Newton-Krylov methods for low mach number combustion

Fully coupled numerical techniques are used to compute steady state solutions to a combusting, low Mach number compressible flow through a channel. The nonlinear governing equations are discretized on a staggered mesh via integration over discrete finite volumes. The resulting nonlinear algebraic equations are linearized with Newton’s method and solved with a preconditioned Krylov algorithm. Both Incomplete Lower-Upper (ILU) factorization and domain-based multiplicative Schwan preconditioning strategies are considered. The selected Krylov solver is the popular Generalized Minimum RESidual (GMRES) algorithm. Because GMRES requires the Jacobian matrix only in the form of matrix vector products, which can be approximated with finite difference projections, some expensive Jacobian evaluations can be avoided. This ‘matrix-free’ implementation is exploited within the context of pseudo-transient Newton-Krylov calculations using the Switched Evolution Relaxation (SER) algorithm to control time step size. For the selected model problem, an order of magnitude reduction in CPU time is observed when the standard, fixed time step, pseudo-transient, Newton-Krylov algorithm is replaced with a variable time step, matrix-free implementation using the SER algorithm. Additionally, the domainbased multiplicative Schwarz preconditioning strategy was found to be more effective than ILU preconditioning at lower Mach numbers.