Richardson Extrapolation-Based High Accuracy High Efficiency Computation for Partial Differential Equations

In this dissertation, Richardson extrapolation and other computational techniques are used to develop a series of high accuracy high efficiency solution techniques for solving partial differential equations (PDEs). A Richardson extrapolation-based sixth-order method with multiple coarse grid (MCG) updating strategy is developed for 2D and 3D steady-state equations on uniform grids. Richardson extrapolation is applied to explicitly obtain a sixth-order solution on the coarse grid from two fourth-order solutions with different related scale grids. The MCG updating strategy directly computes a sixth-order solution on the fine grid by using various combinations of multiple coarse grids. A multiscale multigrid (MSMG) method is used to solve the linear systems resulting from fourth-order compact (FOC) discretizations. Numerical investigations show that the proposed methods compute high accuracy solutions and have better computational efficiency and scalability than the existing Richardson extrapolation-based sixth order method with iterative operator based interpolation. Completed Richardson extrapolation is explored to compute sixth-order solutions on the entire fine grid. The correction between the fourth-order solution and the extrapolated sixth-order solution rather than the extrapolated sixth-order solution is involved in the interpolation process to compute sixth-order solutions for all fine grid points. The completed Richardson extrapolation does not involve significant computational cost, thus it can reach high accuracy and high efficiency goals at the same time. There are three different techniques worked with Richardson extrapolation for computing fine grid sixth-order solutions, which are the iterative operator based interpolation, the MCG updating strategy and the completed Richardson extrapolation. In order to compare the accuracy of these Richardson extrapolation-based sixth-order methods, truncation error analysis is conducted on solving a 2D Poisson equation. Numerical comparisons are also carried out to verify the theoretical analysis. Richardson extrapolation-based high accuracy high efficiency computation is extended to solve unsteady-state equations. A higher-order alternating direction implicit (ADI) method with completed Richardson extrapolation is developed for solving unsteady 2D convection-diffusion equations. The completed Richardson extrapolation is used to improve the accuracy of the solution obtained from a high-order ADI method in spatial and temporal domains simultaneously. Stability analysis is given to show the effects of Richardson extrapolation on stable numerical solutions from the underlying ADI method

Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier–Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas–Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are “quasi-unconditionally stable” in the following sense: each algorithm is stable for all couples (h,Δt)of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form (0,M_h)×(0,M_t). In other words, for each fixed value of Δt below a certain threshold, the Navier–Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second-order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier–Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier–Stokes solvers for which second-order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions

A dual-potential formulation of the Navier-Stokes equations

A dual potential formulation for numerically solving the Navier-Stokes equations is developed and presented. The velocity field is decomposed using a scalar and vector potential. Vorticity and dilatation are used as the dependent variables in the momentum equations. Test cases in two dimensions verify the capability to solve flows using approximations from potential flow to full Navier-Stokes simulations. A three-dimensional incompressible flow formulation is also described;An interesting feature of this approach to solving the Navier-Stokes equations is the decomposition of the velocity field into a rotational part (vector potential) and an irrotational part (scalar potential). The Helmholtz decomposition theorem allows this splitting of the velocity field. This approach has had only limited use since it increases the number of dependent variables in the solution. However, it has often been used for incompressible flows where the solution scheme is known to be fast and accurate. This research extends the usage of this method to fully compressible Navier-Stokes simulations by using the dilatation variable along with vorticity;A time-accurate, iterative algorithm is used for the uncoupled solution of the governing equations. Several levels of flow approximation are available within the framework of this method. Potential flow, Euler and full Navier-Stokes solutions are possible using the dual potential formulation. Solution efficiency can be enhanced in a straightforward way. For some flows, the vorticity and/or dilatation may be negligible in certain regions (e.g., far from a viscous boundary in an external flow). It is possible to drop the calculation of these variables then and optimize the solution speed. Also, efficient Poisson solvers are available for the potentials;The relative merits of non-primitive variables versus primitive variables for solution of the Navier-Stokes equations are also discussed

