GridRipper

GridRipper is a code developed by Péter Csizmadia, András László and István Rácz for solving hyperbolic systems of partial differential equations (PDEs) numerically. For treating the angular dependence, it applies a spectral approach based on spherical harmonics. At the same time, the remaining time-radial sector is handled by a fourth-order precision adaptive mesh refinement library. The code is not actively developed today, but the ideas built into it still flourish.

SW-GridRipper

SW-GridRipper is our effort to revitalize the GridRipper project. Here “SW” refers to the fact that the current code improves on GridRipper in the sense that the spectral part now is advanced to use spin-weighted spherical harmonics. This feature allows us to represent the angular dependence of tensorial quantities properly. However, some other features of GridRipper which are not relevant for our use, such as adaptive mesh refinement, are not implemented yet.

SW-GridRipper currently hosts two code bases: TeuSolver, which is optimized to solve Teukolsky’s equation and Constraint Solver, which solves the evolutionary form of the constraints.

TeuSolver

As a pilot project for testing the use of spin-weighted spherical harmonics in a restricted use case, we have implemented a code solving Teukolsky’s master equation (TME).

As for the analytic setup, we have used conformal compactification and the hyperboloidal initial value problem. These allowed us to study the evolution of the perturbations directly on the black hole horizon and at future null infinity. Additionally complex conserved currents constructed from the solutions are used to verify our results and characterize the dynamics.

On the numerics side, applying the spin-weighted spherical harmonics expansion, the resulting equations for the expansion coefficients are integrated using the “method of lines” and a standard fourth-order Runge-Kutta integrator.

TeuSolver also incorporates innovative ideas from a simplified version of GridRipper written by Gábor Zsolt Tóth.

Constraint Solver

The framework handling the spectral part of the code is extended to take care of more complicated angular dependence naturally occurring in generic spacetimes. We applied this new feature in solving the evolutionary form of the constraints.

The evolutionary form of the constraints is a novel approach based on a 2+1 slicing of the initial data surface. Expressing the constraints in terms of geometrically distinguished variables, one arrives at a system with two hyperbolic and a parabolic PDEs or a pair of hyperbolic PDEs and an algebraic equation depending on the choice of variables we want to solve the constraints (for the details follow this link).

In the corresponding circumstances, the spherical harmonic expansion results in a system of coupled nonlinear ordinary differential equations solved by an adaptive Runge-Kutta-Fehlberg algorithm.

Although the code base is not open source yet, its documentation can be read here.

Authors and contributors

  • Péter Csizmadia — C++: AMR part of GridRipper

  • Károly Zoltán Csukás — C++: Teusolver and Constraint solver

  • András László — C++: spectral part of GridRipper

  • János Östör

  • Máté Somodi

  • Gábor Zsolt Tóth — C++: initial version of Teusolver

  • István Rácz — ideas and theoretical background

Publications related to the project

  • P Csizmadia “Testing a new mesh refinement code in the evolution of a spherically symmetric Klein-Gordon field”, 2006, Int J Mod Phys D15, 107-119, doi:10.1142/S0218271806007614 arXiv:hep-th/0505036

  • P Csizmadia “Fourth order AMR and nonlinear dynamical systems in compactified space”, 2007, Class Quantum Grav 24, S369-S379, doi:10.1088/0264-9381/24/12/S23

  • P Csizmadia, A László, I Rácz “Linear waves on fixed Kerr background and their relevance in jet formation”, 2010, J Phys: Conf Ser 218, 012007

  • P Csizmadia, A László, I Rácz “On the Use of Multipole Expansion in Time Evolution of Non-linear Dynamical Systems and Some Surprises Related to Superradiance”, 2012, Class Quantum Grav 30, 015010, doi:10.1088/0264-9381/30/1/015010 arXiv:1207.5837

  • K Z Csukás, I Rácz, G Zs Tóth “Numerical investigation of the dynamics of linear spin s fields on a Kerr background I. Late time tails of spin s = ±1, ±2 fields”, 2019, Phys Rev D 100, 104025, doi:10.1103/PhysRevD.100.104025, arXiv:1905.09082

  • K Z Csukás, I Rácz “Numerical investigation of the dynamics of linear spin s fields on a Kerr background II: Superradiant scattering”, 2021, Phys Rev D 103, 084035, doi:10.1103/PhysRevD.103.084035, arXiv:2101.05530

  • K Z Csukás, I Rácz “Numerical investigations of the asymptotics of solutions to the evolutionary form of the constraints”, 2020, Class Quantum Grav 37, 155006, doi:10.1088/1361-6382/ab8fce, arXiv:1911.02900

Acknowledgement

The development of the original GridRipper was supported in parts by the OTKA grant K67942. SW-GridRipper recieved funding from NKFIH Grant No. K-115434 (PI István Rácz) and No. K-116505, the COST Action Gravitational waves, black holes and fundamental physics CA16104 and by the POLONEZ programme of the National Science Centre of Poland (under the project No. 2016/23/P/ST1/04195, PI István Rácz) which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 665778.