Nonlinear Krylov-Secant Solvers ∗ Héctor Klíe, and Mary F. Wheeler The Center for Subsurface Modeling (CSM) The Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin, Austin, TX, 78712 Contents 1 Introduction 2 2 Newton-Krylov Framework 3 3 The Family of Secant Solvers 5 3.1 Broyden’s Method ........................................... 7 3.2 The Nonlinear Eirola-Nevanlinna (NEN) Method .......................... 7 4 Krylov-Secant Updates 9 4.1 Secant Updates constrained to the Krylov subspace ......................... 9 4.2 Krylov Secant vs. Standard Secant Update ............................. 12 4.3 The Nonlinear Krylov-Eirola-Nevanlinna Algorithm ......................... 14 4.4 A high-order Newton–Krylov algorithm ............................... 15 5 Implementation issues 15 5.1 Preconditioning ............................................ 15 5.2 Globalization Strategy and Forcing Terms .............................. 16 6 Numerical Experiments 17 7 Conclusions and Further Work 19 Abstract This report describes a new family of Newton-Krylov methods for solving nonlinear systems of equations arising from the solution of Richards’ equation and in fully implicit formulations in air-water sys- tems. The basic approach is to perform secant (Broyden) updates restricted to the Krylov subspace generated by the GMRES iterative solver. This approach is introduced as Krylov-secant methods. One of the most attractive features of these methods is their performance of sequence of rank-one updates without explicitly recalling the com- putation or action of the Jacobian matrix. Implications of these up- dates in line-search globalization strategies, computation of dynamic * Keywords and phrases: Krylov-secant, Broyden’s method, Newton-Krylov, GMRES, Arnoldi factorization, rank-one updates; Author’s e-mail: {klie,mfw}@ices.utexas.edu; UT, Austin. 1