Solving Partial Differential Equations Numerically on Manifolds with Arbitrary Spatial Topologies

TL;DR

This paper introduces a multi-cube numerical method for solving elliptic and hyperbolic PDEs on manifolds with arbitrary topologies, enabling accurate simulations on complex geometries.

Contribution

It develops a novel multi-cube framework that represents any 3D manifold as non-overlapping cubes with face-matching conditions for PDE solutions.

Findings

Successfully solves elliptic equations on T^3, S^2 x S^1, S^3 topologies

Demonstrates hyperbolic PDE solutions on R x T^3, R x S^2 x S^1, R x S^3

Uses pseudo-spectral methods for high-accuracy numerical results

Abstract

A multi-cube method is developed for solving systems of elliptic and hyperbolic partial differential equations numerically on manifolds with arbitrary spatial topologies. It is shown that any three-dimensional manifold can be represented as a set of non-overlapping cubic regions, plus a set of maps to identify the faces of adjoining regions. The differential structure on these manifolds is fixed by specifying a smooth reference metric tensor. Matching conditions that ensure the appropriate levels of continuity and differentiability across region boundaries are developed for arbitrary tensor fields. Standard numerical methods are then used to solve the equations with the appropriate boundary conditions, which are determined from these inter-region matching conditions. Numerical examples are presented which use pseudo-spectral methods to solve simple elliptic equations on multi-cube representations of manifolds with the topologies T^3, S^2 x S^1 and S^3. Examples are also presented of numerical solutions of simple hyperbolic equations on multi-cube manifolds with the topologies R x T^3, R x S^2 x S^1 and R x S^3.