Numerical performance of the parabolized ADM (PADM) formulation of General Relativity

TL;DR

This paper introduces a new parabolic extension of the ADM formulation of general relativity, called PADM, which improves numerical stability and constraint control in simulations.

Contribution

The paper presents the first numerical implementation and analysis of the PADM formulation, demonstrating its stability, accuracy, and improved constraint handling over existing methods.

Findings

PADM is numerically stable and convergent.

PADM has better control of constraint-violating modes.

PADM is second-order accurate in numerical tests.

Abstract

In a recent paper the first coauthor presented a new parabolic extension (PADM) of the standard 3+1 Arnowitt, Deser, Misner formulation of the equations of general relativity. By parabolizing first-order ADM in a certain way, the PADM formulation turns it into a mixed hyperbolic - second-order parabolic, well-posed system. The surface of constraints of PADM becomes a local attractor for all solutions and all possible well-posed gauge conditions. This paper describes a numerical implementation of PADM and studies its accuracy and stability in a series of standard numerical tests. Numerical properties of PADM are compared with those of standard ADM and its hyperbolic Kidder, Scheel, Teukolsky (KST) extension. The PADM scheme is numerically stable, convergent and second-order accurate. The new formulation has better control of the constraint-violating modes than ADM and KST.