Stable 3-level leapfrog integration in numerical relativity

TL;DR

This paper introduces an improved leapfrog integration method, called 'deloused leapfrog', that enhances stability and efficiency for numerical relativity simulations involving complex differential equations.

Contribution

The paper presents a novel 'deloused leapfrog' algorithm that reduces memory use and enables adaptive timesteps while preventing numerical instabilities in relativistic simulations.

Findings

The deloused leapfrog method stabilizes highly relativistic problems.

It reduces CPU costs by five to eight times compared to Crank-Nicholson.

The method is effective in various gravitational scenarios.

Abstract

The 3-level leapfrog time integration algorithm is an attractive choice for numerical relativity simulations since it is time-symmetric and avoids non-physical damping. In Newtonian problems without velocity dependent forces, this method enjoys the advantage of long term stability. However, for more general differential equations, whether ordinary or partial, delayed onset numerical instabilities can arise and destroy the solution. A known cure for such instabilities appears to have been overlooked in many application areas. We give an improved cure ("deloused leapfrog") that both reduces memory demands (important for 3+1 dimensional wave equations) and allows for the use of adaptive timesteps without a loss in accuracy. We show both that the instability arises and that the cure we propose works in highly relativistic problems such as tightly bound geodesics, spatially homogeneous spacetimes, and strong gravitational waves. In the gravitational wave test case (polarized waves in a Gowdy spacetime) the deloused leapfrog method was five to eight times less CPU costly at various accuracies than the implicit Crank-Nicholson method, which is not subject to this instability.