On the application of Algorithmic Differentiation to Newton solvers

Newton solvers have the attractive property of quadratic convergence but they require derivative information. An efficient way of computing derivatives is by Algorithmic Differentiation (AD) also known as automatic differentiation or computational differentiation. AD allows us to evaluate derivatives usually at a cheap cost without the truncation errors associated with finite-differencing. Recent years witnessed an intense activity to produce tools enabling systematic calculation of derivatives. Efficient and reliable AD tools for evaluating derivatives have been published. In this paper, we sketch some of the main theory at the heart of AD, review some of the best AD codes currently available and put into context the use of AD for iterative solution methods of nonlinear systems or adjoint equations. Our aim is to direct scientists and engineers confronted with the need of exactly calculating derivatives to the use of AD as a highly useful tool and those AD tools which they could try primarily. Moreover, we show that the use of AD increases the performance of the quadratically convergence solution of a parabolised Navier-Stokes equations.