Local defect correction techniques : analysis and application to combustion

Combustion processes are of fundamental importance both for industry and for ordinary life. Numerical simulations may be used as a design tool for the development of more efficient burners with a lower exhaust of polluting gases. In the mathematical description of a flame, we consider it a flowing gas mixture in which chemical reactions take place. The mathematical model follows from the conservation laws for mass and momentum (the flow equations) and the conservation laws for the mass fractions of the species in the mixture and for energy (the combustion equations). The system of equations describing a flame has several characteristics that make numerical flame simulations a challenge even with present day computer technology. The system is a set of nonlinear partial differential equations, which may describe complex flows. The chemical source terms are strongly nonlinear. Detailed chemical models contain many reactions and many chemical species. Reactions may occur with very different time scales. Finally, there are large differences in geometric scales: the dependent variables have large gradients in the flame zone and are relatively smooth elsewhere. In this thesis, we focus on the latter problem and study a method for adaptive grid refinement. Rather than using a truly nonuniform grid, we present a method called local defect correction (LDC) that is based on local uniform grid refinement. Advantages of the LDC method include the usage of simple data structures and simple accurate discretization stencils. In the LDC method, the discretization on the composite grid is based on a combination of standard discretizations on several uniform grids with different grid sizes that cover different parts of the domain. At least one grid, the coarse grid, should cover the entire domain, and its grid size should be chosen in agreement with the relatively smooth behavior of the solution outside the high activity areas. Apart from this global coarse grid, one or several local fine grids are used which are also uniform. Each of the local grids covers only a (small) part of the domain and contains a high activity region. The grid sizes of the local grids are chosen in agreement with the behavior of the continuous solution in that part of the domain. The LDC method is an iterative process: a basic global discretization is improved by local discretizations defined in subdomains. The update of the coarse grid solution is achieved by adding a defect correction term to the right hand side of the coarse grid problem. At each iteration step, the process yields a discrete approximation of the continu ous solution on the composite grid. The discrete problem that is actually being solved is an implicit result of the iterative process. The LDC method was originally introduced in combination with finite difference discretizations. In a straightforward generalization of the LDC algorithm to finite volume discretizations, the discrete conservation property, which is one of the main attractive features of a finite volume method, does not hold for the composite grid approximation. We present a modified LDC method, which is based on a special form of the defect correction term used in the right hand side of the coarse grid problem. Due to this finite volume adapted defect correction term, the conservation property is preserved in the discretization on the composite grid. Several properties hold for the fixed point of the LDC iteration. Therefore, it is important to study the convergence behavior of the algorithm. We derive an upper bound for the norm of the iteration matrix for the model problem of Poisson’s equation on the unit square with Dirichlet boundary conditions. If we use trigonometric interpolation on the interface between the fine and coarse grids, we can derive an upper bound for the (infinity) norm of the iteration matrix M of the form kMk1 ?? CH2, in which C is a constant and H is the grid size of the global coarse grid. Numerical experiments show that the asymptotic behavior of kMk1 is indeed as predicted by the theoretical results. The LDC technique is extended by adding adaptivity, multi-level refinement, domain decomposition and regridding to the standard algorithm. The local fine grid is no longer a priori chosen: based on a weight function that measures the smoothness of the solution of the partial differential equation under consideration, high activity areas are determined and flagged for refinement. The flagged boxes are covered with a rectangular patch. In the patch, a local fine grid is chosen. The solution procedure may be applied recursively, i.e., the rectangular patches used to cover high activity areas in the coarse grid may be refined themselves. The usage of a single rectangular patch to cover all flagged boxes in a grid may cause refinement of a large number of unflagged boxes. To remedy this inefficiency we combine the adaptive multi-level LDC algorithm with domain decomposition. Finally, we note that refining a grid and solving the boundary value problemon the new composite grid may cause the area of high activity to move. When this happens, areas of the grid may be refined unnecessarily or areas may not be refined whereas they do need refinement. Therefore, we formulate a regridding procedure. We apply our proposed adaptive multi-level LDC algorithm with domain decomposition to a combustion problem, namely the simulation of an axisymmetric laminar Bunsen flame. We use a simple one-step chemistry model. We outline the discretization of the system of partial differential equations with the finite difference method and sketch the solution process used on the individual tensor-product grids. Our simulation results show that all dependent variables except for the nitrogen mass fraction have large gradients in the flame zone. A remarkable characteristic of the Bunsen flame problem is that the size of the flame increases on the finer grids. The structure of the flame is similar on all grids. We verify the increase of the flame length using Richardson extrapolation. The numerical results are compared to those found in literature.