Zhen Zhong's Blog

Einstein.jl: a high-performance suite for solving PDEs in general relativity

Fri, 03 Jan 2025 00:00:00 GMT

After developing hundreds of PDE solvers for various projects, I decided to create Einstein.jl because I couldn't find a library that was both highly convenient and effective for solving partial differential equations (PDEs) in the context of general relativity (GR).

When solving GR PDEs in spherical symmetry or even axisymmetry, the requirements are often far more demanding than in 3+1 cases. These include the need for very long-time simulations, conservation of quantities, highly accurate late-time behavior, and precise computation of quasinormal modes. While I have addressed these challenges individually in the past, I grew tired of copying and pasting solutions from old repositories. Instead, I decided to develop a comprehensive, feature-rich, and highly efficient library. This is my answer to those challenges:

::github{repo="AuroraDysis/Einstein.jl"}

In summary, Einstein.jl is a high-performance suite designed to compute arbitrary-precision solutions of PDEs in GR. It is planned to support solving the following built-in equations (some of which were developed for previous projects and will be ported to this library):

Spatial Discretization
- [x] Chebyshev collocation at Chebyshev points of the first and second kinds (Most of algorithms are translated from Chebfun)
- [x] Finite difference method
- [x] Hermite finite difference method
- [x] Rectangular collocation method (Most of algorithms are translated from Chebfun)
- [x] Ultraspherical spectral method (Most of algorithms are translated from Chebfun)
  - For boundary value problems, I recommend using ApproxFun.jl.
- [ ] Ultraspherical rectangular collocation
General Utilities
- [x] Correctly rounded floating-point dot/sum using xsum or Kahan summation
- [x] Spin-weighted spheroidal harmonics using the Cook-Zalutskiy spectral method
Quasinormal Modes
- [ ] Compute quasinormal modes for the Schwarzschild black hole using the Regge-Wheeler-Zerilli equation.
  - [ ] Continued fraction method to determine the eigenvalue
  - [x] Ultraspherical spectral method in hyperboloidal coordinates to determine the eigenfunction.
  - [x] Dolan and Ottewill Regge poles expansion method
  - [ ] WKB approximation
  - [ ] Direct integration method
- [x] Compute quasinormal modes for the Kerr black hole using the Teukolsky equation.
  - [x] Continued fraction method for the radial equation (translated from duetosymmetry/qnm)
  - [x] Cook-Zalutskiy spectral method for the angular sector (translated from duetosymmetry/qnm)
  - [x] Direct integration method for the radial equation
  - [x] Ultraspherical spectral method in hyperboloidal coordinates to determine the eigenfunction.
- Utilities
  - [x] Modified Lentz method for continued fractions
  - [ ] WKB approximation for general potentials
Time Domain
- Spherical Symmetry
  - [ ] Regge-Wheeler-Zerilli equation
    - [ ] Hyperboloidal coordinates
    - [ ] Kerr-Schild coordinates
    - [ ] Tortoise coordinates
  - [ ] Klein-Gordon equation
    - [ ] Hyperboloidal coordinates
    - [ ] Kerr-Schild coordinates
    - [ ] Tortoise coordinates
  - [ ] Einstein equations with a scalar field
    - [ ] Z4 formulation
    - [ ] Hyperboloidal coordinates
- Axisymmetry
  - [ ] Teukolsky equation
    - [ ] Hyperboloidal coordinates
    - [ ] Kerr-Schild coordinates
    - [ ] Tortoise coordinates
  - [ ] Klein-Gordon equation
    - [ ] Hyperboloidal coordinates
    - [ ] Kerr-Schild coordinates
    - [ ] Tortoise coordinates

While the ChebyshevSuite is inspired by algorithms from Chebfun, it has been significantly enhanced in this package to improve performance and support arbitrary-precision calculations.

Acknowledgments

This package is inspired by the following projects, and some of the algorithms are translated / adapted from them:

The author would like to thank the developers of these projects for their contributions to the scientific computing community.

Other Related Projects

lucass-carneiro/QuasinormalModes.jl: A Julia package for computing discrete eigenvalues of second-order ODEs: This package focuses on computing quasinormal modes using the Asymptotic Iteration Method. However, as far as I know, no one has proven that the Asymptotic Iteration Method is guaranteed to converge. So why not use more reliable methods? In any case, it is good to have one more method to compare against.
JLRipley314/TeukolskyQNMFunctions.jl: Computes quasinormal modes and eigenfunctions of the Teukolsky equation in HPHC coordinates.: Justin Ripley has developed a package that computes the quasinormal modes of the Teukolsky equation based on his own paper. We also follow his paper for the Teukolsky equation in hyperboloidal coordinates.

