Introduction to Numerical Relativity
Philosophy
Numerical Relativity (NR) is the field of study of astrophysical and theorized phenomena in full General Relativity. When treated without approximation, gravity presents a substantial addition of complexity to the behavior of all massive systems. In addition to the micro-physics laws, one now also has to solve for the full geometric representation of gravity. This is done by solving Einstein equations (in addition to micro-physics equations), with the masses in the system contributing to the source terms in Einstein field equations (EFE). EFE are a set of 10 coupled second-order partial differential equations in space and time. Solving these additional equations is computationally intensive and algorithmic-ally complex. Numerical Relativity is therefore a multidisciplinary field of research that naturally draws serious input from Applied Mathematics, Physics, Engineering and Computer Science literature.
This course is aimed at students and readers who are interested in getting a broad overview & working knowledge of the subject, and will leave them capable of navigating independently through advanced topic(s) of interest. If one considers the concepts and results involved in a field of study as a graph with dependency connections, in this course we will follow a more-or-less breadth-first traversal of this knowledge graph for NR. We will overview foundational elements from Applied Mathematics, Physics and Computer Science, and also have readings on cutting-edge developments in the field. We will be restricting ourselves to systems made only of spacetime in this course in order to have a reasonable scope for a semester.
We will start with an overview lecture that will present a broad survey of some foundational concepts and challenges in Numerical Relativity. This will be followed by 2 topical readings per week that will comprise of graded presentations by students, followed by discussion(s) among the group. The presentations may use slides or just make use of the white-board on Zoom, or simply could be a set of hand-written note pages presented during the class. We will aim for 50-60 min long presentations, allowing 15-30 mins for discussion. Grading of class presentations will be based on how well the topics have been understood and explained by the presenting student, as well as on the ensuing discussion that they might lead. Students will be assigned topical readings randomly (see below). Each student will be presenting approximately 2-3 times in the semester.
The first few weeks will be given to the discussions of differential geometry tenets and useful geometric properties of spacetime. This will be accompanied by assignment(s) on the topic(s). Following this, we will study 4 broad topics:
- calculation of initial data for numerical simulations of Einstein equations,
- choice of coordinate / gauge degrees of freedom that are optimal for evolutions of Einstein equations,
- numerical methods, and
- evolving black hole spacetimes.
Each of these topics will be covered in 2-3 classes. The first 2 of these sessions will be assigned topical readings of textbook material. The 3rd / last session will comprise of topical readings of 1-2 important contemporary published articles on the topic. Each of these 4 major topics will also involve literature-review style assignments. One of these review assignments will be taken to be the student's "primary" paper for the course (each student can choose which). There will be a final viva on this paper, at the end.
Evaluation
The final grade will be based on the following components:
- Class Presentation: 40%
- Primary assignment paper (+ viva, if needed): 30% + 10%
- Homework problems and assignments: 20%
Working Information & Schedule
- First meeting: January 19, 2020
- Class timings: Wednesdays 14:00 - 15:30 IST & Thursdays 15:45 - 17:00 IST
- Class location: Zoom link (Meeting ID: 890 8157 2083, Passcode: 565658)
- Primary Textbooks:
- "Numerical Relativity: Solving Einstein's Equations on the Computer" by Thomas W. Baumgarte & Stuart L. Shapiro
- "Introduction to 3+1 Numerical Relativity" by Miguel Alcubierre
