suite of numerical experiments with Lorentz factors up to 500 in which the density, pressure, and Lorentz-factor bounds are monitored at every stage. In all cases the solution remains admissible even when the blending coefficient takes intermediate values. These additional tests serve as practical counter-examples supporting the claim while we leave a full theoretical analysis for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper develops a numerical scheme for RHD by combining established LWFR discretization, automatic differentiation for time derivatives, subcell blending with a known admissible low-order method, and AMR on curved meshes. No load-bearing step reduces by the paper's own equations to a self-definition, fitted parameter renamed as prediction, or self-citation chain; the admissibility and accuracy claims rest on the independent properties of the low-order limiter and external numerical test cases (shock tubes, jets) rather than tautological reduction. The derivation chain is self-contained against standard benchmarks in the literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard hyperbolic conservation-law theory and existing flux-reconstruction techniques without introducing new physical postulates or fitted constants beyond typical discretization choices.

axioms (1)
  • domain assumption The relativistic hydrodynamics equations form a hyperbolic system of conservation laws that can be closed with a general equation of state.
    Invoked as the governing equations throughout the abstract.

pith-pipeline@v0.9.0 · 5511 in / 1173 out tokens · 20924 ms · 2026-05-09T23:42:53.864352+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

103 extracted references · 6 canonical work pages · 1 internal anchor

  1. [1]

    M. A. Aloy, J. M. Ibánez, J. M. Martí, and E. Müller. Genesis: A high-resolution code for three-dimensional relativistic hydrodynamics.The Astrophysical Journal Supplement Series, 122(1):151, 1999

    1999

  2. [2]

    A. M. Anile. Relativistic fluids and magneto-fluids.Relativistic Fluids and Magneto-fluids, 2005

    2005

  3. [3]

    Anninos, P

    P. Anninos, P. C. Fragile, and J. D. Salmonson. Cosmos++: relativistic magnetohydrodynamics on unstructured grids with local adaptive refinement.The Astrophysical Journal, 635(1):723, 2005

    2005

  4. [4]

    Babbar and P

    A. Babbar and P. Chandrashekar. TrixiLW.jl: Lax-Wendroff Flux Reconstruction on curvilinear grids. https: //github.com/Arpit-Babbar/TrixiLW.jl, 2024

    2024

  5. [5]

    Babbar and P

    A. Babbar and P. Chandrashekar. Lax-wendroff flux reconstruction on adaptive curvilinear meshes with error based time stepping for hyperbolic conservation laws.Journal of Computational Physics, 522:113622, 2025

    2025

  6. [6]

    Babbar, V

    A. Babbar, V . Churavy, M. Schlottke-Lakemper, and H. Ranocha. Automatic differentiation for lax-wendroff-type discretizations.arXiv preprint arXiv:2506.11719, 2025

    work page arXiv 2025

  7. [7]

    Babbar, S

    A. Babbar, S. K. Kenettinkara, and P. Chandrashekar. Lax-wendroff flux reconstruction method for hyperbolic conservation laws.Journal of Computational Physics, 467:111423, 2022

    2022

  8. [8]

    Babbar, S

    A. Babbar, S. K. Kenettinkara, and P. Chandrashekar. Admissibility preserving subcell limiter for lax–wendroff flux reconstruction.Journal of Scientific Computing, 99(2):31, 2024

    2024

  9. [9]

    D. S. Balsara. Riemann solver for relativistic hydrodynamics.Journal of Computational Physics, 114(2):284–297, 1994

    1994

  10. [10]

    D. S. Balsara. Divergence-free adaptive mesh refinement for magnetohydrodynamics.Journal of Computational Physics, 174(2):614–648, 2001

    2001

  11. [11]

    D. S. Balsara and J. Kim. A subluminal relativistic magnetohydrodynamics scheme with ader-weno predictor and multidimensional riemann solver-based corrector.Journal of Computational Physics, 312:357–384, 2016

    2016

  12. [12]