Stability of laminar jets and Clarke-Riley flames

Open flows like jets and flames are sensitive to perturbations, and experience different instabilities. Submerged laminar axisymmetric jets with density sufficiently smaller than that of the ambient become globally unstable, exhibiting a self-sustained oscillatory behavior, when the Reynolds number is above a critical value that depends on the jet-to-ambient density ratio and on the outlet velocity profile. This phenomenon is analyzed in chapter 2 by direct numerical simulations of the unsteady axisymmetric equations of motion governing the velocity and density fields using the finite element method. It is shown that the unforced space-time propagation of non-linear disturbances defines two different asymptotic states that may be reached at large times: either a globally stable flow where the disturbances vanish, or a globally unstable state in which the jet oscillates with a characteristic intrinsic frequency. In the latter case, the numerical oscillation amplitudes are shown to fit the Stuart-Landau model for a supercritical Hopf bifurcation, and the corresponding neutral curve is in good agreement with experiments and linear stability theory. Globally stable jets are noise amplifiers, in the sense that they usually respond to harmonic forcing by amplifying the forcing energy. If the energy gain of small disturbances is sufficiently large, the jet undergoes a transition from an asymptotically stable state to a convectively unstable state, where perturbations experience growth downstream of their spatial origin. This is the main topic of chapter 3, devoted to a global linear frequency response analysis of axisymmetric laminar jets for different Reynolds numbers, density ratios, outlet velocity profiles, and azimuthal modes. The study allows to define a critical Reynolds number, and to compute its value for each configuration, discussing its agreement with previous experimental and numerical results. The value of the critical Reynolds number obtained in the present analysis is affected by the geometry and the type of forcing. Thus, we consider both optimal and uniform forcing, and also vary the forcing region. In addition, we consider the effect of the injection geometry, considering that the jet emerges either from a circular injector tube far from any wall, or from a circular orifice on a wall. Finally, the buoyancy-driven laminar flow associated with the Burke-Schumann diffusion flame that develops from the edge of a semi-infinite horizontal fuel surface, burning in a quiescent oxidizing atmosphere, has a self-similar steady structure. Chapter 4 considers fuels with non-unity Lewis numbers and gas mixtures with a realistic power-law dependence of the different transport properties. The problem is formulated in terms of chemistry-free Shvab-Zel’dovich variables that use linear combinations of the temperature and reactant mass fractions. The resulting self-similar solution is used as a base flow to perform a local stability analysis. A critical local Grashof number is found, above which the flame develops Görtler-like counter-rotating streamwise vortices. The analysis provides the dependence of the critical Grashof number on the relevant flame parameters.Los flujos abiertos, como los chorros y las llamas, son sensibles a las perturbaciones y están sujetos a diversas inestabilidades. Los chorros sumergidos con densidad menor que la del ambiente se vuelven globalmente inestables cuando el número de Reynolds supera cierto valor crítico que depende de la relación de densidad y de la forma del perfil de velocidad a la salida. Este fenómeno se analiza en el capítulo 2 mediante simulaciones numéricas directas de las ecuaciones de conservación que gobiernan los campos de velocidad y densidad, utilizando el método de elementos finitos. Se demuestra que la propagación de perturbaciones no lineales define dos estados posibles a tiempo largo: o bien el flujo es asintóticamente estable, o bien es globalmente inestable, lo que se manifiesta en la aparición de auto-oscilaciones con una frecuencia característica. En este último caso, las amplitudes de oscilación numéricas se ajustan al modelo de Stuart-Landau para bifurcaciones de Hopf supercríticas, y la curva neutra correspondiente muestra un buen acuerdo con los experimentos y la teoría de la estabilidad lineal. Los chorros globalmente estables son amplificadores de ruido, dado que suelen responder al forzado armónico aumentando su energía. Si la ganancia de energía es suficientemente grande, tiene lugar una transición desde un estado asintóticamente estable a un estado convectivamente inestable, caracterizado por el crecimiento de las perturbaciones aguas abajo de su origen espacial. Este es el tema del capítulo 3, dedicado a un análisis de la respuesta global lineal en frecuencia para diferentes números de Reynolds, relaciones de densidad, perfiles de velocidad a la salida, y modos azimutales. El estudio permite definir un número de Reynolds crítico, y calcular su valor para cada configuración, comparando el resultado con observaciones experimentales y numéricas anteriores. El número de Reynolds crítico predicho se ve afectado por el tipo de forzado y por la geometría. Por tanto, se estudian tanto el forzado óptimo como el uniforme, y dos soportes espaciales distintos para la región de forzado. Además, se tienen en cuenta dos geometrías de inyección distintas, una correspondiente a un inyector alejado de paredes, y otra en la que el chorro emerge de un orificio circular en una pared. Por último, en el capítulo 4 se estudia el flujo laminar inducido por la flotabilidad de una llama de difusión de Burke-Schumann que se desarrolla desde el borde de una superficie horizontal semi-infinita de combustible, y que se quema en una atmósfera oxidante en reposo, que posee una estructura estacionaria autosemejante. Se consideran combustibles con números de Lewis distintos de la unidad, y mezclas de gases con una dependencia realista de las propiedades de transporte con la temperatura. El problema se formula mediante las variables de Shvab-Zel’dovich, que hacen uso de combinaciones lineales de la temperatura y las fracciones másicas. La solución autosemejante se usa como flujo base para realizar un análisis de estabilidad local, que permite calcular un número de Grashof local crítico como función de los parámetros relevantes, por encima del cual se desarrollan vórtices tipo Görtler.This doctoral dissertation has been supported by a 4-year PIF contract of the Pre-doctoral Researcher Training Program of Universidad Carlos III de Madrid and under projects DPI2014-59292-C3-1-P and DPI2015-71901-REDT awarded by the Spanish Ministry of Economy and Competitiveness.Programa Oficial de Doctorado en Mecánica de FluidosPresidente: Jesús Carlos Martínez Bazán.- Secretario: María Inmaculada Iglesias Estrade.- Vocal: Luis Parras Anguit