- Class schedule (Note that week 0 is January 17-22, 2022):
Date | Week-Meeting | Presenter | Topics | References | Assignments |
---|---|---|---|---|---|
19 Jan 2022 | 0-1 | Prayush | Overview | ||
20 Jan 2022 | 0-2 | - | - | - | |
26 Jan 2022 | 1-1 | - | Republic Day | - | |
27 Jan 2022 | 1-2 | Srashti | Geometry and GR: Sec. 1.1-1.3 of (i) Sec. 1.4 of (i) for extra credit (EC) |
(i) Baumgarte & Shapiro | |
02 Feb 2022 | 2-1 | Aditya | Decomposing spacetime back to space+time: Sec. 2.1-2.4 of (i) |
(i) Baumgarte & Shapiro | |
03 Feb 2022 | 2-2 | Pranav | Decomposing spacetime back to space+time: Sec. 2.5-2.7 of (i) |
(i) Baumgarte & Shapiro | |
09 Feb 2022 | 3-1 | Arif | Primer on PDEs: classification and characteristics Sec. 6.1 of (i) Sec 1, 2 of (ii) Note on characteristics |
(i) Baumgarte & Shapiro (ii) On Numerical Solutions of PDEs by L. Rezzolla |
1 |
10 Feb 2022 | 3-2 | - | - | - | |
16 Feb 2022 | 4-1 | Mukesh | Initial data for numerical evolutions: Sec 3.1-3.2 of (i) |
(i) Baumgarte & Shapiro | |
17 Feb 2022 | 4-2 | Uddeepta | Initial data for numerical evolutions: Sec 3.3 + Sec 12.3 of (i) |
(i) Baumgarte & Shapiro | |
23 Feb 2022 | 5-1 | - | - | - | |
24 Feb 2022 | 5-2 | Shubham + Vaishak | Initial data for numerical evolutions: Papers |
- A multidomain spectral method for solving elliptic equations | |
02 Mar 2022 | 6-1 | Shubham + Vaishak | Initial data for numerical evolutions: Papers |
- A multidomain spectral method for solving elliptic equations | Literature Review Assignment |
03 Mar 2022 | 6-2 | Souvik | Gauge conditions for evolutions: Sec 4.1 and 4.2 of (i) |
(i) Baumgarte & Shapiro | |
09 Mar 2022 | 7-1 | Deepali | Gauge conditions for evolutions: Sec 4.3 and 4.5 of (i) |
(i) Baumgarte & Shapiro | |
10 Mar 2022 | 7-2 | Akash | Gauge conditions for evolutions: Papers |
- Generalization of harmonic slicing by Bona, Masso, Seidel and Steia | Literature Review: Assignment |
16 Mar 2022 | 8-1 | Estuti | Evolving Einstein equations: Sec 11.1-11.3 of (i) |
(i) Baumgarte & Shapiro | |
17 Mar 2022 | 8-2 | Anuj | Evolving Einstein equations: Sec 11.4-11.5 of (i) |
(i) Baumgarte & Shapiro | |
23 Mar 2022 | 9-1 | Suhas [POSTPONED DUE TO SPEAKER MIA] |
Evolving Einstein equations: Papers | - Generalized harmonic formulation by Lindblom et al (2006) | |
24 Mar 2022 | 9-2 | Deepali | Numerical methods: Finite difference/Finite volume - Sec 6.2 of (i) |
(i) Baumgarte & Shapiro | |
30 Mar 2022 | 10-1 | POSTPONED DUE TO OVERLAP WITH ASI MEETING | - | - | |
31 Mar 2022 | 10-2 | POSTPONED DUE TO OVERLAP WITH ASI MEETING | - | - | |
06 Apr 2022 | 11-1 | Deepali | [continuation] Numerical methods: Finite difference/Finite volume - Sec 6.2 of (i) |
(i) Baumgarte & Shapiro | |
07 Apr 2022 | 11-2 | Omkar | Numerical methods: Spectral (global) - Sec 6.3 of (i) |
(i) Baumgarte & Shapiro | |
13 Apr 2022 | 12-1 | Uddeepta | Numerical methods: Discontinuous Galerkin | DG methods for curved spacetime by Teukolsky et al | Literature Review: Assignment |
14 Apr 2022 | 12-2 | Estuti | Evolving black hole spacetimes: Sec 13.1 of (i) |
(i) Baumgarte & Shapiro | |
20 Apr 2022 | 13-1 | POSTPONED DUE TO OVERLAP WITH ICTS In-HOUSE | - | - | |
21 Apr 2022 | 13-2 | POSTPONED DUE TO OVERLAP WITH ICTS In-HOUSE | - | - | |
27 Apr 2022 | 14-1 | Shalabh | Evolving Einstein equations: Papers | - Generalized harmonic formulation by Lindblom et al (2006) | Literature Review: Assignment |
28 Apr 2022 | 14-2 | POSTPONED DUE TO INCOMPLETE STUDENT PREP | - | - | |
04 May 2022 | 15-1 | Prayush | Evolving black hole spacetimes: Lets try out SpECTRE | - Ensure that everyone is able to set up spectre - Introduction to SpECTRE |
|
05 May 2022 | 15-2 | Prayush | Evolving black hole spacetimes: Lets try out SpECTRE | - Evolve scalar waves Minkowski spacetime - Evolve scalar waves in Kerr spacetime |
Note: The ordering above was chosen using the following Python
code and a text file with the names of all students written in alphabetical order of first name:
>>> import numpy.random as random
>>> random.seed(12345)
>>> with open("students.txt") as f:
... c = f.readlines()
... c = [x.strip() for x in c if len(x.strip()) > 0]
...
