Newton's Method for Steady and Unsteady Reacting Flows

Wensheng Shen, and Jun Zhang
Laboratory for High Performance Scientific Computing and Computer Simulation
Department of Computer Science
University of Kentucky
Lexington, KY 40506-0046, USA

Fuqian Yang
Department of Chemical and Materials Engineering
University of Kentucky
Lexington, KY 40506-0046, USA


This paper describes the application of Newton's method to low Mach number steady and unsteady laminar diffusion flames, which are characterized by low flow speed and highly variable density. The newly-emerged vorticity-velocity formulation of the Navier-Stokes equations is used for both steady and unsteady compres sible flows to avoid staggered mesh discretization. The nonlinear Navier-Stokes equations are discretized using finite difference method, and a second-order backward Euler scheme is applied for the time derivatives. Central difference is used for diffusion terms to achieve better accuracy, and a monotonicity-preserving upwind difference is used for convective ones. We use an unequal-sized single grid mesh for unsteady flow and a three level multigrid method for steady flow. The coupled nonlinear system is solved via the damped Newton's method for both steady and unsteady flows. The Newton Jacobian matrix is formed numerically, and the resulting linear system is ill-conditioned and is solved by the iterative solver Bi-CGSTAB with the Gauss-Seidel preconditioner.

Key words: Newton's method, diffusion flame, iterative solver.

Mathematics Subject Classification:

Download the PDF file wensheng5.pdf.
Technical Report No. 448-05, Department of Computer Science, University of Kentucky, Lexington, KY, 2005.

This research work was supported in part by NSF under grants CCR-0092532, ACR-0202934, and CCF-0527967, in part by DOE under grant DE-FG02-02ER45961, and in part by the Kentucky Science and Engineering Foundation.