Numerical studies of incompressible viscous flow in a driven cavity

A series of project papers is presented in computational fluid dynamics. The work was performed during the 1973-74 academic year at Old Dominion University. Each paper briefly examines a numerical method(s) that can be applied to the Navier-Stokes equations governing incompressible flow in a driven cavity. Solutions obtained with a cubic spline procedure are also included

The Sixth Copper Mountain Conference on Multigrid Methods, part 2

The Sixth Copper Mountain Conference on Multigrid Methods was held on April 4-9, 1993, at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

The dynamics of liquids in moving containers: Numerical models for viscous unsteady free surface flows

The transportation and control of liquid masses poses a problem of immense practical interest. Slosh forces generated by the motion of the liquid can easily interfere with the safe operation of the vehicle. The successful design and execution of such operations depends upon not only the understanding, but also the ability to predict the dynamic behavior of liquids in moving containers;The numerical simulation of liquid sloshing in moving containers is considered in this study. Numerical models are developed and applied to both two and three dimensional flows. The motion of the vehicle can be quite general, given by the superposition of several rectilinear and angular time varying accelerations. The Navier-Stokes equations are recast in a non-inertial coordinate frame which follows the motion of the container. Singularities produced by the onset of sudden motions are removed from the formulation using an asymptotic analysis. A Poisson equation is used for the pressure calculation. The position of the free surface is determined by a kinematic condition. An implicit second order accurate finite difference method is used for the solution of the governing equations;A method that simplifies the coupling of the dynamics of the liquid with those of the moving vehicle is introduced. It relies on the concept of an apparent mass for the liquid, which is formulated in a manner that measures the resistance of the liquid mass to sudden changes in the acceleration of the vehicle. It enables the solution of the solid and liquid equations based on a simple explicit, rather than an implicit, coupling scheme which significantly reduces the computational requirements;Cases of liquid sloshing in containers of rectangular, cylindrical, and spherical geometry are considered. Detailed information on the flowfield and the free surface position is given for several representative cases. Effects due to the forcing conditions, liquid viscosity, surface tension, and liquid geometry, are considered in a parametric study. Information on sloshing frequencies and damping rates is included. An excellent comparison of the present numerical result is demonstrated, where possible, with previous analytical and experimental works