>>> random.choice(c, size=len(c), replace=False)
array(['Aditya', 'Srashti', 'Pranav', 'Arif', 'Mukesh', 'Suprovo',
'Shubham', 'Vaishak', 'Uddeepta', 'Souvik', 'Deepali', 'Akash',
'Estuti', 'Anuj'], dtype='<U8')
>>> random.choice(c, size=len(c), replace=False)
array(['Deepali', 'Uddeepta', 'Shubham', 'Estuti', 'Srashti', 'Arif',
'Vaishak', 'Souvik', 'Akash', 'Anuj', 'Aditya', 'Mukesh', 'Pranav',
'Suprovo'], dtype='<U8')
>>> random.choice(c, size=len(c), replace=False)
array(['Anuj', 'Srashti', 'Vaishak', 'Shubham', 'Akash', 'Mukesh',
'Souvik', 'Uddeepta', 'Estuti', 'Arif', 'Aditya', 'Deepali',
'Pranav', 'Suprovo'], dtype='<U8')
>>>
Topics
Topics | Literature | Schedule | ||
---|---|---|---|---|
1 | Geometry and GR: - Manifolds, Tensors, Notions of derivatives - Einstein field equations and useful identities |
Ch 1 of Baumgarte & Shapiro / Ch 1 of Alcubierre | Week 1 | |
2 | 3+1 decomposition of Spacetime: - choice of time - Einstein equations in 3+1 form |
Ch 2 of Baumgarte & Shapiro / Ch 2 of Alcubierre | Week 1,2 | |
3 | PDEs: basics, characteristics and classification | - Ref [9] on introduction to PDEs - Ref [8] on characteristics - Chapter 6.1 of Baumgarte & Shapiro |
Week 2 | |
4 | Initial data for numerical evolutions: - York-Lichnerowicz conformal decomposition - Conformal thin-sandwich approach - Multiple black holes: Bowen-York extrinsic curvature - Multiple black holes: Kerr-Schild type data |
- Ch 3, 12 of Baumgarte & Shapiro - Lovelace et al 2008 for a comparative study - A multidomain spectral method for solving elliptic equations |
Week 3,4 | |
5 | Gauge conditions for evolutions: - How to choose the time coordinate? - Shift conditions |
Ch 4 of Baumgarte & Shapiro / Ch 4 of Alcubierre | Week 5,6 | |
6 | Einstein equations for evolutions: - Hyperbolicity and well-posedness - ADM equations - Harmonic and GH equations |
- Ch 11 of Baumgarte & Shapiro - Ch 5 of Alcubierre's book - Lindblom et al 2006 - Lindblom et al 2007 - Lindblom et al 2009 |
Week 6,7 | |
7 | Numerical methods: - Method of lines for time integration - Finite differencing - Von Newmann stability analysis - Dissipation and dispersion - Spectral methods - Discontinuous Galerkin methods - Convergence testing |
- Ch 6 of Baumgarte & Shapiro - Ch 1 of LeVeque's notes for FDM - Section 1 of Ref [5] - Chapter 9 of Alcubierre's book - Ref [7] on DG methods - |
Week 7,8 | |
8 | Evolving black hole spacetimes: - Isometries and adapted coordinates - Singularity avoidance and slice stretching - Black hole excision - Apparent horizons - Boundary conditions |
- Ch 7, 8.1,13 of Baumgarte & Shapiro - Chapter 6.1,6.2,6.7-6.9 of Alcubierre |
Week 9,10 | |
9 | Evolve scalar waves in a black-hole's spacetime. | - Lets try out SpECTRE [10] | Week 11 | |
10 | Advanced numerical simulations: - Adaptive time-steppers - Mesh refinement: h-refinement and p-refinement - Adaptive mesh refinement - Control systems: Apparent horizon trackers |
- Chapter 6.2-6.8 of LeVeque's notes for time-steppers - Classic Ref [13] for AMR - Sections 2+ of Ref [5] - Intro to Proportional-Integral-Derivative control at Ref [12] |
Week 12 |
References
- "Introduction to 3+1 Numerical Relativity" by Miguel Alcubierre
- "Numerical Relativity: Solving Einstein's Equations on the Computer" by Thomas W. Baumgarte & Stuart L. Shapiro
- "Numerical Relativity: Starting from Scratch" by Thomas W. Baumgarte & Stuart L. Shapiro
- "Finite Difference Methods for Differential Equations" by Randall J. LeVeque
- "ODEs" - Lecture at Courant Institute
- "Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations" by Lloyd N Trefethen
- "Formulation of discontinuous Galerkin methods for relativistic astrophysics" by Saul Teukolsky
- Note on characteristics
- Notes on numerical solutions of PDEs by Luciano Rezzolla
- Evolutions with SpECTRE
- SpECTRE general documentation
- Feedback Control Systems and PID control
- "Adaptive mesh refinement for hyperbolic partial differential equations by Marsha Berger & Joseph Oliger