Adaptive Mesh Refinement

Thu, 12 Dec 2024 00:00:00 GMT

Introduction

In numerical analysis, adaptive mesh refinement (AMR) is a method of adapting the accuracy of a solution within certain sensitive or turbulent regions of simulation, dynamically and during the time the solution is being calculated [^1].

Tutorials

Here are some tutorials on adaptive mesh refinement (AMR) and related topics.

I found two excellent mini-codes on GitHub that are very useful for learning AMR. They are simple and easy to understand.

::github{repo="JLRipley314/one-dim-amr"}

::github{repo="IanHawke/toy-amr"}

AMR Libraries

Here is a list of AMR libraries, some of which are actively maintained and others that are not. I will try to keep this list updated.

AMR Resources, a website of AMR software that uses the original Berger/Oliger/Colella adaptive mesh refinement software, by Donna Calhoun
[1610.08833] A Survey of High Level Frameworks in Block-Structured Adaptive Mesh Refinement Packages

AMReX

AMReX is a software framework designed to accelerate scientific discovery for applications solving partial differential equations on block-structured meshes. Its massively parallel adaptive mesh refinement (AMR) algorithms focus computational resources and allow scalable performance on heterogeneous architectures so that scientists can efficiently resolve details in large simulations.

::github{repo="AMReX-Codes/amrex"}

MPI-AMRVAC

MPI-AMRVAC is a parallel adaptive mesh refinement framework aimed at solving (primarily hyperbolic) partial differential equations by a number of different numerical schemes. The emphasis is on (near) conservation laws and on shock-dominated problems in particular. A number of physics modules are included; the hydrodynamics and the magnetohydrodynamics module are most frequently used. Users can add their own physics module or modify existing ones. The framework supports 1D to 3D simulations, in a number of different geometries (Cartesian, cylindrical, spherical).

::github{repo="amrvac/amrvac"}

SAMRAI

SAMRAI (Structured Adaptive Mesh Refinement Application Infrastructure) is an object-oriented C++ software library that enables exploration of numerical, algorithmic, parallel computing, and software issues associated with applying structured adaptive mesh refinement (SAMR) technology in large-scale parallel application development. SAMRAI provides software tools for developing SAMR applications that involve coupled physics models, sophisticated numerical solution methods, and which require high-performance parallel computing hardware. SAMRAI enables integration of SAMR technology into existing codes and simplifies the exploration of SAMR methods in new application domains.

::github{repo="LLNL/SAMRAI"}

Uintah

Asynchronous Many Task Framework for set of libraries and applications for simulating and analyzing complex chemical and physical reactions modeled by partial differential equations on structured adaptive grids using hundreds to hundreds of thousands of processors.

::github{repo="Uintah/Uintah"}

Parthenon

Parthenon -- a performance portable block-structured adaptive mesh refinement framework

::github{repo="parthenon-hpc-lab/parthenon"}

p4est

p4est is a C library to manage a collection (a forest) of multiple connected adaptive quadtrees or octrees in parallel.

::github{repo="cburstedde/p4est"}

Chombo (Not actively maintained)

Chombo provides a set of tools for implementing finite difference and finite volume methods for the solution of partial differential equations on block-structured adaptively refined rectangular grids. Both elliptic and time-dependent modules are included. Chombo supports calculations in complex geometries with both embedded boundaries and mapped grids, and Chombo also supports particle methods. Most parallel platforms are supported, and cross-platform self-describing file formats are included.

Chombo - Software for Adaptive Solutions of Partial Differential Equations - Chombo - Berkeley Lab Commons

Clawpack

Clawpack is a collection of finite volume methods for linear and nonlinear hyperbolic systems of conservation laws. Clawpack employs high-resolution Godunov-type methods with limiters in a general framework applicable to many kinds of waves. Clawpack is written in Fortran and Python.

Clawpack documentation — Clawpack 5.11.x documentation

Trixi.jl

Trixi.jl is a numerical simulation framework for conservation laws written in Julia.

::github{repo="trixi-framework/Trixi.jl"}

AMReX Resources

I personally find that AMReX is the best AMR library, featuring modern code infrastructure and a large community. I took some time to study it and have collected some resources here, which I hope will be useful for you as well.

AMReX Tutorials

AMReX Examples

[^1]: Adaptive mesh refinement - Wikipedia.