Steady-state behavior of large water distribution systems: Algebraic multigrid method for the fast solution of the linear step

The Newton-based global gradient algorithm (GGA) (also known as the Todini and Pilati method) is a widely used method for computing the steady-state solution of the hydraulic variables within a water distribution system (WDS). The Newton-based computation involves solving a linear system of equations arising from the Jacobian of the WDS equations. This step is the most computationally expensive process within the GGA, particularly for large networks involving up to O(105) variables. An increasingly popular solver for large linear systems of the M-matrix class is the algebraic multigrid (AMG) method, a hierarchical-based method that uses a sequence of smaller dimensional systems to approximate the original system. This paper studies the application of AMG to the steady-state solution of WDSs through its incorporation as the linear solver within the GGA. The form of the Jacobian within the GGA is proved to be an M-matrix (under specific criteria on the pipe resistance functions), and thus able to be solved using AMG. A new interpretation of the Jacobian from the GGA is derived, enabling physically based interpretations of the AMG's automatically created hierarchy. Finally, extensive numerical studies are undertaken where it is seen that AMG outperforms the sparse Cholesky method with node reordering (the solver used in EPANET2), incomplete LU factorization (ILU), and PARDISO, which are standard iterative and direct sparse linear solvers.