Differential equations on metric graphs: continuum and numerical methods

Abstract

Continuum modelling has proven exceedingly useful throughout science by assuming the infinite divisibility of a given system. However, high-density metric graphs—representing a range of natural and engineered systems—have seen limited modelling in this way. A metric graph is what one might expect intuitively: a spiderweb, neuronal network, water-supply system, or the British Rail. Physical continuity allows for one-dimensional differential equations along each edge. At vertices, solutions satisfy flux-balance conditions. We study the eigenvalues and eigenfunctions of the Laplace operator on an increasingly dense graph within a prescribed embedding space, e.g., a spiderweb filling the unit disc. The continuous, linear eigenvalue problem on the edges reduces to a discrete, nonlinear eigenvalue problem on the vertices. Numerically, solutions require a novel matrix-determinant root-finding algorithm. Alternatively, the discrete system becomes the eigenvalue equation of a continuous partial differential operator, resembling a Laplace--Beltrami operator but with several notable differences. Rather than a Riemannian metric, we derive a distinct symmetric tensor that scales linearly with distance. Rather than the determinant-based volume form, we find an analogous matrix-trace-based distance form. Our findings open the possibility for a new manifold geometry similar to geodesic structure but made from underlying “graph material”. We discuss a series of examples of high-density networks, comparing PDE solutions to numerical solutions of vertex systems. We start with trivial cases and proceed to inhomogeneous and anisotropic media. We develop continuum models for random graphs, hierarchical networks, and cases of nonuniform edge conductances and capacities.