A matrix free Newton /Krylov method for coupling complex multi-physics subsystems

Yunlin Xu, Purdue University

Abstract

A general approach is proposed for solving coupled nonlinear subsystems using Matrix Free Newton/Krylov (MFNK) methods. A system of nonlinear equations is defined based on a Fixed Point Iteration (FPI) scheme which does not require the reformulation of the coupled subsystems into a larger system of equations. This makes it possible for each of the subsystems to retain their respective solvers for the individual field equations. The MFNK method is applied to the new nonlinear system related to the FPI in order to find the fixed point and the solution of the coupled subsystems. An innovative method based on an analysis of the truncation error and the round-off error introduced by the finite difference approximation is developed to estimate the optimal perturbation size for matrix free Newton/Krylov methods. Numerical examples are provided to demonstrate the advantage of the optimal perturbation size compared to the perturbation size determined by an empirical formula. ^ Both local and global convergence of the FPI and the MFNK methods are analyzed. It is shown that even for the cases in which the corresponding FPI diverges, the MFNK can achieve q-quadratic but at least q-linear local convergence. Analysis of global convergence strategies indicates that Line Searches and Model trust region strategies can be implemented efficiently for MFNK. ^ Finally, various parallel models were investigated for MFNK. The most promising results were achieved using a peer-to-peer algorithm in which the MFNK algorithm is applied separately to each subsystem and locally the vector operations in MFNK are parallelized. Results indicate that when the computational load is balanced in each of the subsystems, MFNK can achieve at least linear speedup, and in some cases super-linear speedup is possible. ^ The nonlinear system based on a FPI and the method for estimating the optimal finite different size developed in this work represent original contributions to the field. The globally convergent strategies and the high performance computing implementation of MFNK are applications of existing methods to this general approach. ^

Degree

Ph.D.

Advisors

Major Professor: Thomas J. Downar, Purdue University.

Subject Area

Engineering, Nuclear

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