NRPy+ Tutorial

Zachariah B. Etienne $\leftarrow$ Please feel free to email comments, revisions, or errata!

PART 1: Basic Functionality of NRPy+, a First Application

NRPy+ Basics

• NRPy+: Introduction & Motivation ([email protected] web page)
• Basic C Code Output, NRPy+'s Parameter Interface
• Numerical Grids
• Indexed Expressions (e.g., tensors, pseudotensors, etc.)
• Finite Difference Derivatives
  • Instructional module: How NRPy+ Computes Finite Difference Derivative Coefficients
  • Start-to-Finish Example: Finite-Difference Playground: A Complete C Code for Analyzing NRPy+-Based Finite Differences

PART 2: Basic Physics Applications

Leveraging NRPy+ to Numerically Solve PDEs

• Application: The Scalar Wave Equation in Cartesian Coordinates, with Plane-Wave Initial Data
  • Start-to-Finish Example: Numerically Solving the Scalar Wave Equation: A Complete C Code
  • Solving the Wave Equation with the **Einstein Toolkit** (Courtesy Patrick Nelson)
    • **IDScalarWaveNRPy**: Plane-wave initial data module for the scalar wave equation
    • **WaveToyNRPy**: Solving the scalar wave equation, using the method of lines
• Application (in progress): Two Vector-Potential Formulations of Maxwell's Equations in Cartesian Coordinates. (Formulations based on Illustrating Stability Properties of Numerical Relativity in Electrodynamics by Knapp, Walker, and Baumgarte.) (Courtesy Patrick Nelson)
  • **IDMaxwellNRPy**: An **Einstein Toolkit** initial data thorn for Maxwell's equations
  • **MaxwellEvol**: Solving Maxwell's equations in the **Einstein Toolkit** using the method of lines

Diagnostic Modules: Gravitational Wave Extraction

• Application: The Weyl Scalars and Invariants in Cartesian Coordinates (Courtesy Patrick Nelson)
  • **WeylScal4NRPy**: An **Einstein Toolkit** Diagnostic Thorn (Courtesy Patrick Nelson)

Diagnostic Modules: GRMHD/GRFFE

• Application: Computing the 4-Velocity Time-Component $u^0$, the Magnetic Field Measured by a Comoving Observer $b^{\mu}$, and the Poynting Vector $S^i$
  • **sbPoynETNRPy**: Evaluating $b^\mu$ and $S^i$ in the **Einstein Toolkit**, using HydroBase variables as input

PART 3: Solving PDEs in Curvilinear Coordinate Systems

• Moving beyond Cartesian Grids: Reference Metrics
• Application: The Scalar Wave Equation in Curvilinear Coordinates, using a Reference Metric
  • Start-to-Finish Example: Numerically Solving the Scalar Wave Equation in Curvilinear Coordinates: A Complete C Code
• Application: Maxwell's Equations in Curvilinear Coordinates, using a Reference Metric (Courtesy Ian Ruchlin)
• Application: General Relativity in the BSSN Formalism, in singular curvilinear coordinates
  • Brill-Lindquist: Setting up two black hole initial data
  • ADM Cartesian to BSSN Curvilinear Initial Data Conversion
  • Start-to-Finish Example: Two-black-hole initial data validation: numerical error convergence to zero
  • Start-to-Finish Example: Colliding black holes!
  • Application (in progress) Adding the $8\pi T^{\mu\nu}$ source terms to the BSSN Equations