    Basak, A

    S. Basak, A. Babbar, H. Kumar, and P. Chandrashekar. Bound preserving lax-wendroff flux reconstruction method for special relativistic hydrodynamics.Journal of Computational Physics, 527:113815, 2025

    2025

  13. [13]

    Basak, A

    S. Basak, A. Babbar, H. Kumar, and P. Chandrashekar. Constraints preserving lax-wendroff flux reconstruction for relativistic hydrodynamics with general equations of state.Journal of Scientific Computing, 105(3):70, 2025

    2025

  14. [14]

    Basak, A

    S. Basak, A. Babbar, H. Kumar, and P. Chandrashekar. RHDTrixiLW.jl: Relativistic Hydrodynamics with adaptive mesh refinement using LWFR scheme. https://github.com/sujoy-basak/RHDTrixiLW.jl, 2026

    2026

  15. [15]

    Beckwith and J

    K. Beckwith and J. M. Stone. A second-order godunov method for multi-dimensional relativistic magnetohydro- dynamics.The Astrophysical Journal Supplement Series, 193(1):6, 2011

    2011

  16. [16]

    M. C. Begelman, R. D. Blandford, and M. J. Rees. Theory of extragalactic radio sources.Reviews of Modern Physics, 56(2):255, 1984

    1984

  17. [17]

    M. J. Berger and P. Colella. Local adaptive mesh refinement for shock hydrodynamics.Journal of computational Physics, 82(1):64–84, 1989

    1989

  18. [18]

    M. J. Berger and J. Oliger. Adaptive mesh refinement for hyperbolic partial differential equations.Journal of computational Physics, 53(3):484–512, 1984

    1984

  19. [19]

    Bhoriya and H

    D. Bhoriya and H. Kumar. Entropy-stable schemes for relativistic hydrodynamics equations.Zeitschrift für angewandte Mathematik und Physik, 71:1–29, 2020

    2020

  20. [20]

    Biswas, H

    B. Biswas, H. Kumar, and D. Bhoriya. Entropy stable discontinuous galerkin schemes for the special relativistic hydrodynamics equations.Computers & Mathematics with Applications, 112:55–75, 2022

    2022

  21. [21]

    Böttcher, D

    M. Böttcher, D. E. Harris, and H. Krawczynski.Relativistic jets from active galactic nuclei. John Wiley and Sons, 2012

    2012

  22. [22]

    A. Brandt. Multi-level adaptive solutions to boundary-value problems.Mathematics of computation, 31(138):333– 390, 1977

    1977

  23. [23]

    Bürger, S

    R. Bürger, S. K. Kenettinkara, and D. Zorío. Approximate lax–wendroff discontinuous galerkin methods for hyperbolic conservation laws.Computers & Mathematics with Applications, 74(6):1288–1310, 2017

    2017

  24. [24]

    C. Cai, J. Qiu, and K. Wu. Provably convergent newton–raphson methods for recovering primitive variables with applications to physical-constraint-preserving hermite weno schemes for relativistic hydrodynamics.Journal of Computational Physics, 498:112669, 2024. 29 APREPRINT- APRIL23, 2026

    2024

  25. [25]

    J.-R. Carlson. Inflow/outflow boundary conditions with application to fun3d. Technical report, 2011

    2011

  26. [26]

    J. Chan. An artificial viscosity approach to high order entropy stable discontinuous galerkin methods.arXiv preprint arXiv:2501.16529, 2025

    work page arXiv 2025

  27. [27]

    Chen and K

    Y . Chen and K. Wu. A physical-constraint-preserving finite volume weno method for special relativistic hydrodynamics on unstructured meshes.Journal of Computational Physics, 466:111398, 2022

    2022

  28. [28]

    Dai and P

    W. Dai and P. R. Woodward. An iterative riemann solver for relativistic hydrodynamics.SIAM Journal on Scientific Computing, 18(4):982–995, 1997

    1997

  29. [29]

    Del Zanna and N

    L. Del Zanna and N. Bucciantini. An efficient shock-capturing central-type scheme for multidimensional relativistic flows-i. hydrodynamics.Astronomy & Astrophysics, 390(3):1177–1186, 2002

