Composing Scalable Nonlinear Algebraic Solvers

Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners. A similar set of concepts may be used for nonlinear algebraic systems, where nonlinear composition of different nonlinear solvers may significantly improve the time to solution. We describe the basic concepts of nonlinear composition and preconditioning and present a number of solvers applicable to nonlinear partial differential equations. We have developed a software framework in order to easily explore the possible combinations of solvers. We show that the performance gains from using composed solvers can be substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table

Structural dynamics branch research and accomplishments for fiscal year 1987

This publication contains a collection of fiscal year 1987 research highlights from the Structural Dynamics Branch at NASA Lewis Research Center. Highlights from the branch's four major work areas, Aeroelasticity, Vibration Control, Dynamic Systems, and Computational Structural Methods, are included in the report as well as a complete listing of the FY87 branch publications

On a Moser-Steffensen type method for nonlinear systems of equations

This paper is devoted to the construction and analysis of a Moser–Steffensen iterative scheme. The method has quadratic convergence without evaluating any derivative nor inverse operator. We present a complete study of the order of convergence for systems of equations, hypotheses ensuring the local convergence, and finally, we focus our attention to its numerical behavior. The conclusion is that the method improves the applicability of both Newton and Steffensen methods having the same order of convergence.Peer ReviewedPostprint (author's final draft

On a Moser–Steffensen type method for nonlinear systems of equations

This paper is devoted to the construction and analysis of a Moser–Steffensen iterative scheme. The method has quadratic convergence without evaluating any derivative nor inverse operator. We present a complete study of the order of convergence for systems of equations, hypotheses ensuring the local convergence, and finally, we focus our attention to its numerical behavior. The conclusion is that the method improves the applicability of both Newton and Steffensen methods having the same order of convergence

Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%