Spectral solutions of PDEs on networks

TL;DR

This paper compares spectral, finite difference, and discontinuous Galerkin methods for solving linear PDEs on metric graphs, highlighting their accuracy, convergence, and implementation details for eigenproblems and wave equations.

Contribution

It introduces and evaluates three numerical methods for PDEs on networks, emphasizing spectral accuracy and flexibility in boundary conditions.

Findings

Spectral method achieves exponential convergence and machine precision eigenvalues.

Discontinuous Galerkin method handles arbitrary polynomial orders and complex boundary conditions.

Finite difference method with ghost cells maintains second-order accuracy at vertices.

Abstract

To solve linear PDEs on metric graphs with standard coupling conditions (continuity and Kirchhoff's law), we develop and compare a spectral, a second-order finite difference, and a discontinuous Galerkin method. The spectral method yields eigenvalues and eigenvectors of arbitary order with machine precision and converges exponentially. These eigenvectors provide a Fourier-like basis on which to expand the solution; however, more complex coupling conditions require additional research. The discontinuous Galerkin method provides approximations of arbitrary polynomial order; however computing high-order eigenvalues accurately requires the respective eigenvector to be well-resolved. The method allows arbitrary non-Kirchhoff flux conditions and requires special penalty terms at the vertices to enforce continuity of the solutions. For the finite difference method, the standard one-sided second-order finite difference stencil reduces the accuracy of the vertex solution to $ O(h^{3/2})$. To preserve overall second-order accuracy, we used ghost cells for each edge. For all three methods we provide the implementation details, their validation, and examples illustrating their performance for the eigenproblem, Poisson equation, and the wave equation.