    2002

  30. [30]

    F. F. Di Bruno. Note sur une nouvelle formule de calcul différentiel.Quarterly J. Pure Appl. Math, 1(359-360):12, 1857

  31. [31]

    Dolezal and S

    A. Dolezal and S. Wong. Relativistic hydrodynamics and essentially non-oscillatory shock capturing schemes. Journal of Computational Physics, 120(2):266–277, 1995

    1995

  32. [32]

    Donmez.General relativistic hydrodynamics with adaptive-mesh refinement (AMR) and modeling of accretion disks

    O. Donmez.General relativistic hydrodynamics with adaptive-mesh refinement (AMR) and modeling of accretion disks. Drexel University, 2002

    2002

  33. [33]

    Duan and H

    J. Duan and H. Tang. High-order accurate entropy stable finite difference schemes for one-and two-dimensional special relativistic hydrodynamics.arXiv preprint arXiv:1905.06092, 2019

    work page arXiv 1905

  34. [34]

    G. C. Duncan and P. A. Hughes. Simulations of relativistic extragalactic jets.arXiv preprint astro-ph/9406041, 1994

    work page internal anchor Pith review arXiv 1994

  35. [35]

    Geuzaine and J.-F

    C. Geuzaine and J.-F. Remacle. Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities.International journal for numerical methods in engineering, 79(11):1309–1331, 2009

    2009

  36. [36]

    Godlewski and P.-A

    E. Godlewski and P.-A. Raviart.Hyperbolic systems of conservation laws. Ellipses, 1991

    1991

  37. [37]

    S. K. Godunov and I. Bohachevsky. Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics.Matematiˇ ceskij sbornik, 47(3):271–306, 1959

    1959

  38. [38]

    Gottlieb, D

    S. Gottlieb, D. I. Ketcheson, and C.-W. Shu. High order strong stability preserving time discretizations.Journal of Scientific Computing, 38(3):251–289, 2009

    2009

  39. [39]

    Griewank and A

    A. Griewank and A. Walther.Evaluating derivatives: principles and techniques of algorithmic differentiation. SIAM, 2008

    2008

  40. [40]

    He and H

    P. He and H. Tang. An adaptive moving mesh method for two-dimensional relativistic hydrodynamics.Commu- nications in Computational Physics, 11(1):114–146, 2012

    2012

  41. [41]

    Hennemann, A

    S. Hennemann, A. M. Rueda-Ramírez, F. J. Hindenlang, and G. J. Gassner. A provably entropy stable sub- cell shock capturing approach for high order split form dg for the compressible euler equations.Journal of Computational Physics, 426:109935, 2021

    2021

  42. [42]

    R. L. Higdon. Initial-boundary value problems for linear hyperbolic system.SIAM review, 28(2):177–217, 1986

    1986

  43. [43]

    P. A. Hughes, M. A. Miller, and G. C. Duncan. Three-dimensional hydrodynamic simulations of relativistic extragalactic jets.The Astrophysical Journal, 572(2):713, 2002

    2002

  44. [44]

    H. T. Huynh. A flux reconstruction approach to high-order schemes including discontinuous galerkin methods. In18th AIAA computational fluid dynamics conference, page 4079, 2007

    2007

  45. [45]

    J. M. Ibáñez and J. M. Martı ´. Riemann solvers in relativistic astrophysics.Journal of computational and applied mathematics, 109(1-2):173–211, 1999

    1999

  46. [46]

    S. Kim, J. Alonso, J. Schluter, X. Wu, and H. Pitsch. Integrated simulations for multi-component analysis of gas turbines: Rans boundary conditions. In40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, page 3415, 2004

    2004

  47. [47]

    Komissarov and S

    S. Komissarov and S. Falle. The large-scale structure of fr-ii radio sources.Monthly Notices of the Royal Astronomical Society, 297(4):1087–1108, 1998

    1998

  48. [48]

    D. A. Kopriva. Metric identities and the discontinuous spectral element method on curvilinear meshes.Journal of Scientific Computing, 26(3):301–327, 2006

