  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.
