An implicit compact scheme solver with application to chemically reacting flows

A novel, stable, implicit compact scheme solver that is higher order in space, suitable for modeling steady-state and time-dependent phenomena on nonuniform grids for one-dimensional configurations, is presented. Several properties of compact scheme discretizations are introduced to develop efficient algorithms for Jacobian matrix generation and Jacobian-vector multiplication using a new component form for Jacobian operations. Composite nonuniform grids are introduced that enable the implicit compact scheme solver to achieve sixth order accuracy. A robust Newton's method is employed with explicit generation of Jacobian matrices. Superior resolution characteristics of the implicit compact scheme solver are demonstrated with several steady-state and time-dependent problems for the Burgers equation. The example of the solution of stiff flame problem is given. An analysis of spectral properties of Jacobian matrices is presented, which shows that the condition number and the eigenvalue distributions behave similarly to those found in Jacobians associated with low-order discretizations. Two sparsification strategies are developed for the systematic approximation of a dense Jacobian aimed at the practical implementation of linear system preconditioning through partial Jacobians.