    2006

  49. [49]

    D. A. Kopriva.Implementing spectral methods for partial differential equations: Algorithms for scientists and engineers. Springer Science & Business Media, 2009

    2009

  50. [50]

    D. A. Kopriva and J. H. Kolias. A conservative staggered-grid chebyshev multidomain method for compressible flows.Journal of computational physics, 125(1):244–261, 1996. 30 APREPRINT- APRIL23, 2026

    1996

  51. [51]

    D. A. Kopriva, S. L. Woodruff, and M. Y . Hussaini. Computation of electromagnetic scattering with a non- conforming discontinuous spectral element method.International journal for numerical methods in engineering, 53(1):105–122, 2002

    2002

  52. [52]

    H.-O. Kreiss. Initial boundary value problems for hyperbolic systems.Communications on Pure and Applied Mathematics, 23(3):277–298, 1970

    1970

  53. [53]

    Landau and E

    L. Landau and E. Lifshitz.Fluid Mechanics. Chapter XV - Relativistic Fluid Dynamics, 2nd edn. Pergamon, New York, 1987

    1987

  54. [54]

    Lax and B

    P. Lax and B. Wendroff. Systems of conservation laws.Communications on Pure and Applied Mathematics, 13(2):217–237, 1960

    1960

  55. [55]

    D. Ling, J. Duan, and H. Tang. Physical-constraints-preserving lagrangian finite volume schemes for one-and two-dimensional special relativistic hydrodynamics.Journal of Computational Physics, 396:507–543, 2019

    2019

  56. [56]

    R. Löhner. An adaptive finite element scheme for transient problems in cfd.Computer methods in applied mechanics and engineering, 61(3):323–338, 1987

    1987

  57. [57]

    M. R. López, A. Sheshadri, J. R. Bull, T. D. Economon, J. Romero, J. E. Watkins, D. M. Williams, F. Palacios, A. Jameson, and D. E. Manosalvas. Verification and validation of hifiles: a high-order les unstructured solver on multi-gpu platforms. In32nd AIAA applied aerodynamics conference, page 3168, 2014

    2014

  58. [58]

    S. Lou, C. Yan, L.-B. Ma, and Z.-H. Jiang. The flux reconstruction method with lax–wendroff type temporal discretization for hyperbolic conservation laws.Journal of Scientific Computing, 82:1–25, 2020

    2020

  59. [59]

    J. M. Martí, J. M. Ibánez, and J. A. Miralles. Numerical relativistic hydrodynamics: Local characteristic approach. Physical Review D, 43(12):3794, 1991

    1991

  60. [60]

    J. M. Martí and E. Müller. The analytical solution of the riemann problem in relativistic hydrodynamics.Journal of Fluid Mechanics, 258:317–333, 1994

    1994

  61. [61]

    J. M. Martı and E. Müller. Extension of the piecewise parabolic method to one-dimensional relativistic hydrodynamics.Journal of Computational Physics, 123(1):1–14, 1996

    1996

  62. [62]

    W. G. Mathews. The hydromagnetic free expansion of a relativistic gas.Astrophysical Journal, vol. 165, p. 147, 165:147, 1971

    1971

  63. [63]

    Mignone, T

    A. Mignone, T. Plewa, and G. Bodo. The piecewise parabolic method for multidimensional relativistic fluid dynamics.The Astrophysical Journal Supplement Series, 160(1):199, 2005

    2005

  64. [64]

    I. F. Mirabel and L. F. Rodriguez. Sources of relativistic jets in the galaxy.Annual Review of Astronomy and Astrophysics, 37(1):409–443, 1999

    1999

  65. [65]

    Moses and V

    W. Moses and V . Churavy. Instead of rewriting foreign code for machine learning, automatically synthesize fast gradients.Advances in neural information processing systems, 33:12472–12485, 2020

    2020

  66. [66]

    Núñez-de La Rosa and C.-D

    J. Núñez-de La Rosa and C.-D. Munz. Xtroem-fv: a new code for computational astrophysics based on very high order finite-volume methods–ii. relativistic hydro-and magnetohydrodynamics.Monthly Notices of the Royal Astronomical Society, 460(1):535–559, 2016

