On the solution of nonlinear two-point boundary value problems on successively refined grids

The solution of nonlinear two-point boundary value problems by adaptive finite difference methods ordinarily proceeds from a coarse to a fine grid. Grid points are inserted in regions of high spatial activity and the coarse grid solution is then interpolated onto the finer mesh. The resulting nonlinear difference equations are often solved by Newton's method. As the size of the mesh spacing becomes small enough. Newton's method converges with only a few iterations. In this paper we derive an estimate that enables us to determine the size of the critical mesh spacing that assures us that the interpolated solution for a class of two-point boundary value problems will lie in the domain of convergence of Newton's method on the next finer grid. We apply the estimate in the solution of several model problems

Stability and convergence of linear parabolic mixed initial-boundary value problems on nonuniform grids

Many problems in the physical sciences can be reduced to the solution of a system of time-dependent partial differential equations. Of particular interest to us are problems in combustion and heat and mass transfer, i.e., hydrocarbon ignition and catalytic combustion. The governing equations in these applications can be formulated as a system of parabolic mixed initial-boundary value problems. The numerical stiffness that results from solving these problems on a discrete mesh combined with the inherent stiffness of the disparate decay rates of the various chemical species necessitate the use of implicit time differencing methods. The problems also produce solutions that contain regions of high spatial activity, i.e., sharp peaks and steep fronts. Although an equispaced mesh could be used in the calculations, it is often more efficient to employ a nonuniform adaptive grid in the anticipation that the high activity regions will be better resolved. Important questions in such studies are the effects that adaptive time and space steps (both fixed and variable numbers of points) have on the stability and convergence of the parabolic solver. We investigate these issues in this paper for a class of linear mixed initial-boundary value problems

Estimates for the inverse of tridiagonal matrices arising in boundary-value problems

By considering tridiagonal matrices as three-term recurrence relations with Dirichlet boundary conditions, one can formulate their inverses in terms of Green's functions. This analysis is applied to three-point difference schemes for 1-D problems, and five-point difference schemes for 2-D problems. We derive either an explicit inverse of the Jacobian or a sharp estimate for both uniform and nonuniform grids

High Performance Computing and Numerical Simulation of Flames

which critical fields vary by orders of magnitude. As a result of the fluid dynamics-thermochemistry interaction and its effect on the flame structure, the governing equations are strongly coupled together and are also characterized by the presence of stiff source terms and nonlinearities. However, in spite of these difficulties, the numerical modeling of multidimensional laminar (or turbulent) flames has been recently motivated by the growing demand for high fuel efficiency combined with low pollutant emission. Axisymmetric laminar diffusion flames constitute a problem of practical importance since they are the flame type of several combustion devices. Hence, new robust numerical models of such a system will provide an efficient tool to probe flame structures and investigate the coupled effects of complex transport phenomena with chemical kinetics. In diffusion flames the combustion process is primarily controlled by the rate at which the fuel and oxidizer are brought together in st