An Implicit Finite-Volume Depth-Integrated Model For Coastal Hydrodynamics And Multiple-Sized Sediment Transport

A two-dimensional depth-integrated model is developed for simulating wave-averaged hydrodynamics and nonuniform sediment transport and morphology change in coastal waters. The hydrodynamic model includes advection, wave-enhanced turbulent mixing and bottom friction; wave-induced volume flux; wind, atmospheric pressure, wave, river, and tidal forcing; and Coriolis-Stokes force. The sediment transport model simulates nonequilibrium total-load transport, and includes flow and sediment transport lags, hiding and exposure, bed material sorting, bed slope effects, nonerodible beds, and avalanching. The flow model is coupled with an existing spectral wave model and a newly developed surface roller model. The hydrodynamic and sediment transport models use finite-volume methods on a variety of computational grids including nonuniform Cartesian, telescoping Cartesian, quadrilateral, triangular, and hybrid triangular/quadrilateral. Grid cells are numbered in an unstructured one-dimensional array, so that all grid types are implemented under the same framework. The model uses a second-order fully implicit temporal scheme and first- and second-order spatial discretizations including corrections for grid non-orthogonality. The hydrodynamic equations are solved using an iterative pressure-velocity coupling algorithm on a collocated grid with a momentum interpolation for inter-cell fluxes. The multiple-sized sediment transport, bed change, and bed material sorting equations are solved in a coupled manner but are decoupled from the hydrodynamic equations. The spectral wave and roller models are calculated using finite-difference methods on nonuniform Cartesian grids. An efficient inline steering procedure is developed to couple the flow and wave models. The model is verified using seven analytical solution cases and validated using ten laboratory and five field test cases which cover a wide range of conditions, time and spatial scales. The hydrodynamic model simulates reasonably well long wave propagation, wetting and drying, recirculation flows near a spur-dike and a sudden channel expansion, and wind- and wave generated currents and water levels. The sediment transport model reproduces channel shoaling, erosion due to a clear-water inflow, downstream sediment sorting, and nearshore morphology change. Calculated longshore sediment transport rates are well simulated except near the shoreline where swash processes, which are not included, become dominant. Model sensitivity to the computation grid and calibration parameters is presented for several test cases

Local time stepping methods and discontinuous Galerkin methods applied to diffusion advection reaction equations

Partial differential equations (PDEs), especially the diffusion advection and re action equations (DAREs), are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Due to the difficulty of finding exact solutions, developing efficient numerical methods for simulating the solution of the DAREs is a very important and challenging research topic. In this work, we present the transformation of the DAREs to ordinary differen tial equations (ODEs) using the standard finite element (FE) or the discontinuous Galerkin (DG) spatial discretization method. The resulting system of ODEs is then solved with standard time integrators such as implicit Euler methods, integrat ing factor method, exponential time differencing methods, exponential Rosenbrock methods, orthogonal Runge-Kutta Chebyshev methods. To illustrate the limitations of the FE method, we simulate and invert the cyclic voltammetry models using both spatial discretization methods (i.e. FE and DG) and show numerically that DG is more efficient. In many physical applications, there are special features (such as fractures, walls, corners, obstacles or point loads) which globally, as well as locally, have important effects on the solution. In order to efficiently capture these, we propose two new numerical methods in which the mesh is locally refined in time and space. These new numerical methods are based on the combination of the DG method with local time stepping (LTS) approaches. We then apply these new numerical methods to investigate two physical problems (the cyclic voltammetry model and the transport of solute through porous media). These numerical investigations show that the combination of the DG with the LTS approaches are more efficient compared to the combination of DG with standard time integrators