CurvatureEngine

Overview

This project is a high-performance computational framework for solving geodesic equations and curvature tensors in General Relativity. It includes tools for computing Christoffel symbols, Ricci tensors, Riemann curvature tensors, and the evolution of spacetime using the ADM/BSSN formalism. The code supports multiple metrics, including Schwarzschild and Minkowski, and provides efficient numerical solvers.

Features

• Geodesic Integrator: Computes null and timelike geodesics in curved spacetimes.
• Curvature Tensor Computation in analytical formalism: Computes Christoffel symbols, Riemann, Ricci, and Einstein tensors.
• BSSN Formalism Solver: Implements evolution equations for numerical relativity using a 3+1 decomposition.
• Constraint Equations: Enforces Hamiltonian and momentum constraints to ensure numerical stability.
• Support for Multiple Metrics: Includes Schwarzschild, Minkowski, and Kerr metrics, with extensibility to other spacetimes.
• Parallelized Computation: Optimized for multi-threading and SIMD vectorization to accelerate computations (Parallel computing is not ready atm so monothread atm...).
• High-Precision Numerical Methods: Uses high-order finite difference schemes for derivatives and tensor calculations.

Compilation

This project is optimized for Clang++ with advanced compiler flags for performance:

make

Usage

Run the simulation with:

./CurvatureEngine <options>

Data will be stored in CSV and VTK (for extrinsic curvature tensor components and Geodesics)format for further analysis. The plots are managed in the scripts (all python and for k_to_vtk you need paraview and pvpython)

Future Work (TODO)

• Extend support to Kerr-Newman/dS/adS and other exotic spacetimes.
• Unit tests and debug mode.
• Class template to integrate your own problems
• Optimize for parallel computation and SIMDs.
• Implement relativistic fluid dynamics template.
• Implement GPU acceleration for faster computations.
• Improve numerical stability of high-curvature geodesics and ADM compute (AMR grid and CFL conditions).
• Implement spectral methods (see spectral methods TODO section).
• Implement BSSN formalism (modern and currently wip).
• Add a FFT solver for spectral differentiation (from scratch or with FFTW, maybe on GPU) for vacuum solutions or GRMHD.
• Chebyshev polynomial imp for non-uniform grids (rotating spacetime).
• Add analytical approximations and tests.
• Improve stability (Spectral BSSN/z4c) and reduce constraints violations.
• Klein-Gordon Scalar fields template.
• Gravitational waves and binary black holes template.

Contributions

Contributions are welcome! If you wish to contribute, feel free to open a pull request or issue.

License

GPL-3.0 license. See LICENSE for details.

Contact

For inquiries, reach out via GitHub Issues or email.

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This project is developed as part of personal advanced numerical relativity research and high-performance computing optimizations. The code and physics computations might have inconsistencies (still at the beginning of the project, and I'm not a physicist).