Lattice Boltzmann Model for Numerical Relativity

TL;DR

This paper introduces a lattice Boltzmann model for numerical relativity based on the Bona-Masso formulation, demonstrating its stability, accuracy, and parallel scalability in simulating Einstein's equations.

Contribution

It is the first to apply lattice kinetic theory to gravitational problems, offering a new computational approach for numerical relativity.

Findings

Model shows good agreement with analytical solutions.

Increasing relaxation time enhances stability but reduces accuracy.

Parallel scaling law demonstrates efficient computation.

Abstract

In the Bona-Masso formulation, Einstein equations are written as a set of flux conservative first order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for Numerical Relativity. Our model is validated with well-established tests, showing good agreement with analytical solutions. Furthermore, we show that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improves. Finally, in order to show the potential of our approach a linear scaling law for parallelisation with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.