Toy models for Cauchy-Characteristic Extraction (CCE) and Matching (CCM)

This repository contains the code used in the paper arXiv:2306.13010. The goal of this work is to analyze the numerical convergence of CCE and CCM for toy models that mimic the hyperbolic structure of the general relativistic PDE systems used in CCE and CCM. The implemented models are the following:

• for the Initial Boundary Value Problem (IBVP): $$\partial_t \phi_1 = - v_{\phi_1} \partial_\rho \phi_1 + a_{z11} \partial_z \phi_1 + a_{z12} \partial_z \psi_{v 1} + a_{z13} \partial_z \psi_1 + b_{11} \phi_1 + b_{12} \psi_{v 1} + b_{13}\psi_1 $$

$$\partial_t \psi_{v1} = - v_{\psi_{v1}} \partial_\rho \psi_{v1} + a_{z21} \partial_z \phi_1 + a_{z22} \partial_z \psi_{v 1} + a_{z23} \partial_z \psi_1 + b_{21} \phi_1 + b_{22} \psi_{v 1} + b_{23}\psi_1 $$

$$\partial_t \psi_1 = v_{\psi_1} \partial_\rho \psi_1 + a_{z31} \partial_z \phi_1 + a_{z32} \partial_z \psi_{v 1} + a_{z33} \partial_z \psi_1 + b_{31} \phi_1 + b_{32} \psi_{v 1} + b_{33}\psi_1 $$

• for the Characteristic Initial Boundary Value Problem (CIBVP): $$\partial_x \phi_2 = a_{z11} \partial_z \phi_2 + a_{z12} \partial_z \psi_{v 2} + a_{z13} \partial_z \psi_2 + b_{11} \phi_2 + b_{12} \psi_{v 2} + b_{13}\psi_2 $$

$$\partial_x \psi_{v2} = a_{z21} \partial_z \phi_2 + a_{z22} \partial_z \psi_{v 2} + a_{z23} \partial_z \psi_2 + b_{21} \phi_2 + b_{22} \psi_{v 2} + b_{23}\psi_2 $$

$$\partial_u \psi_2 = v_{\psi_2} \partial_\rho \psi_2 + a_{z31} \partial_z \phi_2 + a_{z32} \partial_z \psi_{v 2} + a_{z33} \partial_z \psi_2 + b_{31} \phi_2 + b_{32} \psi_{v 2} + b_{33}\psi_2 $$

The fields $\phi_1, \psi_{v1}, \phi_2, \psi_{v2}$ are left-moving, whereas $\psi_1, \psi_2$ right-moving. The paremeters $v_{\phi_1}, v_{\psi_{v1}}, v_{\psi_1}, v_{\psi_2}$ control the speeds of the fields and should receive only positive values to maintain the direction of propagation of the fields, as well as the correct prescription of boundary data. The speeds of $\phi_2, \psi_{v2}$ are fixed to $1$. The parameters $a_{zij}, b_{ij}$, with $i,j=1,2,3$, control the angular principal part and sources of the systems, respectively, and can be tuned independently for the IBVP and CIBVP.

In the paper, the speeds of propagation are fixed to $\phi_1 = \psi_{v1} = \psi_1 = 1$ and $\psi_2 = 0.5$. Furthermore, the models explored are the symmetric hyperbolic $a_{z12}=a_{z21}=a_{z33}=1$, and the weakly hyperbolic $a_{z21}=a_{z33}=1$, with the rest of the $a_{zij}$ vanishing in each case. Regarding source terms, the following cases are explored:

• homogeneous i.e. $b_{ij} = 0$ for all $i,j$
• inhomogeneous with only non-vanishing $b_{13}=1$
• inhomogeneous with only non-vanishing $b_{32}=1$

The code is written in the Julia programming languange as a module, and tested in Julia version 1.8.5.

Installation

After you have installed Julia and downloaded the repository "model_CCE_CCM_public", change to your local directory where the repository is saved. The module can be installed using Julia's REPL mode. To enter Julia's REPL open your terminal and run "julia". Then do:

julia> ]
pkg> add .

