# Lattice Boltzmann model for the simulation of the wave equation in curvilinear coordinates

## TL;DR

This paper develops a lattice-Boltzmann model for simulating wave equations in curvilinear coordinates, enabling accurate analysis of vibrational modes in complex geometries like cylinders and tori.

## Contribution

It introduces a novel lattice-Boltzmann approach for scalar wave propagation in curvilinear coordinates, expanding the method's applicability beyond traditional rectangular systems.

## Key findings

- Accurately reproduces theoretical vibrational modes in complex geometries
- Demonstrates the model's effectiveness in cylindrical, trumpet, and toroidal shapes
- Shows potential for broad applications in physics and engineering

## Abstract

Since its origins, lattice-Boltzmann methods have been restricted to rectangular coordinates, a fact which jeopardises the applications to problems with cylindrical or spherical symmetries and complicates the implementations with complex geometries. However, M. Mendoza recently proposed in his doctoral thesis a general procedure (based on Christoffel symbols) to construct lattice-Boltzmann models on curvilinear coordinates, which has shown very good results for hydrodynamics on cylindrical and spherical coordinates. In this work, we construct a lattice-Boltzmann model for the propagation of scalar waves in curvilinear coordinates, and we use it to determine the vibrational modes inside cylinders, trumpets and tori. The model correctly reproduces the theoretical expectations for the vibrational modes, and exemplifies the wide range of future applications of lattice-Boltzmann models on general curvilinear coordinates.