On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations

The two-dimensional unsteady coupled Burgers' equations with moderate to severe gradients, are solved numerically using higher-order accurate finite difference schemes; namely the fourth-order accurate compact ADI scheme, and the fourth-order accurate Du Fort Frankel scheme. The question of numerical stability and convergence are presented. Comparisons are made between the present schemes in terms of accuracy and computational efficiency for solving problems with severe internal and boundary gradients. The present study shows that the fourth-order compact ADI scheme is stable and efficient

A bounded upwinding scheme for computing convection-dominated transport problems

A practical high resolution upwind differencing scheme for the numerical solution of convection-dominated transport problems is presented. The scheme is based on TVD and CBC stability criteria and is implemented in the context of the finite difference methodology. The performance of the scheme is investigated by solving the 1D/2D scalar advection equations, 1D inviscid Burgers’ equation, 1D scalar convection–diffusion equation, 1D/2D compressible Euler’s equations, and 2D incompressible Navier–Stokes equations. The numerical results displayed good agreement with other existing numerical and experimental data

A RBF partition of unity collocation method based on finite difference for initial-boundary value problems

Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection-diffusion and pseudo-parabolic equations

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 /

Coupled/combined compact IRBF schemes for fluid flow and FSI problems

The thesis is concerned with the development of compact approximation methods based on Integrated Radial Basis Functions (IRBFs) and their applications in fluid flows and FSI problems. The contributions include (i) new compact IRBF stencils where first- and second-order derivatives are included; (ii) a preconditioning technique where a preconditioner to enhance the stability of the flat IRBF solutions; and, (iii) the incorporation of the proposed stencils into the immersed boundary methods. Numerical experiments show the present schemes generally produce more accurate solutions and better convergence rates than existing methods (e.g. FDM, high-order compact FDM and compact IRBF methods)

A space-time discretization procedure for wave propagation problems

Higher order compact algorithms are developed for the numerical simulation of wave propagation by using the concept of a discrete dispersion relation. The dispersion relation is the imprint of any linear operator in space-time. The discrete dispersion relation is derived from the continuous dispersion relation by examining the process by which locally plane waves propagate through a chosen grid. The exponential structure of the discrete dispersion relation suggests an efficient splitting of convective and diffusive terms for dissipative waves. Fourth- and eighth-order convection schemes are examined that involve only three or five spatial grid points. These algorithms are subject to the same restrictions that govern the use of dispersion relations in the constructions of asymptotic expansions to nonlinear evolution equations. A new eighth-order scheme is developed that is exact for Courant numbers of 1, 2, 3, and 4. Examples are given of a pulse and step wave with a small amount of physical diffusion