Approximation error of the Lagrange reconstructing polynomial

The reconstruction approach [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] for the numerical approximation of $f'(x)$ is based on the construction of a dual function $h(x)$ whose sliding averages over the interval $[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$ are equal to $f(x)$ (assuming an homogeneous grid of cell-size $\Delta x$). We study the deconvolution problem [Harten A., Engquist B., Osher S., Chakravarthy S.R.: {\em J. Comp. Phys.} {\bf 71} (1987) 231--303] which relates the Taylor polynomials of $h(x)$ and $f(x)$, and obtain its explicit solution, by introducing rational numbers $\tau_n$ defined by a recurrence relation, or determined by their generating function, $g_\tau(x)$, related with the reconstruction pair of ${\rm e}^x$. We then apply these results to the specific case of Lagrange-interpolation-based polynomial reconstruction, and determine explicitly the approximation error of the Lagrange reconstructing polynomial (whose sliding averages are equal to the Lagrange interpolating polynomial) on an arbitrary stencil defined on a homogeneous grid.Comment: 31 pages, 1 table; revised version to appear in J. Approx. Theor

A computational approach to flame hole dynamics

Turbulent diffusion flames at low strain rates sustain a spatially continuous flame surface. However, at high strains, which may be localized in a flow or not, the flame can be quenched due to the increased heat loss away from the reaction zone. These quenched regions are sometimes called flame holes. Flame holes reduce the efficiency of combustion, can increase the production of certain pollutants (e.g. carbon monoxide, soot) as well as limit the overall stability of the flame. We present a numerical algorithm for the calculation of the dynamics of flame holes in diffusion flames. The key element is the solution of an evolution equation defined on a general moving surface. The low-dimensional manifold (the surface) can evolve in time and it is defined implicitly as an iso-level set of an associated Cartesian scalar field. An important property of the method described here is that the surface coordinates or parameterization does not need to be determined explicitly; instead, the numerical method employs an embedding technique where the evolution equation is extended to the Cartesian space, where well-known and efficient numerical methods can be used. In our application of this method, the field defined on the surface represents the chemical activity state of a turbulent diffusion flame. We present a formulation that describes the formation, propagation, and growth of flames holes using edge-flame modeling in laminar and turbulent diffusion flames. This problem is solved using a high-order finite-volume WENO method and a new extension algorithm defined in terms of propagation PDEs. The complete algorithm is demonstrated by tracking the dynamics of flame holes in a turbulent reacting shear layer. The method is also implemented in a generalized unstructured low-Mach number fluid solver (Sandia's SIERRA low Mach Module ``Nalu") and applied to simulate local extinction in a piloted jet diffusion flame configuration

An Accurate and Robust Numerical Scheme for Transport Equations

En esta tesis se presenta una nueva técnica de discretización para ecuaciones de transporte en problemas de convección-difusión para el rango completo de números de Péclet. La discretización emplea el flujo exacto de una ecuación de transporte unidimensional en estado estacionario para deducir una ecuación discreta de tres puntos en problemas unidimensionales y cinco puntos en problemas bidimensionales. Con "flujo exacto" se entiende que se puede obtener la solución exacta en función de integrales de algunos parámetros del fluido y flujo, incluso si estos parámetros son vari- ables en un volumen de control. Las cuadraturas de alto orden se utilizan para lograr resultados numéricos cercanos a la precisión de la máquina, incluso con mallas bastas.Como la discretización es esencialmente unidimensional, no está garantizada una solución con precisión de máquina para problemas multidimensionales, incluso en los casos en que las integrales a lo largo de cada coordenada cartesiana tienen una primitiva. En este sentido, la contribución principal de esta tesis consiste en una forma simple y elegante de obtener soluciones en problemas multidimensionales sin dejar de utilizar la formulación unidimensional. Además, si el problema es tal que la solución tiene precisión de máquina en el problema unidimensional a lo largo de las líneas coordenadas, también la tendrá para el dominio multidimensional.In this thesis, we present a novel discretization technique for transport equations in convection-diffusion problems across the whole range of Péclet numbers. The discretization employs the exact flux of a steady-state one-dimensional transport equation to derive a discrete equation with a three-point stencil in one-dimensional problems and a five-point stencil in two-dimensional ones. With "exact flux" it is meant that the exact solution can be obtained as a function of integrals of some fluid and flow parameters, even if these parameters are variable across a control volume. High-order quadratures are used to achieve numerical results close to machine- accuracy even with coarse grids. As the discretization is essentially one-dimensional, getting the machine- accurate solution of multidimensional problems is not guaranteed even in cases where the integrals along each Cartesian coordinate have a primitive. In this regard, the main contribution of this thesis consists in a simple and elegant way of getting solutions in multidimensional problems while still using the one-dimensional formulation. Moreover, if the problem is such that the solution is machine-accurate in the one-dimensional problem along coordinate lines, it will also be for the multidimensional domain.<br /