A fully coupled, fully implicit simulation method for unsteady flames using Jacobian approximation and clustering

Abstract

Jacobian clustering is proposed to reduce the computational cost associated with matrix operations encountered in the Newton iteration in fully coupled, fully implicit schemes for unsteady reactive flow simulations with detailed chemistry. The iterative solver is based on the Lower-Upper Symmetric Gauss-Seidel (LUSGS) algorithm and sparse matrix technique. The evaluation and sparse Lower-Upper (LU) factorization of the diagonal block of the system Jacobian are performed within clusters rather than individual cells for Computational Fluid Dynamics (CFD) simulations. Cells that are close in the state space are clustered to provide the averaged states for calculating the Jacobians of chemical source terms and transport fluxes. The cells retrieve the factorized sparse matrices from the belonging clusters to perform the necessary iterations. For the purpose of clustering, the spatial dependency of transport Jacobian in the diagonal block is eliminated. To further reduce the computational cost, the sparsity of chemical Jacobian is augmented by removing the insignificant matrix elements. The method is tested in various one-dimensional hydrocarbon flames with both the second order Crank-Nicolson scheme and a third order implicit Runge-Kutta scheme. Various chemical mechanisms with 9 to 111 species are used to test the performance of the iterative solver. Fast convergence of Newton iteration is achieved, and the formal order of accuracy is demonstrated with Jacobian clustering. The overall costs of evaluation and factorization of the block diagonal Jacobian are negligible compared to the cost of calculating transport fluxes and chemical source terms. The averaged costs of Jacobian evaluation, LU factorization and Newton iteration, all increase only linearly with the number of chemical species. The fully coupled, fully implicit Crank-Nicolson scheme with Jacobian clustering shows 4 to 42 times speedup in computational time compared to the decoupled implicit scheme with Strang operator splitting. Jacobian clustering is promising to increase the computational efficiency of high order fully coupled, fully implicit schemes for unsteady reactive flow simulations with detailed chemistry.