CFD Newton Solvers with EliAD: An Elimination Automatic Differentiation Tool We present a matrix interpretation of standard forward and reverse modes of automatic differentiation (AD) in terms of forward- and back-substitution of the extended Jacobian system. We then show how efficiency improvements for Jacobian calculation are achieved by performing Gaussian elimination on the extended Jacobian. We introduce the ELIAD tool, developed to enable such elimination AD and present results demonstrating significant run-time improvements both for individual finite-volume flux Jacobian calculations and for a 2-D parabolised Navier-Stokes (PNS) flow solver.