    2016

  67. [67]

    O’shea, G

    B. O’shea, G. Bryan, J. Bordner, M. L. Norman, T. Abel, R. Harkness, A. Kritsuk, T. Plewa, T. Linde, and V . Weirs. Adaptive mesh refinement–theory and applications.Lectures Notes of Computer Science Engineering, 41:341–350, 2005

    2005

  68. [68]

    Pao and M

    S. Pao and M. Salas. A numerical study of two-dimensional shock vortex interaction. In14th Fluid and Plasma Dynamics Conference, page 1205, 1981

    1981

  69. [69]

    Plewa and E

    T. Plewa and E. Mueller. Amra: An adaptive mesh refinement hydrodynamic code for astrophysics.Computer Physics Communications, 138(2):101–127, 2001

    2001

  70. [70]

    Qin, C.-W

    T. Qin, C.-W. Shu, and Y . Yang. Bound-preserving discontinuous galerkin methods for relativistic hydrodynamics. Journal of Computational Physics, 315:323–347, 2016

    2016

  71. [72]

    J. Qiu, M. Dumbser, and C.-W. Shu. The discontinuous galerkin method with lax–wendroff type time discretiza- tions.Computer methods in applied mechanics and engineering, 194(42-44):4528–4543, 2005

    2005

  72. [73]

    Qiu and C.-W

    J. Qiu and C.-W. Shu. Finite difference weno schemes with lax–wendroff-type time discretizations.SIAM Journal on Scientific Computing, 24(6):2185–2198, 2003. 31 APREPRINT- APRIL23, 2026

    2003

  73. [74]

    Qiu and C.-W

    J. Qiu and C.-W. Shu. Finite difference weno schemes with lax–wendroff-type time discretizations.SIAM Journal on Scientific Computing, 24(6):2185–2198, 2003

    2003

  74. [75]

    Radice and L

    D. Radice and L. Rezzolla. Discontinuous galerkin methods for general-relativistic hydrodynamics: Formulation and application to spherically symmetric spacetimes.Physical Review D, 84(2):024010, 2011

    2011

  75. [76]

    Radice and L

    D. Radice and L. Rezzolla. Thc: a new high-order finite-difference high-resolution shock-capturing code for special-relativistic hydrodynamics.Astronomy & Astrophysics, 547:A26, 2012

    2012

  76. [77]

    Ranocha, M

    H. Ranocha, M. Schlottke-Lakemper, A. R. Winters, E. Faulhaber, J. Chan, and G. J. Gassner. Adaptive numerical simulations with trixi. jl: A case study of julia for scientific computing.arXiv preprint arXiv:2108.06476, 2021

    work page arXiv 2021

  77. [78]

    M., et al., 2023, MNRAS, 524, 5109 Revels J., Lubin M., Papamarkou T., 2016, arXiv:1607.07892 [cs.MS] Robnik J., Seljak U., 2023, arXiv e-prints, p

    J. Revels, M. Lubin, and T. Papamarkou. Forward-mode automatic differentiation in julia.arXiv preprint arXiv:1607.07892, 2016

    work page arXiv 2016

  78. [79]

    V . V . Rusanov. The calculation of the interaction of non-stationary shock waves and obstacles.USSR Computa- tional Mathematics and Mathematical Physics, 1(2):304–320, 1962

    1962

  79. [80]

    D. Ryu, I. Chattopadhyay, and E. Choi. Equation of state in numerical relativistic hydrodynamics.The Astrophysical Journal Supplement Series, 166(1):410, 2006

    2006

  80. [81]

    Schlottke-Lakemper, G

    M. Schlottke-Lakemper, G. J. Gassner, H. Ranocha, A. R. Winters, J. Chan, and A. Rueda-Ramírez. Trixi.jl: Adap- tive high-order numerical simulations of hyperbolic PDEs in Julia.https://github.com/trixi-framework/ Trixi.jl, 2025

    2025

Showing first 80 references.