  Algebraic multigrid (AMG) is often an effective solver for symmetric positive
definite (SPD) linear systems resulting from the discretization of general
elliptic PDEs, or the spatial discretization of parabolic PDEs. However,
convergence theory and most variations of AMG rely on $A$ being SPD. Hyperbolic
PDEs, which arise often in large-scale scientific simulations, remain a
challenge for AMG, as well as other fast linear solvers, in part because the
resulting linear systems are often highly nonsymmetric. Here, a novel
convergence framework is developed for nonsymmetric, reduction-based AMG, and
sufficient conditions derived for $\ell^2$-convergence of error and residual.
In particular, classical multigrid approximation properties are connected with
reduction-based measures to develop a robust framework for nonsymmetric,
reduction-based AMG.
  Matrices with block-triangular structure are then recognized as being
amenable to reduction-type algorithms, and a reduction-based AMG method is
developed for upwind discretizations of hyperbolic PDEs, based on the concept
of a Neumann approximation to ideal restriction ($n$AIR). $n$AIR can be seen as
a variation of local AIR ($\ell$AIR) introduced in previous work, specifically
targeting matrices with triangular structure. Although less versatile than
$\ell$AIR, setup times for $n$AIR can be substantially faster for problems with
high connectivity. $n$AIR is shown to be an effective and scalable solver of
steady state transport for discontinuous, upwind discretizations, with
unstructured meshes, and up to 6th-order finite elements, offering a
significant improvement over existing AMG methods. $n$AIR is also shown to be
effective on several classes of `nearly triangular' matrices, resulting from
curvilinear finite elements and artificial diffusion.
