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A Convergent Hybridizable Discontinuous Galerkin Method for Einstein--Scalar Equations

Mukul Dwivedi, Andreas Rupp

Abstract

We propose and analyze a hybridized discontinuous Galerkin (HDG) method for the spherically symmetric Einstein--scalar system in Bondi gauge. After rewriting the model as a local first-order PDE--ODE system by introducing suitable scaled variables, we construct a semidiscrete scheme in which the element unknowns are computed locally and the coupling is carried by traces on the mesh skeleton. In the present radial setting, these traces can be eliminated recursively, so that only the main evolution variable is advanced in time, while the metric variables are recovered from discrete constraint relations. We prove local semidiscrete well-posedness, derive a global \(L^2\)--stability estimate, establish an optimal order \(L^2\) error bound for the main evolution variable for polynomial degree \(k\ge 1\), and obtain reconstruction error estimates for the metric variables and the associated mass functional. Numerical experiments verify the predicted spatial convergence rate and illustrate qualitative features of the Einstein--scalar dynamics, including large-data collapse profiles and smooth-pulse evolution.

A Convergent Hybridizable Discontinuous Galerkin Method for Einstein--Scalar Equations

Abstract

We propose and analyze a hybridized discontinuous Galerkin (HDG) method for the spherically symmetric Einstein--scalar system in Bondi gauge. After rewriting the model as a local first-order PDE--ODE system by introducing suitable scaled variables, we construct a semidiscrete scheme in which the element unknowns are computed locally and the coupling is carried by traces on the mesh skeleton. In the present radial setting, these traces can be eliminated recursively, so that only the main evolution variable is advanced in time, while the metric variables are recovered from discrete constraint relations. We prove local semidiscrete well-posedness, derive a global --stability estimate, establish an optimal order error bound for the main evolution variable for polynomial degree , and obtain reconstruction error estimates for the metric variables and the associated mass functional. Numerical experiments verify the predicted spatial convergence rate and illustrate qualitative features of the Einstein--scalar dynamics, including large-data collapse profiles and smooth-pulse evolution.

Paper Structure

This paper contains 12 sections, 10 theorems, 120 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A hybridizable discontinuous Galerkin discretization
  3. Mesh, skeleton notation, and discrete spaces
  4. HDG method
  5. Local well--posedness
  6. Stability
  7. Optimal error analysis
  8. Numerical experiments
  9. Example 1
  10. Example 2
  11. Example 3
  12. Conclusion

Key Result

Lemma 1.1

Assume $g$ and $\tilde{g}$ satisfy eq:ES_g--eq:ES_gtilde with the outer normalization $g(b)=1$. Then for every fixed $t$ and all $r\in(0,b]$, $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Convergence history for Example \ref{['subsec:numerical-example1']}. Panel (a) shows the error of $u_h$ with guide slopes $k+1$, while panel (b) shows the error of $g_h$ with guide slopes $k+2$.
  • Figure 2: Solutions of the Einstein--scalar system for Example \ref{['subsec:numerical-example1']}. In each row, from left to right, the three panels show $\tilde{u}_h$, $\tilde{g}_h$, and $g_h$.
  • Figure 3: Solutions of the Einstein--scalar system for Example \ref{['subsec:numerical-example2']}. In each row, from left to right, the three panels show $\tilde{u}_h$, $\tilde{g}_h$, and $g_h$.
  • Figure 4: Solutions of the Einstein--scalar system for the Gaussian-pulse test of Example \ref{['subsec:numerical-example3']}. In each row, from left to right, the three panels show $\tilde{u}_h$, $\tilde{g}_h$, and $g_h$.
  • Figure 5: Time history of the numerical Bondi-mass proxy for Example \ref{['subsec:numerical-example3']}.

Theorems & Definitions (21)

  • Lemma 1.1: Bounds for the metric coefficients
  • proof
  • Theorem 2.1: Local semidiscrete well-posedness
  • proof
  • Lemma 2.2: Exact discrete constraint reconstruction
  • proof
  • Theorem 3.1: $L^2$-stability
  • proof
  • Corollary 3.2: Global well-posedness
  • proof
  • ...and 11 more