Anderson acceleration. Convergence analysis and applications to equilibrium chemistry

In this paper, we study theoretically and numerically the Anderson acceleration method. First, we extend the convergence results of Anderson's method for a small depth to general nonlinear cases. More precisely, we prove that the Type-I and Type-II Anderson(1) are locally q-linearly convergent if the fixed point map is a contraction with a Lipschitz constant small enough. We then illustrate the effectiveness of the method by applying it to the resolution of chemical equilibria. This test case has been identified as a challenging one because of the high nonlinearity of the chemical system and stiffness of the transport phenomena. The Newton method (usually Newton-Raphson) has been adopted by quite all the equilibrium and reactive transport codes. But the often ill-conditioned Jacobian matrix and the choice of a bad initial data can lead to convergence problems, especially if solute transport produces sharp concentrations profiles. Here we propose to combine the Anderson acceleration method with a particular formulation of the equilibrium system called the method of positive continued fractions (usually used as preconditioning). As shown by the numerical simulations, this approach makes it possible to considerably improve the robustness of the resolution of chemical equilibria algorithms, especially since it is coupled with a strategy to monitor the depth of the Anderson acceleration method in order to control the condition number.