The algorithm for linear PDEs usually boils down to some kind of meshing/discretization (often the hard part) to make a (usually sparse) linear system to solve

The algorithm for linear PDEs usually boils down to some kind of meshing/discretization (often the hard part) to make a (usually sparse) linear system to solve using standard numerical methods. In the basic 1d, first-order case, it winds up being exactly the same thing as equivalent finite difference method.