Usage

This section gives the recipe to create the convergence plots presented in the paper.

• Open your terminal
• Change to the directory "./examples"
• Let's assume you want to run the CCM model for noisy given data, that is the script "./examples/run_ccm_noise.jl". To do so, you need to run in your terminal

julia run_ccm_noise.jl

To perform the convergence tests, you need to run the same script for different resolutions. The resolution of a run is controlled by the parameter "D" in the script. There are more parameters in the script that control e.g. the degree of hyperbolicity of the IBVP and CIBVP, their source terms etc. See "./examples/REAME.md" for more details, as well as the comments within the script.

The norms (L2, q, and H1) are calculated with the Julia scripts in "./tools" and plotted with the Jupyter notebooks in the same directory. For example, to reproduce Fig.8-10, you first need to run the noisy CCM script for SYMH-SYMH, WH-WH, SYMH-WH, and SYMH B1-WH B2, for resolutions D=0,1,2,3,4 for each case. Then, you need to run scripts "./tools/L2_norm_exact_ccm.jl" and "dev_norms_exact_ccm.jl", one time for each case (you need to appropriately change the path that points to each directory with the relevant data, in each of the two scripts). Then, you can produce the figures by running the Jupyter notebook "./tools/all_tests_noise_ccm.ipynb", where you have to change again the paths to point to the directories where the norms are saved. See "./tools/README.md" for more details.

Convergence plots: boundary data via time derivative

The paper presents convergence tests for noisy given data, for a numerical algorithm were the boundary data are provided as an injection on the relevant function. Another technique to prescribe boundary data is to provide the time derivative of the relevant function, and not the function itself. The results however are qualitatively the same in both cases. The convergence tests for the second prescription of boundary data (via time derivative) are:

1. homogeneous IBVP: SYMH top and WH bottom (compare to FIG. 5 in the paper)

2. homogeneous CIBVP: SYMH top and WH bottom (compare to FIG.6 in the paper)

3. homogeneous SYMH-WH CCE: the WH CIBVP part (compare to FIG. 7 in the paper)

4. homogeneous CCM: SYMH-SYMH top and WH-WH bottom (compare to FIG. 8 in the paper)

5. homogeneous CCM: SYMH-WH (compare to FIG. 9 in the paper)

6. inhomogeneous CCM: SYMH B1-WH B2 (compare to FIG. 10 in the paper)

7. for comparison with 6): SYMH B1-SYM B2 left, WH B1-WH B2 middle, and on the right the CIBVP part for CCE with SYMH B1-WH B2 (compare to FIG. 11 in the paper)

Comparing the above with the ones presented in the paper, one can see that the method used to prescribe boundary data does not affect the results (at least for the two methods explored).

Convergence plots: different CCE scenarios

In subsection III.C of the paper, different scenarios for CCE are discussed. In figure 7 of the paper however, we presented only the results of the convergence test for the scenario where the IBVP is SYMH and the CIBVP WH. Here were present the same test for SYMH-SYMH and WH-WH CCE as well. The boundary data are given with pure injection.

1. homogeneous SYMH-SYMH CCE: the SYMH CIBVP part (compare to FIG. 7 in the paper)

2. homogenenous WH-WH CCE: the WH CIBVP part (compare to FIG.7 in the paper)

Convergence plots: smooth data

In the paper we focus on tests with noisy given data, since these are important to detect the effect of weak hyperbolicity in numerical experiments. We further argued that one might miss this effect with only smooth data, as for instance was seen in arxiv 2007.06419. For completeness, we perform smooth convergence tests for both boundary data methods, for the setup that exhibit complete loss of convergence with noisy data, that is CCM with SYMH B1-WH B2. We present below the self convergence rate in the L2 norm. In the directory "./tools" one can find a Jupyter notebook that performs a pointwise convergence analysis.

1. Boundary data provided via injection:

2. Boundary data provided via time derivative: